A six-variable quadratic-product inequality with three nonnegative variables

Proved

WorkbookSource.base_41597

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

lean-workbooksource-checked

$(a^2+b^2+c^2)(x^2+y^2+z^2+xy+xz+yz)\geq2(x+y+z)(xbc+yac+zab)$ is also true for all non-negatives $x$ , $y$ and $z$ and all reals $a$ , $b$ and $c$ .

Source: InternLM Lean-Workbook, record lean_workbook_41597 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_41597 (a b c x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : (a^2 + b^2 + c^2) * (x^2 + y^2 + z^2 + x*y + x*z + y*z) ≥ 2 * (x + y + z) * (x*b*c + y*a*c + z*a*b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_41597; Apache-2.0

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