Group_Abelian_iff_Cross_Cancellation_Property

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let G be a group. Then G is abelian iff G has the cross cancellation property.

Why this node was retired

The posted statement is

theorem Group_Abelian_iff_Cross_Cancellation_Property {G : Type _} [Group G] : (∀ a b : G, a * b = b * a) ↔ (∀ a b c d : G, a * b = c * d → a * d = c * b) := by sorry

The displayed cross-cancellation formula is false even in the additive integers; the intended definition is not substantiated by available source text.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

A concrete valid replacement is: a group is abelian iff ∀a b c d, ab=cd → ac⁻¹=db⁻¹. This is an algebraic replacement, not a verified reconstruction of the unavailable source definition of cross cancellation.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.