lean_workbook_plus_79400

⚠️ 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 $\frac{1}{(t-s) \binom{t-s-1}{r}} = \frac{1}{(t+1-s)\binom{t-s}{r}} + \frac{1}{(t+1-s)\binom{t-s}{r+1}}$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_79400 : ∀ t s r : ℕ, t + 1 - s ≠ 0 ∧ t - s ≠ 0 →
    1 / ((t - s) * Nat.choose (t - s - 1) r) = 1 / ((t + 1 - s) * Nat.choose (t - s) r) +
    1 / ((t + 1 - s) * Nat.choose (t - s) (r + 1))   :=  by sorry

Natural division replaces a reciprocal binomial identity, and the hypotheses do not ensure the binomial denominators are nonzero.

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

Proposed corrected statement

Use rational division with s<t and 0≤r<t−s, then prove the displayed identity.

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.