A weighted four-variable cyclic difference ratio sum is nonnegative

Proved

WorkbookSource.base_36191

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

lean-workbooksource-checked

Let $a,b,c,d>0$ ,prove that: $\frac{a-b}{a+2b+c}+\frac{b-c}{b+2c+d}+\frac{c-d}{c+2d+a}+\frac{d-a}{d+2a+b}\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_36191 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36191 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a - b) / (a + 2 * b + c) + (b - c) / (b + 2 * c + d) + (c - d) / (c + 2 * d + a) + (d - a) / (d + 2 * a + b) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36191; Apache-2.0

View graph