A mixed rational sum and product is at least two

Proved

WorkbookSource.base_772

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{b}{c+a}+\frac{c}{a+b}+\left( \frac{c}{a+b}+\frac{a}{b+c}\right)\left( \frac{a}{b+c}+\frac{b}{c+a}\right)\ge 2

Source: InternLM Lean-Workbook, record lean_workbook_772 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_772 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b / (c + a) + c / (a + b) + ((c / (a + b) + a / (b + c)) * (a / (b + c) + b / (c + a)))) ≥ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_772; Apache-2.0

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