A squared pairwise-sum bound with a cubic product

Proved

WorkbookSource.base_56296

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

lean-workbooksource-checked

Prove that $2+\frac{1}{3}(xy+zx+yz)^2 \geq xy+zx+yz+2xyz$ with $x,y,z\geq 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_56296 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_56296 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : 2 + (1 / 3) * (x * y + y * z + z * x) ^ 2 ≥ x * y + y * z + z * x + 2 * x * y * z  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_56296; Apache-2.0

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