lean_workbook_plus_34239

⚠️ 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 the sequence defined by $F_n = \frac{\left(\frac{1+\sqrt 5}{2}\right)^n - \left(\frac{1-\sqrt 5}{2}\right)^n}{\sqrt 5}$ satisfies the Fibonacci sequence, i.e., $F_0=0$, $F_1=1$, and $F_n+F_{n+1}=F_{n+2}$ for all non-negative integers $n$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_34239 (F : ℕ → ℝ) (h₀ : F 0 = 0) (h₁ : F 1 = 1) (h₂ : ∀ n, F n + F (n + 1) = F (n + 2)) : ∀ n, F n = (1 / √5) * ((1 + √5) / 2)^n - ((1 - √5) / 2)^n   :=  by sorry

The factor1/√5 multiplies only the first exponential term.

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

Proposed corrected statement

Prove Fₙ=(((1+√5)/2)ⁿ−((1−√5)/2)ⁿ)/√5 with the entire difference grouped, under the Fibonacci initial values and 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.