diff --git a/samples/expansions_6.json b/samples/expansions_6.json
index 6c8347b..7df9fd3 100644
--- a/samples/expansions_6.json
+++ b/samples/expansions_6.json
@@ -1 +1,14960 @@
-[{"children_for_tactic": [[{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}]], "effects": [{"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}, {"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}], "env_durations": [0, 0], "expander_duration": 1, "generation_duration": 1, "log_critic": -0.5, "priors": [0.500134289264679, 0.4998657703399658], "tactics": [{"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "thm": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}}, {"children_for_tactic": [[{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}]], "effects": [{"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}, {"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}, {"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}, {"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}, {"children": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)\nsimp at h\u2081 h\u2082 h\u2083 h\u2084"}], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}}], "env_durations": [0, 0, 0, 0, 0], "expander_duration": 1, "generation_duration": 1, "log_critic": -0.5, "priors": [0.20181716978549957, 0.20181716978549957, 0.20181716978549957, 0.20181716978549957, 0.19273139536380768], "tactics": [{"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083 h\u2084"}], "thm": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}}, {"children_for_tactic": [[], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []], "effects": [{"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 2, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all"}}, {"children": [], "goal": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}, "tac": {"duration": 0, "is_valid": true, "unique_string": "simp_all [h\u2081, h\u2082, h\u2083]"}}], "env_durations": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "expander_duration": 1, "generation_duration": 1, "log_critic": -0.5, "priors": [0.016283227130770683, 0.016283227130770683, 0.014078863896429539, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.014078863896429539, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.014078863896429539, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.014078863896429539, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.014078863896429539, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.016283227130770683, 0.014864221215248108, 0.014864221215248108, 0.014049867168068886, 0.014864221215248108, 0.013513033278286457, 0.014143623411655426, 0.014143623411655426, 0.016359707340598106, 0.014143623411655426, 0.014143623411655426, 0.016359707340598106, 0.0028445173520594835], "tactics": [{"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all 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{"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 2, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [mul_assoc]"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all [h\u2081, h\u2082, h\u2083]"}], "thm": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}}]
\ No newline at end of file
+[
+  {
+    "children_for_tactic": [
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp_all"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+        }
+      ]
+    ],
+    "effects": [
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082\nsimp_all"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      }
+    ],
+    "env_durations": [
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+    ],
+    "expander_duration": 1,
+    "generation_duration": 1,
+    "log_critic": -0.5,
+    "priors": [
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+      0.003913207910954952,
+      0.016951587051153183,
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+      0.017056837677955627,
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+      0.017056837677955627,
+      0.017056837677955627,
+      0.017056837677955627
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+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2083 := hS (a * b) a"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2083 := hS (a * b) a"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2083 := hS (a * b) a"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
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+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2083 := hS (a * b) a"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "simp_all"
+      }
+    ],
+    "thm": {
+      "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+      "ctx": {
+        "namespaces": []
+      },
+      "hypotheses": [],
+      "past_tactics": [
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "intro a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2081 := hS b a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2082 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        }
+      ],
+      "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+    }
+  },
+  {
+    "children_for_tactic": [
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS b (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ],
+      [
+        {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2084 := hS a (a * b)"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+        }
+      ]
+    ],
+    "effects": [
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nsimp at h\u2081 h\u2082 h\u2083"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082 h\u2083"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS b (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS b (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2084 := hS a (a * b)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"
+          }
+        ],
+        "goal": {
+          "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+          "ctx": {
+            "namespaces": []
+          },
+          "hypotheses": [],
+          "past_tactics": [
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "intro a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2081 := hS b a"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2082 := hS a b"
+            },
+            {
+              "duration": 0,
+              "is_valid": true,
+              "unique_string": "have h\u2083 := hS (a * b) a"
+            }
+          ],
+          "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+        },
+        "tac": {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2084 := hS a (a * b)"
+        }
+      },
+      {
+        "children": [
+          {
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+      {
+        "duration": 0,
+        "is_valid": true,
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+      },
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+      {
+        "duration": 0,
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+      },
+      {
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+      {
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+      },
+      {
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+        "is_valid": true,
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+      },
+      {
+        "duration": 0,
+        "is_valid": true,
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+      },
+      {
+        "duration": 0,
+        "is_valid": true,
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+      },
+      {
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+      },
+      {
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+      },
+      {
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+      },
+      {
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+      },
+      {
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+      },
+      {
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+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      },
+      {
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+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      },
+      {
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+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      },
+      {
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+        "is_valid": true,
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+      },
+      {
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+        "is_valid": true,
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+      },
+      {
+        "duration": 0,
+        "is_valid": true,
+        "unique_string": "have h\u2084 := hS a (a * b)"
+      }
+    ],
+    "thm": {
+      "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+      "ctx": {
+        "namespaces": []
+      },
+      "hypotheses": [],
+      "past_tactics": [
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "intro a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2081 := hS b a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2082 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      ],
+      "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+    }
+  }
+]
\ No newline at end of file
diff --git a/samples/search_6.json b/samples/search_6.json
index e8ceb61..0ec47e0 100644
--- a/samples/search_6.json
+++ b/samples/search_6.json
@@ -1 +1,55959 @@
-{"ancestors": {"theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n": [["", 0]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 58]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS (b * a) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 14]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 34]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 23]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 32]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 20]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 45]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a (b * a)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 33]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 31]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 33]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 36]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 4]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 21]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 6]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 30]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 8]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 1]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 17]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 15]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 26]]}, "backedup_hashes": [], "currently_expanding": ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"], "done": false, "expansion_count": 6, "initial_minimum_proof_size": {"DEPTH": 9223372036854775807, "SIZE": 9223372036854775807, "TIME": 9223372036854775807}, "minimum_proof_size": {"DEPTH": 9223372036854775807, "SIZE": 9223372036854775807, "TIME": 9223372036854775807}, "nodes": [{"children_for_tactic": [[{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 1, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)"}]], "counts": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "exploration": 