An eighth-degree cyclic product inequality for nonnegative variables

Proved

WorkbookCorrected.base_24501

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

corrected-formalizationlean-workbooksource-checked

Prove that for $x,y,z\geq 0$ , $-(xy+xz+yz)^2(xy^3+yz^3+x^3z)+(x+y+z)(x^2+y^2+z^2)(x^2z^3+x^3y^2+y^3z^2)\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_24501 (Apache-2.0). Natural-language proposition preserved; raw Lean transcription corrected: Restore the nonnegative domain assumptions for x, y, z explicitly stated in the natural-language source and omitted by the raw Lean transcription. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.base_24501 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : -(x * y + x * z + y * z) ^ 2 * (x * y ^ 3 + y * z ^ 3 + x ^ 3 * z) + (x + y + z) * (x ^ 2 + y ^ 2 + z ^ 2) * (x ^ 2 * z ^ 3 + x ^ 3 * y ^ 2 + y ^ 3 * z ^ 2) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24501; Apache-2.0

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