A shifted ratio sum bounds a triple-product expression

Proved

WorkbookSource.plus_23017

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{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1} \geq \frac{3(1+abc)}{4}

Source: InternLM Lean-Workbook, record lean_workbook_plus_23017 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_23017 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a + b + c = 3) : (a / (a + 1) + b / (b + 1) + c / (c + 1)) ≥ (3 * (1 + a * b * c)) / 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_23017; Apache-2.0

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