0.3, "in_minimum_proof": {"DEPTH": false, "SIZE": false, "TIME": false}, "in_proof": false, "is_solved_leaf": false, "killed_tactics": [], "log_critic_value": -0.5, "log_w": [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], "minimum_proof_size": {"DEPTH": 9223372036854775807, "SIZE": 9223372036854775807, "TIME": 9223372036854775807}, "minimum_tactic_length": {"DEPTH": [], "SIZE": [], "TIME": []}, "minimum_tactics": {"DEPTH": [], "SIZE": [], "TIME": []}, "old_critic_value": 0.0, "policy": {"exploration": 0.3, "type": 1}, "priors": [0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.004972024820744991, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.004972024820744991, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.004972024820744991, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016815531998872757, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.016750147566199303, 0.0051857042126357555, 0.016750147566199303, 0.0051857042126357555], "q_value_solved": 2, "reset_mask": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true], "solved": false, "solving_tactics": [], "tactic_expandable": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true], "tactic_init_value": 0.0, "tactics": [{"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 1, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (b * a)"}], "theorem": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}, "virtual_counts": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]}, {"children_for_tactic": [[{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 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h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "simp_all"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082"}], "theorem": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}, "virtual_counts": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]}, {"children_for_tactic": [[{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}]], "counts": [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "exploration": 0.3, "in_minimum_proof": {"DEPTH": false, "SIZE": false, "TIME": false}, "in_proof": false, "is_solved_leaf": false, "killed_tactics": [], "log_critic_value": -0.5, "log_w": [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 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"ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc, hS] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc, hS] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a (b * a)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a (b * a)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a (b * a)"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "simp [mul_assoc] at h\u2081"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": 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Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS (b * a) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS (b * a) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS (b * a) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2080 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS (b * a) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h1 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"}]], "counts": [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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[{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}], [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": 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\u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", 58]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS (b * a) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 14]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 34]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 23]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 32]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 20]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", 45]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a (b * a)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 33]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 31]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 33]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 63], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 6], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 30], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 36]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 4]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 21]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nhave h\u2084 := hS b (b * a)": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 62], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 60], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 17], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 6]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 61], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 58], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 54], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 53], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 51], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 49], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 47], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 46], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 44], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 42], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 40], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 33], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 32], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 13], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 11], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 37], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 26], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 8], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 5], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 35], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 4], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 3], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 2], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 1], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 0], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 14], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 15], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 16], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 20], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 21], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 22], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 23], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 25], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 38], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 27], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 39], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 28], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 29], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", 30]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 57], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 50], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 45], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 55], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 36], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 34], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 24], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 31], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 12], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 10], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 8]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 41], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 19], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 9], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 7], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", 1]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc, hS] at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 18], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 17]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 59], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 56], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 52], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 43], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 15]], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081": [["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 48], ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", 26]]}, "policy": {"exploration": 0.3, "type": 1}, "propagate_needed": false, "root": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}, "seed": 42, "simulations": [{"children_for_theorem": {"10041070152938961489": {"previous": 842976703719519088, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}]}, "12116204433008218031": {"previous": 16342763735530018773, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}]}, "16342763735530018773": {"previous": 16579496881221230816, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}]}, "16579496881221230816": {"previous": 0, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}]}, "17735310507799149018": {"previous": 10041070152938961489, "value": []}, "842976703719519088": {"previous": 12116204433008218031, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}]}}, "depth": {"10041070152938961489": {"previous": 842976703719519088, "value": 4}, "12116204433008218031": {"previous": 16342763735530018773, "value": 2}, "16342763735530018773": {"previous": 16579496881221230816, "value": 1}, "16579496881221230816": {"previous": 0, "value": 0}, "17735310507799149018": {"previous": 10041070152938961489, "value": 5}, "842976703719519088": {"previous": 12116204433008218031, "value": 3}}, "expansions": 1, "parent_for_theorem": {"10041070152938961489": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}}, "12116204433008218031": {"previous": 16342763735530018773, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}}, "16579496881221230816": {"previous": 0, "value": null}, "17735310507799149018": {"previous": 10041070152938961489, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}}}, "root": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}, "seen": null, "solved": null, "tactic_ids": {"10041070152938961489": {"previous": 842976703719519088, "value": 35}, "12116204433008218031": {"previous": 16342763735530018773, "value": 45}, "16342763735530018773": {"previous": 16579496881221230816, "value": 9}, "16579496881221230816": {"previous": 0, "value": 49}, "842976703719519088": {"previous": 12116204433008218031, "value": 49}}, "tactics": {"10041070152938961489": {"previous": 842976703719519088, "value": {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}}, "12116204433008218031": {"previous": 16342763735530018773, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}}, "16579496881221230816": {"previous": 0, "value": {"duration": 0, "is_valid": true, "unique_string": "intro a b"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}}}, "theorems": {"10041070152938961489": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a 