A cyclic pair-sum product ratio bounds an alternating reciprocal sum

Proved

WorkbookSource.base_30696

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

lean-workbooksource-checked

Let $a,b,c,d>0$ ,prove that:

$\frac{(a+b)(b+c)(c+d)(d+a)}{(a+b+c+d)abcd} \geq 4(\frac{1}{c+a}+\frac{1}{b+d})$ BQ

Source: InternLM Lean-Workbook, record lean_workbook_30696 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_30696 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a + b) * (b + c) * (c + d) * (d + a) / ((a + b + c + d) * a * b * c * d) ≥ 4 * (1 / (c + a) + 1 / (b + d))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_30696; Apache-2.0

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