A four-variable cyclic quadratic difference ratio sum is nonnegative

Proved

WorkbookSource.base_31243

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

lean-workbooksource-checked

Let $a,b,c,d>0,$ prove that:

$\frac{b(-d+2b-c)}{b+c+d}+\frac{c(-d+2c-a)}{c+d+a}+\frac{d(2d-b-a)}{d+a+b}+\frac{a(2a-b-c)}{a+b+c}\geq 0.$

Source: InternLM Lean-Workbook, record lean_workbook_31243 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_31243 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (b * (-d + 2 * b - c) / (b + c + d) + c * (-d + 2 * c - a) / (c + d + a) + d * (2 * d - b - a) / (d + a + b) + a * (2 * a - b - c) / (a + b + c)) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31243; Apache-2.0

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