lean_workbook_plus_40283

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Suppose, by way of contradiction, that for some positive integer $k,$ all $a_n$ with $n\ge k$ have the same parity. Since $2\le a_{n + 1} - a_n \le 4$ , it follows that $a_{n + 1} - a_n \in \{2,4\},$ for all $n\ge k.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_40283  (a : ℕ → ℕ)
  (h₀ : 0 < a 0)
  (h₁ : ∀ n, 2 ≤ a (n + 1) - a n ∧ a (n + 1) - a n ≤ 4) :
  ∃ k, ∀ n, n ≥ k → Odd (a n) ∨ Even (a n)   :=  by sorry

Odd(a_n)∨Even(a_n) says each term has some parity, not that the tail has the same parity. Every natural sequence satisfies it.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Prove the actual local source step: if 2≤a_(n+1)−a_n≤4 and a_(n+1)≡a_n (mod 2) for each n≥k, then a_(n+1)−a_n∈{2,4} on that tail. Any claim ruling out eventual constant parity needs the omitted definition/additional recurrence of a_n.

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.