A sixth-degree bound involving a cubed sum and triangle factors

Proved

WorkbookSource.base_16037

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

lean-workbooksource-checked

Let $a,b,c$ be positive reals .Prove that

(a+b+c)^3(-a+b+c)(a-b+c)(a+b-c)\leq27a^2b^2c^2 .

Source: InternLM Lean-Workbook, record lean_workbook_16037 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16037 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b + c) ^ 3 * (-a + b + c) * (a - b + c) * (a + b - c) ≤ 27 * a ^ 2 * b ^ 2 * c ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16037; Apache-2.0

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