lean_workbook_plus_26910

⚠️ 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.

Determine all functions : $f: R$ \ $\{ 0,1{\text{\} }}$ $\to R$ satisfying : $f(x) + f\left( {\frac{1}{{1 - x}}} \right) = \frac{{2(1 - 2x)}}{{x(1 - x)}}$ , $\forall x \ne 0,x \ne 1$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_26910 (f : ℝ → ℝ) (hf : ∀ x ∈ ({0, 1}ᶜ : Set ℝ), f x + f (1 / (1 - x)) = 2 * (1 - 2 * x) / (x * (1 - x))) : ∀ x ∈ ({0, 1}ᶜ : Set ℝ), f x = 1 / x + 1 / (1 - x)   :=  by sorry

The inserted formula is refuted by f(x)=(x+1)/(x−1), which satisfies the equation on x≠0,1.

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

Proposed corrected statement

Restore the original mathematical task without the inserted answer: Determine all functions : $f: R$ \ $\{ 0,1{\text{\} }}$ $\to R$ satisfying : $f(x) + f\left( {\frac{1}{{1 - x}}} \right) = \frac{{2(1 - 2x)}}{{x(1 - x)}}$ , $\forall x \ne 0,x \ne 1$ Require both necessity and sufficiency for a find-all/classification task; no replacement fixed answer is asserted in this audit.

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.