lean_workbook_plus_37280

⚠️ 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.

Solve the recursion, along with proof: \n\n $x_n = \frac {\sqrt {2}}{2}(x_{n - 1} - y_{n - 1}) + 10$ \n\n $y_n = \frac {\sqrt {2}}{2}(x_{n - 1} + y_{n - 1}), \ n \in \mathbb{N}$ \n\n \n\n Write the explicit formula for $x_n, y_n$ in terms of $x_0, y_0$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_37280 (x y : ℕ → ℝ) (hx : x 0 = 10) (hy : y 0 = 0) (hn : ∀ n, x (n + 1) = (Real.sqrt 2 / 2) * (x n - y n) + 10) (hn' : ∀ n, y (n + 1) = (Real.sqrt 2 / 2) * (x n + y n)) : ∀ n, x n = 10 + (Real.sqrt 2) ^ n * (x 0 - y 0) / 2 ∧ y n = (Real.sqrt 2) ^ n * (x 0 + y 0) / 2   :=  by sorry

The invented formula fails at n=0 and incorrectly has exponential growth; the stated matrix formula uniquely specifies the correct closed form.

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

Proposed corrected statement

Restore arbitrary initial values and the genuine rotation recurrence formula v_n=c+R^n(v_0−c), where R is rotation byπ/4 and c solves(I−R)c=(10,0).

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.