A cyclic product ratio bounds a pairwise ratio sum

Proved

WorkbookSource.base_12712

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

lean-workbooksource-checked

Prove the stronger inequality: For any positive real numbers $a,b$ and $c$ , $3+\frac{a(a+c)}{b(b+c)}+\frac{b(b+a)}{c(c+a)}+\frac{c(c+b)}{a(a+b)}\ge 4\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)$

Source: InternLM Lean-Workbook, record lean_workbook_12712 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12712 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 3 + a * (a + c) / (b * (b + c)) + b * (b + a) / (c * (c + a)) + c * (c + b) / (a * (a + b)) ≥ 4 * (a / (b + c) + b / (c + a) + c / (a + b))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12712; Apache-2.0

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