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[], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (b * a) b"}, {"duration": 0, "is_valid": true, "unique_string": "simp at h\u2081 h\u2082 h\u2083"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}}}, "values": null, "virtual_count_added": {"10041070152938961489": {"previous": 842976703719519088, "value": true}, "12116204433008218031": {"previous": 16342763735530018773, "value": true}, "16342763735530018773": {"previous": 16579496881221230816, "value": true}, "16579496881221230816": {"previous": 0, "value": true}, "17735310507799149018": {"previous": 10041070152938961489, "value": false}, "842976703719519088": {"previous": 12116204433008218031, "value": true}}}, {"children_for_theorem": {"12116204433008218031": {"previous": 16342763735530018773, 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"past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}]}, "16579496881221230816": {"previous": 0, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}]}, "3014227773155032910": {"previous": 842976703719519088, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)"}]}, "842976703719519088": {"previous": 12116204433008218031, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) 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: S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}}, "16579496881221230816": {"previous": 0, "value": null}, "3014227773155032910": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}}, "9604053314602174359": {"previous": 3014227773155032910, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}}}, "root": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}, "seen": null, "solved": null, "tactic_ids": {"12116204433008218031": {"previous": 16342763735530018773, "value": 45}, "16342763735530018773": {"previous": 16579496881221230816, "value": 9}, "16579496881221230816": {"previous": 0, "value": 49}, "3014227773155032910": {"previous": 842976703719519088, "value": 43}, "842976703719519088": {"previous": 12116204433008218031, "value": 2}}, "tactics": {"12116204433008218031": {"previous": 16342763735530018773, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}}, "16579496881221230816": {"previous": 0, "value": {"duration": 0, "is_valid": true, "unique_string": "intro a b"}}, "3014227773155032910": {"previous": 842976703719519088, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS b (a * b)"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}}}, "theorems": {"12116204433008218031": {"previous": 16342763735530018773, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}}, "16579496881221230816": {"previous": 0, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}}, "3014227773155032910": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, 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a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}]}, "842976703719519088": {"previous": 12116204433008218031, "value": [{"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}]}}, "depth": {"12116204433008218031": {"previous": 16342763735530018773, "value": 2}, "13920815031433863277": {"previous": 3014227773155032910, "value": 5}, "16342763735530018773": {"previous": 16579496881221230816, "value": 1}, "16579496881221230816": {"previous": 0, "value": 0}, "3014227773155032910": {"previous": 842976703719519088, "value": 4}, "842976703719519088": {"previous": 12116204433008218031, "value": 3}}, "expansions": 1, "parent_for_theorem": {"12116204433008218031": {"previous": 16342763735530018773, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}}, "13920815031433863277": {"previous": 3014227773155032910, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", 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(b * a) = b := by\n"}}, "16579496881221230816": {"previous": 0, "value": null}, "3014227773155032910": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}}}, "root": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}, "seen": null, "solved": null, "tactic_ids": {"12116204433008218031": {"previous": 16342763735530018773, "value": 45}, "16342763735530018773": {"previous": 16579496881221230816, "value": 9}, "16579496881221230816": {"previous": 0, "value": 49}, "3014227773155032910": {"previous": 842976703719519088, "value": 11}, "842976703719519088": {"previous": 12116204433008218031, "value": 2}}, "tactics": {"12116204433008218031": {"previous": 16342763735530018773, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}}, "16579496881221230816": {"previous": 0, "value": {"duration": 0, "is_valid": true, "unique_string": "intro a b"}}, "3014227773155032910": {"previous": 842976703719519088, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}}}, "theorems": {"12116204433008218031": {"previous": 16342763735530018773, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"}}, "13920815031433863277": {"previous": 3014227773155032910, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2084 := hS a (a * b)"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)"}}, "16342763735530018773": {"previous": 16579496881221230816, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"}}, "16579496881221230816": {"previous": 0, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"}}, "3014227773155032910": {"previous": 842976703719519088, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2083 := hS (a * b) a"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"}}, "842976703719519088": {"previous": 12116204433008218031, "value": {"conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b", "ctx": {"namespaces": []}, "hypotheses": [], "past_tactics": [{"duration": 0, "is_valid": true, "unique_string": "intro a b"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2081 := hS b a"}, {"duration": 0, "is_valid": true, "unique_string": "have h\u2082 := hS a b"}], "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"}}}, "values": null, "virtual_count_added": {"12116204433008218031": {"previous": 16342763735530018773, "value": true}, "13920815031433863277": {"previous": 3014227773155032910, "value": false}, "16342763735530018773": {"previous": 16579496881221230816, "value": true}, "16579496881221230816": {"previous": 0, "value": true}, "3014227773155032910": {"previous": 842976703719519088, "value": true}, "842976703719519088": {"previous": 12116204433008218031, "value": true}}}], "simulations_for_theorem": {"theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS a (a * b)": [], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a\nhave h\u2084 := hS b (a * b)": [], "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (b * a) b\nsimp at h\u2081 h\u2082 h\u2083": []}, "unexplored_theorems": ["theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"]}
\ No newline at end of file
+{
+  "ancestors": {
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n": [
+      [
+        "",
+        0
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        63
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        58
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        19
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        41
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        47
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        23
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        2
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        55
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        59
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        3
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        36
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        28
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        63
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        58
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        5
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        53
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        2
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        63
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        44
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        29
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        52
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        28
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        24
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        4
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        26
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        34
+      ]
+    ]
+  },
+  "backedup_hashes": [],
+  "currently_expanding": [
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+  ],
+  "done": false,
+  "expansion_count": 9,
+  "initial_minimum_proof_size": {
+    "DEPTH": 9223372036854775807,
+    "SIZE": 9223372036854775807,
+    "TIME": 9223372036854775807
+  },
+  "minimum_proof_size": {
+    "DEPTH": 9223372036854775807,
+    "SIZE": 9223372036854775807,
+    "TIME": 9223372036854775807
+  },
+  "nodes": [
+    {
+      "children_for_tactic": [
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "unique_string": "intro a b"
+              },
+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
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+              },
+              {
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+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
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+                "is_valid": true,
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+              },
+              {
+                "duration": 0,
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+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
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+                "duration": 0,
+                "is_valid": true,
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+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
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+                "duration": 0,
+                "is_valid": true,
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+              },
+              {
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+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
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+                "duration": 0,
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+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
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+            },
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+                "unique_string": "have h1 := hS b a"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
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+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
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+                },
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "unique_string": "simp at h1 h2"
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+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
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+              }
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+          },
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+            }
+          ],
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
