A product bound for three shifted quadratic factors

Proved

WorkbookSource.base_32699

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

lean-workbooksource-checked

Prove that, for any real numbers $x,y, z$ , $3(x^2-x+1)(y^2-y+1)(z^2-z+1)\geq (xyz)^2+xyz+1$ .

Source: InternLM Lean-Workbook, record lean_workbook_32699 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_32699 (x y z : ℝ) : 3 * (x ^ 2 - x + 1) * (y ^ 2 - y + 1) * (z ^ 2 - z + 1) ≥ (x*y*z) ^ 2 + x*y*z + 1  :=  by sorry

Source