Preserving squares forces nonnegative values

Proved

WorkbookSource.problem_19290

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

lean-workbooksource-checked

For $f(1) = 1$ , $f(x^2) = [f(x)]^2$ . This implies that $f(x)$ is nonnegative whenever $x$ is nonnegative.

Source: InternLM Lean-Workbook, record lean_workbook_19290 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_19290  (f : ℝ → ℝ)
  (h₀ : f 1 = 1)
  (h₁ : ∀ x, f (x^2) = (f x)^2) :
  ∀ x ≥ 0, 0 ≤ f x  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_19290; Apache-2.0

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