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+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
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+                  "unique_string": "simp_all"
+                }
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+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "unique_string": "simp at h1 h2"
+              }
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+          },
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+            "unique_string": "simp_all"
+          }
+        },
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+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
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+                },
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+                }
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "unique_string": "simp at h1 h2"
+              }
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+          },
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+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
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+                },
+                {
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+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
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+                "unique_string": "have h2 := hS a b"
+              },
+              {
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+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
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+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all",
+              "ctx": {
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+                  "is_valid": true,
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
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+                },
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+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        }
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+        "SIZE": false,
+        "TIME": false
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+          "unique_string": "simp_all"
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+          "is_valid": true,
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+          "unique_string": "simp_all"
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+        {
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+          "unique_string": "simp_all"
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+          "is_valid": true,
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+          "unique_string": "simp_all"
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+        },
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+        },
+        {
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+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
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+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
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+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
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+        },
+        {
+          "duration": 0,
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+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
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+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        }
+      ],
+      "theorem": {
+        "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        "ctx": {
+          "namespaces": []
+        },
+        "hypotheses": [],
+        "past_tactics": [
+          {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          },
+          {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          },
+          {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          },
+          {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        ],
+        "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+      },
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+    },
+    {
+      "children_for_tactic": [
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * (b * a)) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2083 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
+        ]
+      ],
+      "counts": [
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+        0,
+        0,
+        0,
+        0
+      ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * (b * a)) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * (b * a)) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2083 := hS (a * b) a"
+          }
+        }
+      ],
+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
+      "is_solved_leaf": false,
+      "killed_tactics": [],
+      "log_critic_value": -0.5,
+      "log_w": [
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0,
+        0.0
+      ],
+      "minimum_proof_size": {
+        "DEPTH": 9223372036854775807,
+        "SIZE": 9223372036854775807,
+        "TIME": 9223372036854775807
+      },
+      "minimum_tactic_length": {
+        "DEPTH": [],
+        "SIZE": [],
+        "TIME": []
+      },
+      "minimum_tactics": {
+        "DEPTH": [],
+        "SIZE": [],
+        "TIME": []
+      },
+      "old_critic_value": 0.0,
+      "policy": {
+        "exploration": 0.3,
+        "type": 1
+      },
+      "priors": [
+        0.029047179967164993,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.015694234520196915,
+        0.029047179967164993,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.015694234520196915,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.015694234520196915,
+        0.015694234520196915,
+        0.010685855522751808,
+        0.015694234520196915,
+        0.0039747110567986965,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.010705706663429737,
+        0.010705706663429737,
+        0.029047179967164993,
+        0.015694234520196915,
+        0.029047179967164993,
+        0.029047179967164993,
+        0.010685855522751808,
+        0.006869869772344828,
+        0.010705706663429737,
+        0.015694234520196915,
+        0.013658224605023861,
+        0.021557528525590897,
+        0.021557528525590897,
+        0.018847310915589333,
+        0.013658224605023861,
+        0.013658224605023861,
+        0.021557528525590897,
+        0.018847310915589333,
+        0.0058678328059613705,
+        0.021557528525590897,
+        0.018847310915589333,
+        0.021557528525590897
+      ],
+      "q_value_solved": 2,
+      "reset_mask": [
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true
+      ],
+      "solved": false,
+      "solving_tactics": [],
+      "tactic_expandable": [
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true
+      ],
+      "tactic_init_value": 0.0,
+      "tactics": [
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h\u2081 h\u2082"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * (b * a)) b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp_all"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h\u2083 := hS (a * b) a"
+        }
+      ],
+      "theorem": {
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+        "ctx": {
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+            "ctx": {
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+                "is_valid": true,
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+              {
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
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+                "is_valid": true,
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
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+              {
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+              },
+              {
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
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+                "is_valid": true,
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
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+              {
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
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+                "is_valid": true,
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
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+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
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+                "unique_string": "have h1 := hS b a"
+              },
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
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+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
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+                "is_valid": true,
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
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+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
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+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
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+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
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+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
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+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
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+                "is_valid": true,
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+                "unique_string": "have h1 := hS b a"
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+                "unique_string": "simp [mul_assoc] at h1"
+              },
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+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+          }
+        ]
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+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
+                "namespaces": []
+              },
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+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
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+                {
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+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
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+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
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+                "unique_string": "intro a b"
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
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+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
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+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
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+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        },
+        {
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+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2",
+              "ctx": {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
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+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
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+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
