A four-variable cubic sum with a normalized product correction

Proved

WorkbookSource.base_7055

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

lean-workbooksource-checked

Let $a$ , $b$ , $c$ and $d$ are positive numbers. Prove that: $a^3+b^3+c^3+d^3+\frac{32abcd}{a+b+c+d}\geq3(abc+abd+acd+bcd)$

Source: InternLM Lean-Workbook, record lean_workbook_7055 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7055 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : a^3 + b^3 + c^3 + d^3 + (32 * a * b * c * d) / (a + b + c + d) ≥ 3 * (a * b * c + a * b * d + a * c * d + b * c * d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_7055; Apache-2.0

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