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teorth merged 1 commit into from
Mar 9, 2026
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add acknowledgment #468

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4 changes: 2 additions & 2 deletions analysis/Analysis/Misc/equational.lean
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Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
import Mathlib.Tactic

/-
A informal proof of the theorem `singleton_law` is provided below. Claude Code was used to formalize this proof using the following steps:
A informal proof of the theorem `singleton_law` is provided below, courtesy of Bruno Le Floch https://leanprover.zulipchat.com/#narrow/channel/458659-Equational/topic/Alternative.20proofs.20of.20E1689.E2.8A.A2E2/near/517189582. Claude Code was used to formalize this proof using the following steps:

Step 0: Formalize the `S` and `f` notation.

Expand Down Expand Up @@ -122,7 +122,7 @@ lemma lemma3 (h : Equation1689 M) (a : M) : ∃ e : M, S e a = a := by
have h_step3 : a ◇ f b c = f a d := by grind
-- Lemma 1 with b←a, c←d gives S a a = a ◇ f a d, i.e., (a ◇ a) ◇ a = a ◇ f a d.
have h_step4 : (a ◇ a) ◇ a = a ◇ f a d := by
simpa only [S] using lemma1 h a a d
simpa using lemma1 h a a d
-- Combine: S(f a d) a = (a ◇ f a d) ◇ f a d = ((a ◇ a) ◇ a) ◇ f a d
-- = ((a ◇ a) ◇ a) ◇ (a ◇ f b c) = ((a ◇ a) ◇ a) ◇ S b a = a.
calc S (f a d) a
Expand Down