A cyclic shifted rational difference sum is nonnegative

Proved

WorkbookSource.base_12977

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

lean-workbooksource-checked

Prove that $\frac{b(1-a)}{a(1+b)}+\frac{c(1-b)}{b(1+c)}+\frac{a(1-c)}{c(1+a)}\ge 0$ given $a,b,c >0$ and $a+b+c=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_12977 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_12977 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (b * (1 - a) / (a * (1 + b)) + c * (1 - b) / (b * (1 + c)) + a * (1 - c) / (c * (1 + a))) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_12977; Apache-2.0

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