A squared cubic expression bounds symmetric quartic products

Proved

WorkbookSource.plus_44728

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

lean-workbooksource-checked

Prove that if a,b,c≥0,then $(a^3+b^3+c^3+3abc)^2$ ≥ $2(ab+bc+ca)(ab(a^2+b^2)+bc(b^2+c^2)+ca(c^2+a^2))$

Source: InternLM Lean-Workbook, record lean_workbook_plus_44728 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_44728 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) : (a ^ 3 + b ^ 3 + c ^ 3 + 3 * a * b * c) ^ 2 ≥ 2 * (a * b + b * c + c * a) * (a * b * (a ^ 2 + b ^ 2) + b * c * (b ^ 2 + c ^ 2) + c * a * (c ^ 2 + a ^ 2))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_44728; Apache-2.0

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