lean_workbook_plus_9827

⚠️ 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.

Derive the inequality ${\left(m-\frac{1}{2}\right)}^2-\frac{1}{4}<n<{\left(m+\frac{1}{2}\right)}^2+\frac{3}{4}$ from the given conditions $m^2\leq F_n<(m+1)^2$ and $F_n = m^2 + r$ with $0 < r < 2m + 1$, where $n$ is the index of $F_n$ in the sequence.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_9827 (m n r : ℕ) (F : ℕ → ℕ) (h₁ : m^2 ≤ F n) (h₂ : F n < (m + 1)^2) (h₃ : F n = m^2 + r) (h₄ : 0 < r) (h₅ : r < 2 * m + 1): (m - 1 / 2)^2 - 1 / 4 < n ∧ n < (m + 1 / 2)^2 + 3 / 4   :=  by sorry

The supplied value-range conditions on an arbitrary F impose no restriction on its index; natural fractions introduce a separate error.

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 real arithmetic in the two bounds and restore the defining relation of the sequence F_n to its index 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.