A squared quadratic sum bounds a cubic triple-product term

Proved

WorkbookSource.base_40907

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

lean-workbooksource-checked

Prove that: $27abc(a^{3}+b^{3}+c^{3}) \leq (a^2+b^2+c^2)^2(a+b+c)^{2}$ where $a,b,c \ge 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_40907 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_40907 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) : 27 * a * b * c * (a ^ 3 + b ^ 3 + c ^ 3) ≤ (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 * (a + b + c) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_40907; Apache-2.0

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