A normalized sixth-degree bound for the pairwise sum

Proved

WorkbookSource.base_10044

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

lean-workbooksource-checked

Let $x,y,z$ be nonnegative and $x + y + z = 1$ . Prove that

$xy + yz + zx\geq 8(x^2 + y^2 + z^2)(x^2y^2 + y^2z^2 + z^2x^2)$

Source: InternLM Lean-Workbook, record lean_workbook_10044 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_10044 (x y z : ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hz : z ≥ 0) (h : x + y + z = 1) : x * y + y * z + z * x ≥ 8 * (x ^ 2 + y ^ 2 + z ^ 2) * (x ^ 2 * y ^ 2 + y ^ 2 * z ^ 2 + z ^ 2 * x ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10044; Apache-2.0

View graph