The total times the cubic sum bounds a cyclic product-ratio expression

Proved

WorkbookSource.base_26997

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

lean-workbooksource-checked

Let $a, b, c,d$ be positive real numbers. Prove that

(a+ b+ c+d)(a^3+ b^3+ c^3+d^3)\geq 4abcd\left(\frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}\right)

Source: InternLM Lean-Workbook, record lean_workbook_26997 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26997 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a + b + c + d) * (a ^ 3 + b ^ 3 + c ^ 3 + d ^ 3) ≥ 4 * a * b * c * d * (b / a + c / b + d / c + a / d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26997; Apache-2.0

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