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+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h2"
+          }
+        }
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+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
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+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
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+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+              {
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+                "unique_string": "simp at h1 h2"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all",
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+              {
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+                "is_valid": true,
+                "unique_string": "simp_all"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
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+            "hypotheses": [],
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+                "is_valid": true,
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
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+                "unique_string": "have h2 := hS a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
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+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
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+                "unique_string": "have h1 := hS b a"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
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+            "hypotheses": [],
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+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
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+            "hypotheses": [],
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+                "is_valid": true,
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+                "unique_string": "have h1 := hS b a"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (a * b) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1 h2"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h3 := hS (b * a) b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+          }
+        ]
+      ],
+      "counts": [
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+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
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+        0,
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+        0,
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+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (a * b) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (a * b) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+              "ctx": {
+                "namespaces": []
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+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1 h2"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1 h2"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b",
+              "ctx": {
+                "namespaces": []
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+              "hypotheses": [],
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                },
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+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h3 := hS (b * a) b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h3 := hS (b * a) b"
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+        }
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+        "SIZE": false,
+        "TIME": false
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+          "is_valid": true,
+          "unique_string": "have h3 := hS (b * a) b"
+        }
+      ],
+      "theorem": {
+        "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        "ctx": {
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+        },
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+            "is_valid": true,
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+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
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+          {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        ],
+        "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+      },
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+    {
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+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a (b * a)"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp_all [mul_assoc]"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
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+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          }
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+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
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+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
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+              },
+              {
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+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
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+                },
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+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
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+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
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+              "past_tactics": [
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+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
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+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
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+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a (b * a)"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a (b * a)"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp_all [mul_assoc]"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp_all [mul_assoc]"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        }
+      ],
+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
+      "is_solved_leaf": false,
+      "killed_tactics": [],
+      "log_critic_value": -0.5,
+      "log_w": [
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+        "SIZE": 9223372036854775807,
+        "TIME": 9223372036854775807
+      },
+      "minimum_tactic_length": {
+        "DEPTH": [],
+        "SIZE": [],
+        "TIME": []
+      },
+      "minimum_tactics": {
+        "DEPTH": [],
+        "SIZE": [],
+        "TIME": []
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+      "old_critic_value": 0.0,
+      "policy": {
+        "exploration": 0.3,
+        "type": 1
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+      "priors": [
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+        0.007499188184738159,
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+        0.006485720165073872,
+        0.03596969693899155,
+        0.03596969693899155,
+        0.03596969693899155,
+        0.00607823533937335,
+        0.010509527288377285,
+        0.03596969693899155,
+        0.00945228524506092,
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+        0.006603240966796875,
+        0.006603240966796875
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+      "q_value_solved": 2,
+      "reset_mask": [
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+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
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+        true,
+        true,
+        true,
+        true,
+        false
+      ],
+      "solved": false,
+      "solving_tactics": [],
+      "tactic_expandable": [
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+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true,
+        true
+      ],
+      "tactic_init_value": 0.0,
+      "tactics": [
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+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a (b * a)"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "simp [mul_assoc] at h1"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
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+          }
+        ],
+        [
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+          }
+        ],
+        [
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
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+          }
+        ],
+        [
+          {
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+            "ctx": {
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+              }
+            ],
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h2 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+          }
+        ]
+      ],
+      "counts": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h1"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h2 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+            "ctx": {
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+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h1"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h2 := hS a b"
+          }
+        }
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+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
+      "is_solved_leaf": false,
+      "killed_tactics": [],
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+      "minimum_tactics": {
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+        "SIZE": [],
+        "TIME": []
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+        "type": 1
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+      "reset_mask": [
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+        {
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+        },
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+        },
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+          "is_valid": true,
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+        },
+        {
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+          "is_valid": true,
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+        },
+        {
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+        },
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+        },
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+          "unique_string": "have h2 := hS a b"
+        },
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+        },
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+        },
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+          "is_valid": true,
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+        },
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+          "unique_string": "have h2 := hS a b"
+        },
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+          "unique_string": "have h2 := hS a b"
+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        },
+        {
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+          "unique_string": "have h2 := hS a b"
+        },
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+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
