A cyclic quartic inequality at positive sum three

Proved

WorkbookSource.plus_35064

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

lean-workbooksource-checked

prove that $\sum_{\text{cyc}} \left(x^4-8x^3y+18x^2y^2+xy^3-12xyz^2 \right) \ge 0$ given $x,y,z>0$ and $x+y+z=3$

Source: InternLM Lean-Workbook, record lean_workbook_plus_35064 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_35064 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (x ^ 4 - 8 * x ^ 3 * y + 18 * x ^ 2 * y ^ 2 + x * y ^ 3 - 12 * x * y * z ^ 2) + (y ^ 4 - 8 * y ^ 3 * z + 18 * y ^ 2 * z ^ 2 + y * z ^ 3 - 12 * y * z * x ^ 2) + (z ^ 4 - 8 * z ^ 3 * x + 18 * z ^ 2 * x ^ 2 + z * x ^ 3 - 12 * z * x * y ^ 2) ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_35064; Apache-2.0

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