lean_workbook_plus_53740

⚠️ 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.

Let $( z_{n})$ the sequence defined by $z_{n+2} =3z_{n+1} -z_{n}$ , $z_{1} =z_{2} =1$ recursively. It is not difficult to show that $y_{n} =z^{2}_{n}$ for all $n\in \mathbb{N}$ by induction on $n$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_53740 (n : ℕ) (f : ℕ → ℕ) (hf: f 1 = 1 ∧ f 2 = 1 ∧ ∀ n, f (n + 2) = 3 * f (n + 1) - f n) : ∀ n, f n = (f 1)^2   :=  by sorry

The source definition of y is absent, so the full corrected theorem needs that context.

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

Proposed corrected statement

Introduce the separate sequence y and its actual defining recurrence, and conclude y(n)=z(n)² at the source indices; do not replace this relation by z(n)=z(1)².

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.