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+        },
+        {
+          "duration": 0,
+          "is_valid": true,
+          "unique_string": "have h2 := hS a b"
+        }
+      ],
+      "theorem": {
+        "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        "ctx": {
+          "namespaces": []
+        },
+        "hypotheses": [],
+        "past_tactics": [
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+            "is_valid": true,
+            "unique_string": "intro a b"
+          },
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+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h1"
+          }
+        ],
+        "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1"
+      },
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+      ]
+    },
+    {
+      "children_for_tactic": [
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h\u2081"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h\u2081"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
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+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
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+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp [mul_assoc] at h\u2081"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
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+                "unique_string": "intro a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        ]
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+      ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h\u2081"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h\u2081"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h\u2081"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h\u2081"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "simp [mul_assoc] at h\u2081"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp [mul_assoc] at h\u2081"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2082 := hS a b"
+          }
+        }
+      ],
+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
+      "is_solved_leaf": false,
+      "killed_tactics": [],
+      "log_critic_value": -0.5,
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+        "SIZE": 9223372036854775807,
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+      },
+      "minimum_tactic_length": {
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+      },
+      "minimum_tactics": {
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+        "SIZE": [],
+        "TIME": []
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+        "type": 1
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+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
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+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
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+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
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+                "is_valid": true,
+                "unique_string": "intro a b"
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2080 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
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+            "hypotheses": [],
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
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+            "hypotheses": [],
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+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
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+                "unique_string": "have h\u2081 := hS b a"
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a",
+            "ctx": {
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+            "hypotheses": [],
+            "past_tactics": [
+              {
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS (b * a) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
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+            "hypotheses": [],
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
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+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS (b * a) a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h1 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        ]
+      ],
+      "counts": [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        4,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        3,
+        2,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2080 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2080 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS (b * a) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS (b * a) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS (b * a) a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS (b * a) a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h1 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h1 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "have h\u2081 := hS b a"
+          }
+        }
+      ],
+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
+      "is_solved_leaf": false,
+      "killed_tactics": [],
+      "log_critic_value": -0.5,
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+        "TIME": []
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+        "type": 1
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
+        ],
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
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+          }
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
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+          }
+        ],
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+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
+        ],
+        [
+          {
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+          }
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+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
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+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+            },
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+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+            },
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+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+            },
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+              }
+            ],
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
+        ],
+        [
+          {
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+              }
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+          }
+        ],
+        [
+          {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
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+              }
+            ],
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+          }
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+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
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+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
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+              "past_tactics": [
+                {
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+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+          }
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+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+                }
+              ],
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+            }
+          ],
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+          }
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+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+                }
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+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
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+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
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+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
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+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
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+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
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+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
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+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
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+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
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+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+              "namespaces": []
+            },
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
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+              },
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+                }
+              ],
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+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
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+              },
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+                {
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+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
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+              },
+              "hypotheses": [],
+              "past_tactics": [
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+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
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+            "past_tactics": [],
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+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
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+            },
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+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
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+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ],
+          "goal": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "intro a b"
+          }
+        }
+      ],
+      "error": false,
+      "exploration": 0.3,
+      "in_minimum_proof": {
+        "DEPTH": false,
+        "SIZE": false,
+        "TIME": false
+      },
+      "in_proof": false,
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+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        58
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        19
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        41
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a (b * a)": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        47
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        23
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nhave h3 := hS (b * a) b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        2
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nhave h3 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        55
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp at h1 h2",
+        59
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nhave h2 := hS a b",
+        3
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        36
