A mixed bilinear inequality with equal sums

Proved

WorkbookSource.plus_28132

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

lean-workbooksource-checked

Prove that: \begin{align*}ax+by+cz+\frac{x^2+y^2+z^2- xy-yz-zx}{3} \geqslant ab+bc+ca. \end{align*} Given: $a,b,c,x,y,z \in \mathbb{R}$ and $a+b+c=x+y+z$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_28132 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_28132 (a b c x y z : ℝ) (ha : a + b + c = x + y + z) : a * x + b * y + c * z + (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) / 3 ≥ a * b + b * c + c * a   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_28132; Apache-2.0

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