A complementary cyclic ratio sum is nonnegative at fixed sum three

Proved

WorkbookSource.plus_63837

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers such that: $a+b+c=3$ .Prove that: $\frac{(1-a)(1-ab)}{a(1+c)}+\frac{(1-b)(1-bc)}{b(1+a)}+\frac{(1-c)(1-ca)}{c(1+b)} \ge 0$

Source: InternLM Lean-Workbook, record lean_workbook_plus_63837 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_63837 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (1 - a) * (1 - a * b) / (a * (1 + c)) + (1 - b) * (1 - b * c) / (b * (1 + a)) + (1 - c) * (1 - c * a) / (c * (1 + b)) ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_63837; Apache-2.0

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