lean_workbook_plus_66022

⚠️ 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 function $f$ ,defined onthe sets of ordered pairs of positive integers,satisfying properties: $f(x,x) = x; f(x,y) = f(y,x)$ and $(x + y)f(x,y) = yf(x,x + y)$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_66022 (f : ℕ × ℕ → ℕ) (hx: ∀ x : ℕ, f (x,x) = x) (hab: ∀ {x y : ℕ}, f (x,y) = f (y,x)) (h : ∀ {x y : ℕ}, (x + y) * f (x,y) = y * (f (x,x + y))) : ∀ {x y : ℕ}, f (x,y) = x * y / (x + y)   :=  by sorry

The inserted harmonic-mean formula fails the diagonal condition; the source solution is lcm.

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 f on positive integer pairs satisfying the three displayed conditions, prove f(x,y)=lcm(x,y), and verify the converse. If retaining natural zero inputs, handle both axes explicitly.

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.