A cyclic product correction bounded by the quadratic sum

Proved

WorkbookSource.base_910

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

For every $a,b,c>0$ prove the inequality $ab\biggl(1-\frac{2c^2}{(a+b)^2}\biggr)+bc\biggl(1-\frac{2a^2}{(b+c)^2}\biggr)+ca\biggl(1-\frac{2b^2}{(c+a)^2}\biggr)\le \frac{1}{2}(a^2+b^2+c^2)$

Source: InternLM Lean-Workbook, record lean_workbook_910 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_910 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a * b * (1 - 2 * c^2 / (a + b)^2) + b * c * (1 - 2 * a^2 / (b + c)^2) + c * a * (1 - 2 * b^2 / (c + a)^2) ≤ (1 / 2) * (a^2 + b^2 + c^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_910; Apache-2.0

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