Prove2Me

A cyclic quintic nonnegativity inequality

Proved
WorkbookSource.base_14032

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

lean-workbooksource-checked

Let x,y,z≥0x,y,z\geq 0 ,prove that: −x(1+x)(1+z)(x+y)(y+z)−y(1+x)(y+1)(y+z)(z+x)−z(1+z)(x+y)(y+1)(z+x)+1/2(1+x)2(y+z)2(x+y)+1/2(y+1)2(z+x)2(y+z)+1/2(1+z)2(x+y)2(z+x)≥0-x(1+x)(1+z)(x+y)(y+z)-y(1+x)(y+1)(y+z)(z+x)-z(1+z)(x+y)(y+1)(z+x)+1/2(1+x)^2(y+z)^2(x+y)+1/2(y+1)^2(z+x)^2(y+z)+1/2(1+z)^2(x+y)^2(z+x)\geq 0

Source: InternLM Lean-Workbook, record lean_workbook_14032 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_14032 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : -x * (1 + x) * (1 + z) * (x + y) * (y + z) - y * (1 + x) * (y + 1) * (y + z) * (z + x) - z * (1 + z) * (x + y) * (y + 1) * (z + x) + 1 / 2 * (1 + x) ^ 2 * (y + z) ^ 2 * (x + y) + 1 / 2 * (y + 1) ^ 2 * (z + x) ^ 2 * (y + z) + 1 / 2 * (1 + z) ^ 2 * (x + y) ^ 2 * (z + x) ≥ 0  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14032; Apache-2.0

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