A squared pair-product sum bounds a triple-product expression

Proved

WorkbookSource.base_984

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

lean-workbooksource-checked

Given $a,b,c\in \mathbb{R}$ and $a+b+c=1$ . Prove that $8abc-8\leqslant ( ab+bc+ac+1)^{2}$

Source: InternLM Lean-Workbook, record lean_workbook_984 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_984 (a b c : ℝ) (h : a + b + c = 1) :
  8 * a * b * c - 8 ≤ (a * b + b * c + c * a + 1)^2  :=  by sorry

Source