lean_workbook_plus_13643

⚠️ 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.

Prove the identity $k \binom{n}{k} = n \binom{n - 1}{k - 1}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_13643 : ∀ n k : ℕ, k * Nat.choose n k = n * Nat.choose (n - 1) (k - 1)   :=  by sorry

The combinatorial identity includes k=0, where truncated k−1 does not mean an index−1 with zero binomial coefficient.

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

Proposed corrected statement

Require k≥1 for the natural-index identity k·choose(n,k)=n·choose(n−1,k−1), or use an extended binomial coefficient that is0 at negative indices.

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.