A square-root bound for three positive inputs

Proved

WorkbookSource.problem_31079

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

lean-workbooksource-checked

Prove that for positive real numbers a, b, and c, the following inequality holds:

\sqrt{2(a^2+b^2)(b^2+c^2)(c^2+a^2)} \geq (a+b)(b+c)(c+a)-4abc.

Source: InternLM Lean-Workbook, record lean_workbook_31079 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_31079 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : Real.sqrt (2 * (a^2 + b^2) * (b^2 + c^2) * (c^2 + a^2)) ≥ (a + b) * (b + c) * (c + a) - 4 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31079; Apache-2.0

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