A piecewise function has no fixed point

Proved

WorkbookSource.plus_76043

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that if $f(x) = \begin{cases}1, & 0\le x \le 1/2,\\0, & 1/2 < x \le 1\end{cases}$ , it does not have a fixed point in $[0,1]$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_76043 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_76043 (f : ℝ → ℝ) (hf: f = fun x => if x ≤ 1/2 then 1 else 0) : ¬ (∃ x, f x = x)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_76043; Apache-2.0

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