lean_workbook_plus_8856

⚠️ 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.

Case 3. $\alpha>0$ .\n\nOriginal equation can be rewriting as $(x-\alpha)^2=2\alpha-x$ and $x\geq\alpha$ .\n\nSo, $x^2-(2\alpha-1)x+\alpha^2-2\alpha=0$ and $x\geq\alpha$ .\n\nTherefore, $x=\frac{2\alpha-1\pm\sqrt{4\alpha+1}}{2}$ . But $\alpha\geq 0$ and $x\geq\alpha$ , hence $x=\frac{2\alpha-1+\sqrt{4\alpha+1}}{2}$ - unique solution of equation.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_8856  (x α : ℝ)
  (f : ℝ → ℝ)
  (h₀ : α > 0)
  (h₁ : ∀ x, f x = (x - α) ^ 2)
  (h₂ : ∀ x, x ≥ α)
  (h₃ : ∃! a, f a = 2 * α - a) :
  x = 2 * α - 1 + Real.sqrt (4 * α + 1)   :=  by sorry

A restriction on a prospective solution is quantified over every real number.

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

Replace the global hypothesis ∀x,x≥α by x≥α and assume the actual equation (x−α)^2=2α−x. The conclusion is x=(2α−1+sqrt(4α+1))/2, including the missing denominator 2.

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.