A cyclic cubic lower bound at fixed sum three

Proved

WorkbookSource.plus_75038

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

lean-workbooksource-checked

For non-negative reals $x, y, z$ such that $x + y + z = 3$ , prove that: $2(x^2 + y^2 + z^2) + x^2z + xy^2 + yz^2 \ge 2xyz + xy + yz + xz + 4$

Source: InternLM Lean-Workbook, record lean_workbook_plus_75038 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75038 (x y z : ℝ) (hx : x + y + z = 3) (hx0 : 0 ≤ x) (hy0 : 0 ≤ y) (hz0 : 0 ≤ z) : 2 * (x ^ 2 + y ^ 2 + z ^ 2) + x ^ 2 * z + x * y ^ 2 + y * z ^ 2 ≥ 2 * x * y * z + x * y + y * z + x * z + 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_75038; Apache-2.0

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