A squared cubic sum bounds two pairwise sums

Proved

WorkbookSource.plus_21480

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

lean-workbooksource-checked

Prove that for $a, b, c \geq 0$ , $(a^3+b^3+c^3+3abc)^2 \geq 4(ab+bc+ca)(a^2b^2+b^2c^2+c^2a^2)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_21480 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_21480 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) : (a^3 + b^3 + c^3 + 3 * a * b * c)^2 ≥ 4 * (a * b + b * c + c * a) * (a^2 * b^2 + b^2 * c^2 + c^2 * a^2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_21480; Apache-2.0

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