A product of shifted squares bounds a squared pairwise sum

Proved

WorkbookSource.plus_28475

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

lean-workbooksource-checked

Prove that: $(a^{2}+2)(b^{2}+2)(c^{2}+2) \geq \frac{3}{2}(ab+bc+ca)^{2}$ holds for all reals $a$ , $b$ , $c$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_28475 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_28475 (a b c : ℝ) : (a^2 + 2) * (b^2 + 2) * (c^2 + 2) ≥ 3 / 2 * (a * b + b * c + c * a)^2   :=  by sorry

Source