A shifted cyclic ratio sum bounds a normalized total

Proved

WorkbookSource.plus_47518

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

lean-workbooksource-checked

Prove that $\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}\ge \frac{3(a+b+c)}{3+a+b+c}$ given $a,b,c$ are positive reals.

Source: InternLM Lean-Workbook, record lean_workbook_plus_47518 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_47518 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + 1) + b / (c + 1) + c / (a + 1)) ≥ 3 * (a + b + c) / (3 + a + b + c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_47518; Apache-2.0

View graph