A cubic pair-product bound at fixed sum two

Proved

WorkbookSource.plus_2144

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

lean-workbooksource-checked

Let $x,y,z \geq 0$ such that $x + y + z = 2$ Prove that $x^3y^3 + y^3z^3 + z^3x^3 + 2xyz \leq 1$

Source: InternLM Lean-Workbook, record lean_workbook_plus_2144 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_2144 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) (h : x + y + z = 2) : x^3*y^3 + y^3*z^3 + z^3*x^3 + 2*x*y*z ≤ 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_2144; Apache-2.0

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