lean_workbook_plus_20668
Disproved⚠️ Retired — specification defect
The Lean statement below does not encode the problem shown on this page, so its
Disprovedstatus carries no information about that problem. Do not import this node or use it as a dependency.
Given the function $ g(x) = f(2x-1) = \frac{2x-1}{-4x^2+4x} $, show that it is a one-to-one mapping from $ (0;1) $ onto $ \mathbb{R} $.
Why this node was retired
The posted statement is
theorem lean_workbook_plus_20668 (f : ℝ → ℝ) (g : ℝ → ℝ) (h₁ : ∀ x, g x = f (2 * x - 1)) (h₂ : ∀ x, f x = (2 * x - 1) / (-4 * x ^ 2 + 4 * x)) : ∀ x, 0 < x ∧ x < 1 → ∀ y, g x = g y → x = y := by sorry
The target defines f by the g formula, composes twice, leaves y unrestricted, and omits surjectivity.
The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.
Proposed corrected statement
Define g(x)=(2x−1)/(−4x²+4x) directly and prove its restriction to(0,1) is bijective ontoℝ.
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.
import Mathlib.Analysis.Complex.Basic
theorem lean_workbook_plus_20668 (f : ℝ → ℝ) (g : ℝ → ℝ) (h₁ : ∀ x, g x = f (2 * x - 1)) (h₂ : ∀ x, f x = (2 * x - 1) / (-4 * x ^ 2 + 4 * x)) : ∀ x, 0 < x ∧ x < 1 → ∀ y, g x = g y → x = y := by sorry