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1",
+        28
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b\nsimp [mul_assoc] at h2": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        63
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp [mul_assoc] at h1\nhave h2 := hS a b",
+        58
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp at h1": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        5
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a\nsimp_all [mul_assoc]": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h1 := hS b a",
+        53
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2080 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        2
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS (b * a) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        6
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        63
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+        44
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        62
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        61
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        59
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        58
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        57
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        56
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        52
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        24
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        4
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        28
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        29
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * (b * a)) b": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        52
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        55
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        50
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        45
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        39
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        36
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        33
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        0
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        38
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        3
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        5
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        6
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        25
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        8
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        46
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        11
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        13
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        16
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        53
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        18
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        31
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        21
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        28
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        49
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        41
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        35
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        34
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        32
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        30
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        1
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        2
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        15
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        42
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        7
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        44
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        9
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        48
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        10
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        12
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        14
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        17
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        19
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        22
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        23
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        24
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp_all": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        47
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        43
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        37
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        54
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        29
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        27
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        51
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        26
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        20
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+        4
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp [mul_assoc] at h\u2081": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        60
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        40
+      ],
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        26
+      ]
+    ],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nsimp at h\u2081": [
+      [
+        "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+        34
+      ]
+    ]
+  },
+  "policy": {
+    "exploration": 0.3,
+    "type": 1
+  },
+  "propagate_needed": false,
+  "root": {
+    "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+    "ctx": {
+      "namespaces": []
+    },
+    "hypotheses": [],
+    "past_tactics": [],
+    "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+  },
+  "seed": 42,
+  "simulations": [
+    {
+      "children_for_theorem": {
+        "12116204433008218031": {
+          "previous": 16342763735530018773,
+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+            }
+          ]
+        },
+        "16342763735530018773": {
+          "previous": 16579496881221230816,
+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ]
+        },
+        "16579496881221230816": {
+          "previous": 0,
+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                }
+              ],
+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+            }
+          ]
+        },
+        "3014227773155032910": {
+          "previous": 842976703719519088,
+          "value": []
+        },
+        "842976703719519088": {
+          "previous": 12116204433008218031,
+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "intro a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2082 := hS a b"
+                },
+                {
+                  "duration": 0,
+                  "is_valid": true,
+                  "unique_string": "have h\u2083 := hS (a * b) a"
+                }
+              ],
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+          }
+        }
+      },
+      "root": {
+        "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        "ctx": {
+          "namespaces": []
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+        "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+      },
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
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+          }
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+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
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+          }
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+          }
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+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a",
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+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a"
+          }
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+        "842976703719519088": {
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+            "ctx": {
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+              },
+              {
+                "duration": 0,
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+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        }
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+      }
+    },
+    {
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+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+              "ctx": {
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+                },
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+                }
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
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+        "16342763735530018773": {
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+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
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+                  "is_valid": true,
+                  "unique_string": "have h\u2081 := hS b a"
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+              "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+            }
+          ]
+        },
+        "16579496881221230816": {
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+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+              "ctx": {
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+            }
+          ]
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+        "842976703719519088": {
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+          "value": [
+            {
+              "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
+              "ctx": {
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+                },
+                {
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+                  "is_valid": true,
+                  "unique_string": "simp at h\u2081 h\u2082"
+                }
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+            }
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+      "parent_for_theorem": {
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+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
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+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
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+          }
+        },
+        "14195976025840234014": {
+          "previous": 842976703719519088,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
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+              {
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+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
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+          }
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+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
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+            },
