lean_workbook_plus_41309

⚠️ 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.

Given the recurrence relation $x_{n+1} = 3x_n + 4y_n$, $y_{n+1} = 2x_n + 3y_n$ with initial values $x_1 = 4$, $y_1 = 1$, find the solutions $(x_n, y_n)$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_41309 (x y : ℕ → ℕ) (hx : x 1 = 4) (hy : y 1 = 1) (hn: ∀ n, x (n + 1) = 3 * x n + 4 * y n) (hn2: ∀ n, y (n + 1) = 2 * x n + 3 * y n) : ∀ n, x n = 2 ^ n + 2 * 3 ^ n ∧ y n = 2 ^ n + 3 ^ n   :=  by sorry

The formal zero recurrence forces 1=2x₀+3y₀ in naturals, making the hypotheses impossible.

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

Impose the two recurrence rules only for n≥1, keep x₁=4,y₁=1, and find their true closed forms (or use powers of [[3,4],[2,3]]). Remove the unsupported powers of 2 and 3.

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.