A quartic bound on a cyclic antisymmetric sum

Proved

WorkbookSource.base_1581

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

lean-workbooksource-checked

Prove that

$3\sum_{cyc}x^4+7\left(\sum_{cyc}x^2y^2-\sum_{cyc}x^2yz\right)\geq 9\left(\sum_{cyc}x^3y-\sum_{cyc}xy^3\right)$

for $x,y,z\in\mathbb{R}$ .

Source: InternLM Lean-Workbook, record lean_workbook_1581 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1581 (x y z : ℝ) : 3 * (x ^ 4 + y ^ 4 + z ^ 4) + 7 * (x ^ 2 * y ^ 2 + y ^ 2 * z ^ 2 + z ^ 2 * x ^ 2 - (x ^ 2 * y * z + y ^ 2 * z * x + z ^ 2 * x * y)) ≥ 9 * (x ^ 3 * y + y ^ 3 * z + z ^ 3 * x - (x * y ^ 3 + y * z ^ 3 + z * x ^ 3))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1581; Apache-2.0

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