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+          }
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+          "value": null
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+        "842976703719519088": {
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+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
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+          }
+        }
+      },
+      "root": {
+        "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+        "ctx": {
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+        "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
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+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "simp at h\u2081 h\u2082"
+          }
+        }
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+      "theorems": {
+        "12116204433008218031": {
+          "previous": 16342763735530018773,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a",
+            "ctx": {
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+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a"
+          }
+        },
+        "14195976025840234014": {
+          "previous": 842976703719519088,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082",
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+                "unique_string": "have h\u2081 := hS b a"
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+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "simp at h\u2081 h\u2082"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082"
+          }
+        },
+        "16342763735530018773": {
+          "previous": 16579496881221230816,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b"
+          }
+        },
+        "16579496881221230816": {
+          "previous": 0,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+          }
+        },
+        "842976703719519088": {
+          "previous": 12116204433008218031,
+          "value": {
+            "conclusion": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "intro a b"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2081 := hS b a"
+              },
+              {
+                "duration": 0,
+                "is_valid": true,
+                "unique_string": "have h\u2082 := hS a b"
+              }
+            ],
+            "unique_string": "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b"
+          }
+        }
+      },
+      "values": null,
+      "virtual_count_added": {
+        "12116204433008218031": {
+          "previous": 16342763735530018773,
+          "value": true
+        },
+        "14195976025840234014": {
+          "previous": 842976703719519088,
+          "value": false
+        },
+        "16342763735530018773": {
+          "previous": 16579496881221230816,
+          "value": true
+        },
+        "16579496881221230816": {
+          "previous": 0,
+          "value": true
+        },
+        "842976703719519088": {
+          "previous": 12116204433008218031,
+          "value": true
+        }
+      }
+    }
+  ],
+  "simulations_for_theorem": {
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nhave h\u2083 := hS (a * b) a": [],
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n\nintro a b\nhave h\u2081 := hS b a\nhave h\u2082 := hS a b\nsimp at h\u2081 h\u2082": []
+  },
+  "unexplored_theorems": [
+    "theorem putnam_2001_a1 (S : Type*) [Mul S] (hS : \u2200 a b : S, (a * b) * a = b) : \u2200 a b : S, a * (b * a) = b := by\n"
+  ]
+}
\ No newline at end of file
diff --git a/samples/test.json b/samples/test.json
index 5976435..df3ff5d 100644
--- a/samples/test.json
+++ b/samples/test.json
@@ -73,6 +73,63 @@
         0,
         0
       ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "C",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [],
+              "unique_string": "C"
+            }
+          ],
+          "goal": {
+            "conclusion": "B",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "B"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "TACC"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "C",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [],
+              "unique_string": "C"
+            }
+          ],
+          "goal": {
+            "conclusion": "B",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "B"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "TACD"
+          }
+        }
+      ],
+      "error": false,
       "exploration": 0.3,
       "in_minimum_proof": {
         "DEPTH": false,
@@ -178,6 +235,63 @@
         2,
         1
       ],
+      "effects": [
+        {
+          "children": [
+            {
+              "conclusion": "B",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [],
+              "unique_string": "B"
+            }
+          ],
+          "goal": {
+            "conclusion": "A",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "A"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "TACA"
+          }
+        },
+        {
+          "children": [
+            {
+              "conclusion": "B",
+              "ctx": {
+                "namespaces": []
+              },
+              "hypotheses": [],
+              "past_tactics": [],
+              "unique_string": "B"
+            }
+          ],
+          "goal": {
+            "conclusion": "A",
+            "ctx": {
+              "namespaces": []
+            },
+            "hypotheses": [],
+            "past_tactics": [],
+            "unique_string": "A"
+          },
+          "tac": {
+            "duration": 0,
+            "is_valid": true,
+            "unique_string": "TACB"
+          }
+        }
+      ],
+      "error": false,
       "exploration": 0.3,
       "in_minimum_proof": {
         "DEPTH": false,
diff --git a/src/graph/htps.cpp b/src/graph/htps.cpp
index a92a24c..79cba03 100644
--- a/src/graph/htps.cpp
+++ b/src/graph/htps.cpp
@@ -772,13 +772,11 @@ bool HTPSNode::has_virtual_count() const {
 }
 
 HTPSNode HTPSNode::from_json(const nlohmann::json &j) {
-    HTPSNode node;
-    node.thm =j["theorem"];
+    TheoremPointer thm = j["theorem"];
     std::vector<std::shared_ptr<tactic>> tactics;
     for (const auto &tac: j["tactics"]) {
         tactics.push_back(tac);
     }
-    node.tactics = tactics;
     std::vector<std::vector<TheoremPointer>> children_for_tactic;
     for (const auto &children: j["children_for_tactic"]) {
         std::vector<TheoremPointer> children_for_tactic_inner;
@@ -787,28 +785,30 @@ HTPSNode HTPSNode::from_json(const nlohmann::json &j) {
         }
         children_for_tactic.push_back(children_for_tactic_inner);
     }
-    node.children_for_tactic = children_for_tactic;
-    node.killed_tactics = j["killed_tactics"].get<std::unordered_set<size_t>>();
-    node.solving_tactics = j["solving_tactics"].get<std::unordered_set<size_t>>();
-    node.tactic_expandable = j["tactic_expandable"].get<std::vector<bool>>();
-    node.minimum_proof_size = MinimumLengthMap::from_json(j["minimum_proof_size"]);
-    node.minimum_tactics = MinimumTacticMap::from_json(j["minimum_tactics"]);
-    node.minimum_tactic_length = MinimumTacticLengthMap::from_json(j["minimum_tactic_length"]);
-    node.in_minimum_proof = MinimumBoolMap::from_json(j["in_minimum_proof"]);
-    node.solved = j["solved"];
-    node.is_solved_leaf = j["is_solved_leaf"];
-    node.in_proof = j["in_proof"];
-    node.old_critic_value = j["old_critic_value"];
+    children_for_tactic = children_for_tactic;
+    std::unordered_set<size_t> killed_tactics = j["killed_tactics"].get<std::unordered_set<size_t>>();
+    std::unordered_set<size_t> solving_tactics = j["solving_tactics"].get<std::unordered_set<size_t>>();
+    std::vector<bool> tactic_expandable = j["tactic_expandable"].get<std::vector<bool>>();
+    auto minimum_proof_size = MinimumLengthMap::from_json(j["minimum_proof_size"]);
+    auto minimum_tactics = MinimumTacticMap::from_json(j["minimum_tactics"]);
+    auto minimum_tactic_length = MinimumTacticLengthMap::from_json(j["minimum_tactic_length"]);
+    auto in_minimum_proof = MinimumBoolMap::from_json(j["in_minimum_proof"]);
+    bool solved = j["solved"];
+    bool is_solved_leaf = j["is_solved_leaf"];
+    bool in_proof = j["in_proof"];
+    double old_critic_value = j["old_critic_value"];
+    double log_critic_value;
     if (!j["log_critic_value"].is_null()) {
-        node.log_critic_value = j["log_critic_value"];
+        log_critic_value = j["log_critic_value"];
     } else {
-        node.log_critic_value = MIN_FLOAT;
+        log_critic_value = MIN_FLOAT;
     }
-    node.priors = static_cast<std::vector<double>>(j["priors"]);
-    node.q_value_solved = j["q_value_solved"];
-    node.policy = j["policy"];
-    node.exploration = j["exploration"];
-    node.tactic_init_value = j["tactic_init_value"];
+    auto priors = static_cast<std::vector<double>>(j["priors"]);
+    QValueSolved q_value_solved = j["q_value_solved"];
+    std::vector<std::shared_ptr<htps::env_effect>> effects = j["effects"];
+    std::shared_ptr<Policy> policy = j["policy"];
+    double exploration = j["exploration"];
+    double tactic_init_value = j["tactic_init_value"];
     std::vector<double> log_w;
     for (const auto &w: j["log_w"]) {
         if (w.is_null()) {
@@ -817,10 +817,25 @@ HTPSNode HTPSNode::from_json(const nlohmann::json &j) {
             log_w.push_back(w);
         }
     }
-    node.log_w = log_w;
-    node.counts = static_cast<std::vector<size_t>>(j["counts"]);
-    node.virtual_counts = static_cast<std::vector<size_t>>(j["virtual_counts"]);
-    node.reset_mask = static_cast<std::vector<bool>>(j["reset_mask"]);
+
+    std::vector<size_t> counts = static_cast<std::vector<size_t>>(j["counts"]);
+    std::vector<size_t> virtual_counts = static_cast<std::vector<size_t>>(j["virtual_counts"]);
+    std::vector<bool> reset_mask = static_cast<std::vector<bool>>(j["reset_mask"]);
+    bool error = j["error"];
+
+    HTPSNode node = {thm, tactics, children_for_tactic, policy, priors, exploration, log_critic_value, q_value_solved, tactic_init_value, effects, error};
+    node.killed_tactics = killed_tactics;
+    node.solving_tactics = solving_tactics;
+    node.tactic_expandable = tactic_expandable;
+    node.minimum_proof_size = minimum_proof_size;
+    node.minimum_tactics = minimum_tactics;
+    node.minimum_tactic_length = minimum_tactic_length;
+    node.in_minimum_proof = in_minimum_proof;
+    node.solved = solved;
+    node.is_solved_leaf = is_solved_leaf;
+    node.in_proof = in_proof;
+    node.old_critic_value = old_critic_value;
+    node.log_critic_value = log_critic_value;
     return node;
 }
 
@@ -850,6 +865,8 @@ HTPSNode::operator nlohmann::json() const {
     j["counts"] = counts;
     j["virtual_counts"] = virtual_counts;
     j["reset_mask"] = reset_mask;
+    j["error"] = error;
+    j["effects"] = effects;
     return j;
 }
 
diff --git a/src/graph/htps.h b/src/graph/htps.h
index d7c8b67..01c87a4 100644
--- a/src/graph/htps.h
+++ b/src/graph/htps.h
@@ -431,8 +431,6 @@ namespace htps {
             reset_HTPS_stats();
         }
 
-        HTPSNode() = default;
-
         /* Reset the HTPS statistics, resetting counts and logW values.
          * */
         void reset_HTPS_stats();