A cyclic ratio upper bound with a triple-product denominator

Proved

WorkbookSource.base_23740

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

lean-workbooksource-checked

For $a, b, c>0, a+b+c=3$ , prove that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le\frac{3}{1+abc}.$

Source: InternLM Lean-Workbook, record lean_workbook_23740 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23740 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a / (a + b) + b / (b + c) + c / (c + a) ≤ 3 / (1 + a * b * c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23740; Apache-2.0

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