A sixth-degree cyclic inequality with a Vandermonde square

Proved

WorkbookSource.base_8651

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $x$ , $y$ and $z$ be real numbers. Prove that: ${({x^2} + {y^2} + {z^2})^3} \ge 8({x^3}{y^3} + {y^3}{z^3} + {x^3}{z^3})+3x^2y^2z^2+3(x-y)^2(x-z)^2(y-z)^2$

Source: InternLM Lean-Workbook, record lean_workbook_8651 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8651 (x y z: ℝ) :  (x^2 + y^2 + z^2)^3 ≥ 8 * (x^3 * y^3 + y^3 * z^3 + x^3 * z^3) + 3 * x^2 * y^2 * z^2 + 3 * (x-y)^2 * (x-z)^2 * (y-z)^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8651; Apache-2.0

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