lean_workbook_plus_11853

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Prove that there does not exist the function f(x) that is defined in $[0;2]$ and satisfies all these conditions:\n\n1) f(x) is bounded in [0,2].\n2) f(x+y) ≥ 1+2y(f(x))^2, ∀x, y ≥ 0, x+y ≤ 2.

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_11853    (f : ℝ → ℝ)
    (h1 : ∃ M, ∀ x ∈ Set.Icc (0:ℝ) 2, abs (f x) ≤ M)
    (h2 : ∀ x y, x ≥ 0 ∧ y ≥ 0 ∧ x + y ≤ 2 → f (x + y) ≥ 1 + 2 * y * (f x)^2) :
    False   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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