A mixed product and cubic bound at unit sum

Proved

WorkbookSource.base_23431

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

lean-workbooksource-checked

Let $a, b, c$ be positive real numbers such that $a+b+c=1$ . Prove that $(a+b)(b+c)(c+a)(a^3+b^3+c^3+\frac{1}{3}) \le \frac{4}{27}$

Source: InternLM Lean-Workbook, record lean_workbook_23431 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23431 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) :  (a + b) * (b + c) * (c + a) * (a ^ 3 + b ^ 3 + c ^ 3 + 1 / 3) ≤ 4 / 27  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23431; Apache-2.0

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