A product of mixed quadratics bounds a squared symmetric cubic sum

Proved

WorkbookSource.plus_56151

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

lean-workbooksource-checked

if $a,b,c \geq 0$ ,prove that $4(a^{2}+ab+c^{2})(b^{2}+bc+a^{2})(c^{2}+ca+b^{2})\ge3(a^2b+a^2c+b^2a+b^2c+c^2a+c^2b)^2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_56151 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_56151 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) : 4 * (a ^ 2 + a * b + c ^ 2) * (b ^ 2 + b * c + a ^ 2) * (c ^ 2 + c * a + b ^ 2) ≥ 3 * (a ^ 2 * b + a ^ 2 * c + b ^ 2 * a + b ^ 2 * c + c ^ 2 * a + c ^ 2 * b) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_56151; Apache-2.0

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