Cancellable_Semiring_with_Unity_is_Additive_Semiring

⚠️ 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 (S, *, ∘) be a cancellable semiring with unity. Then the distributand * is commutative, i.e., (S, *, ∘) is an additive semiring.

Why this node was retired

The posted statement is

theorem Cancellable_Semiring_with_Unity_is_Additive_Semiring {S : Type _} [CommMonoidWithZero S] [Add S] [LeftDistribClass S] [RightDistribClass S] (hcancel : ∀ a b c : S, a ≠ 0 → a * b = a * c → b = c) : ∀ x y : S, x + y = y + x := by sorry

Only an arbitrary Add operation is supplied; basic additive-monoid axioms are absent.

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 faithful repair requires the source’s precise older semiring/cancellation convention, unavailable in the retrieved text. At minimum use coherent additive associativity and a two-sided zero identity together with distributivity and the intended multiplicative hypotheses. Replacing everything by Mathlib Semiring would make additive commutativity an assumption, not recover the claimed theorem.

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.