Prove2Me

A quartic bound for mixed pairwise products

Proved
WorkbookSource.plus_66952

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

lean-workbooksource-checked

Let x,y,z≥0.x,y,z\geq 0. Prove that

xy(x2+y2)+yz(y2+z2)+zx(z2+x2)≤18(x+y+z)4xy(x^2+y^2)+yz(y^2+z^2)+zx(z^2+x^2)\leq \frac{1}{8}(x+y+z)^4

Source: InternLM Lean-Workbook, record lean_workbook_plus_66952 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_66952 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : (x * y * (x ^ 2 + y ^ 2) + y * z * (y ^ 2 + z ^ 2) + z * x * (z ^ 2 + x ^ 2)) ≤ (1 / 8) * (x + y + z) ^ 4   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_66952; Apache-2.0

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