A product of pairwise ratio sums has a symmetric lower bound

Proved

WorkbookSource.base_278

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

lean-workbooksource-checked

Let $a,b,c>0$ , prove that :

\left( \frac{a}{b+c}+\frac{b}{c+a}\right)\left( \frac{b}{c+a}+\frac{c}{a+b}\right)\left( \frac{c}{a+b}+\frac{a}{b+c}\right)\ge \frac{7}{8}\left( \frac{a^2+b^2+c^2}{ab+bc+ca}\right)+\frac{1}{8}

Source: InternLM Lean-Workbook, record lean_workbook_278 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_278 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a)) * (b / (c + a) + c / (a + b)) * (c / (a + b) + a / (b + c)) ≥ 7 / 8 * (a^2 + b^2 + c^2) / (a * b + b * c + c * a) + 1 / 8  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_278; Apache-2.0

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