Banach_Fixed_Point_Theorem
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.
A contraction mapping on a complete metric space has a unique fixed point.
Why this node was retired
The posted statement is
theorem Banach_Fixed_Point_Theorem {α : Type _} [MetricSpace α] [CompleteSpace α] (f : α → α) (q : ℝ) (hq : q ∈ Set.Ico (0 : ℝ) 1) (hf : ∀ x y, dist (f x) (f y) ≤ q * dist x y) : ∃! x, f x = x := by sorry
The empty metric space is complete and satisfies the contraction premise vacuously but contains no fixed point.
The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.
Proposed corrected statement
Add [Nonempty X] and use a contraction factor 0≤k<1, retaining completeness and the global contraction inequality.
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 Banach_Fixed_Point_Theorem {α : Type _} [MetricSpace α] [CompleteSpace α] (f : α → α) (q : ℝ) (hq : q ∈ Set.Ico (0 : ℝ) 1) (hf : ∀ x y, dist (f x) (f y) ≤ q * dist x y) : ∃! x, f x = x := by sorry