A seventh-degree comparison of symmetric product sums

Proved

WorkbookSource.base_43873

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

lean-workbooksource-checked

Prove that for $x,y,z\geq 0$ , $(x+y+z)(x^3y^3+y^3z^3+x^3z^3)-(x^2+y^2+z^2)(xy+xz+yz)xyz\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_43873 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_43873 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : (x + y + z) * (x ^ 3 * y ^ 3 + y ^ 3 * z ^ 3 + x ^ 3 * z ^ 3) - (x ^ 2 + y ^ 2 + z ^ 2) * (x * y + x * z + y * z) * x * y * z ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_43873; Apache-2.0

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