Prove2Me

A four-term product-of-differences ratio sum is nonnegative

Proved
WorkbookSource.plus_60100

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

lean-workbooksource-checked

Let a,b,c,d>0 a,b,c,d > 0 ,prove that: (a−b)(a−c)(a+b)(a+c)+(−c+b)(−d+b)(b+c)(b+d)+(−d+c)(−a+c)(c+d)(a+c)+(d−a)(−b+d)(a+d)(b+d)≥0. {\frac {\left( a - b \right) \left( a - c \right) }{\left( a + b \right) \left( a + c \right) }} + {\frac {\left( - c + b \right) \left( - d + b \right) }{\left( b + c \right) \left( b + d \right) }} + {\frac { \left( - d + c \right) \left( - a + c \right) }{\left( c + d \right) \left( a + c \right) }} + {\frac {\left( d - a \right) \left( - b + d \right) }{\left( a + d \right) \left( b + d \right) }}\geq 0.

Source: InternLM Lean-Workbook, record lean_workbook_plus_60100 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_60100 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a - b) * (a - c) / (a + b) / (a + c) + (-c + b) * (-d + b) / (b + c) / (b + d) + (-d + c) * (-a + c) / (c + d) / (a + c) + (d - a) * (-b + d) / (a + d) / (b + d) ≥ 0   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_60100; Apache-2.0

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