Two squared cubic sums with a symmetric correction

Proved

WorkbookSource.plus_39713

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

lean-workbooksource-checked

Prove that for real numbers $x, y, z$ , the following inequality holds: $(xy^2 + z^2y + zx^2)^2 + (x^2y + y^2z + xz^2)^2 + 6(xy + zx + yz)(x + y + z)xyz \geq 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_39713 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_39713 (x y z : ℝ) : (x * y ^ 2 + z ^ 2 * y + z * x ^ 2) ^ 2 + (x ^ 2 * y + y ^ 2 * z + x * z ^ 2) ^ 2 + 6 * (x * y + z * x + y * z) * (x + y + z) * x * y * z ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_39713; Apache-2.0

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