lean_workbook_plus_44316

⚠️ 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 that if a sequence $\\{G(n) \\}_{n=0}^{\\infty}$ of integers satisfies: $G(0)=0, G(n)=n-G(G(n))\quad (n=1,2,3,...)$ then:\n\n $(a)$ $G(k) \ge G(k-1)$ for any positive integer $k$ \n\n $(b)$ no integer $k$ exists such that $G(k-1)=G(k)=G(k+1)$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_44316 (G : ℕ → ℕ) (hG : G 0 = 0 ∧ ∀ n, G (n + 1) = n - G (G n)) : (∀ k, G k ≥ G (k - 1)) ∧ ¬∃ k, G (k - 1) = G k ∧ G k = G (k + 1)   :=  by sorry

The recurrence n−G(G(n)) was shifted to the LHS at n+1 without shifting the RHS, and the no-three-equal condition includes k=0.

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

Proposed corrected statement

Recover the exact source recurrence (the printed implicit G(n)=n−G(G(n)) differs from the usual Hofstadter recurrence). Require k≥1 in both conclusions and translate every shifted index consistently; do not silently substitute a different recurrence.

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.