A symmetric sixth-power inequality with a triple-product correction

Proved

WorkbookSource.base_1818

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

lean-workbooksource-checked

Prove that $4(x^6+y^6+z^6)+27x^2y^2z^2 \ge 12(x^4yz+y^4zx+z^4xy)+(x^3y^3+y^3z^3+z^3x^3)$ where $x, y, z$ are arbitrary positive real numbers.

Source: InternLM Lean-Workbook, record lean_workbook_1818 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1818 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 4 * (x ^ 6 + y ^ 6 + z ^ 6) + 27 * x ^ 2 * y ^ 2 * z ^ 2 ≥ 12 * (x ^ 4 * y * z + y ^ 4 * z * x + z ^ 4 * x * y) + (x ^ 3 * y ^ 3 + y ^ 3 * z ^ 3 + z ^ 3 * x ^ 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1818; Apache-2.0

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