A four-variable cubic difference ratio sum is nonnegative

Proved

WorkbookSource.base_47517

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

lean-workbooksource-checked

4-var: Let $a,b,c,d>0$ ,prove that: ${\frac {{d}^{3}-abc}{a+b+c}}+{\frac {{a}^{3}-bcd}{b+c+d}}+{\frac {{b}^{3}-acd}{a+c+d}}+{\frac {{c}^{3}-abd}{d+a+b}}\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_47517 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_47517 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (d^3 - a * b * c) / (a + b + c) + (a^3 - b * c * d) / (b + c + d) + (b^3 - a * c * d) / (a + c + d) + (c^3 - a * b * d) / (d + a + b) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47517; Apache-2.0

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