Two equivalent formulations of a cubic mean inequality

Proved

WorkbookSource.problem_8823

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

$4(a^3 + b^3)\geq (a + b)^3\Longleftrightarrow \frac{a^3+b^3}{2}\geq \left(\frac{a+b}{2}\right)^3$

Source: InternLM Lean-Workbook, record lean_workbook_8823 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_8823 (a b : ℝ) : 4 * (a ^ 3 + b ^ 3) ≥ (a + b) ^ 3 ↔ (a ^ 3 + b ^ 3) / 2 ≥ ((a + b) / 2) ^ 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8823; Apache-2.0

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