A cyclic product ratio sum bounds a pairwise reciprocal sum

Proved

WorkbookSource.base_2488

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that

$\frac {ab}{c(c + a)} + \frac {bc}{a(a + b)} + \frac {ca}{b(b + c)}\geq \frac {a}{c + a} + \frac {b}{a + b} + \frac {c}{b + c}.$

Source: InternLM Lean-Workbook, record lean_workbook_2488 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2488 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b) / (c * (c + a)) + (b * c) / (a * (a + b)) + (c * a) / (b * (b + c)) ≥ a / (c + a) + b / (a + b) + c / (b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2488; Apache-2.0

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