A cubic sum inequality at unit pairwise product sum

Proved

WorkbookSource.plus_7320

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

lean-workbooksource-checked

Prove that $(2(x+y+z)-1)(x+y+z-2)+5xyz\geq 0$ for nonnegative reals $x, y, z$ satisfying $xy+yz+zx=1$

Source: InternLM Lean-Workbook, record lean_workbook_plus_7320 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_7320 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) (hab : x * y + y * z + z * x = 1) : (2 * (x + y + z) - 1) * (x + y + z - 2) + 5 * x * y * z ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_7320; Apache-2.0

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