lean_workbook_plus_69074

⚠️ 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.

Find all continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfy\n1) $f(f(x))=x$\n2) $\displaystyle f(x+1)=\frac{f(x)}{f(x)+1}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_69074 (f : ℝ → ℝ) (hf1 : ∀ x, 0 < x → f (f x) = x) (hf2 : ∀ x, 0 < x → f (x + 1) = f x / (f x + 1)) : ∀ x, 0 < x → f x = x   :=  by sorry

The claimed identity solution is wrong and source continuity/positive codomain were omitted; f(x)=1/x satisfies even the source conditions.

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

Proposed corrected statement

Classify continuous f:(0,∞)→(0,∞) satisfying f(f(x))=x and f(x+1)=f(x)/(f(x)+1). Do not assert f(x)=x; the reciprocal is a verified candidate, but this audit does not assert uniqueness.

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.