Ordering_of_Inverses_in_Ordered_Monoid

⚠️ 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 $\left({S, \circ, \preceq}\right)$ be an ordered monoid whose identity is $e$. Let $x, y \in S$ be invertible. Then $x \prec y \iff y^{-1} \prec x^{-1}$

Why this node was retired

The posted statement is

theorem Ordering_of_Inverses_in_Ordered_Monoid (x y : ℚ) (hx : x ≠ 0) (hy : y ≠ 0) (h : x < y) : y⁻¹ < x⁻¹ := by sorry

Arbitrary nonzero rationals of opposite signs are permitted; their multiplication is not an order-compatible group operation on all nonzero rationals.

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

Proposed corrected statement

For rationals 0<x<y, conclude y⁻¹<x⁻¹; a second valid version assumes x<y<0. Or state inversion reverses order in a genuinely ordered group.

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.