A cyclic shifted ratio difference is nonnegative

Proved

WorkbookSource.base_26851

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that $\frac{a(a-b)}{b(a+b)}+\frac{b(b-c)}{c(b+c)}+\frac{c(c-a)}{a(c+a)} \geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_26851 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26851 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * (a - b) / (b * (a + b)) + b * (b - c) / (c * (b + c)) + c * (c - a) / (a * (c + a))) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26851; Apache-2.0

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