A reciprocal pair-product sum bounds eight divided by two pair sums

Proved

WorkbookSource.base_20401

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

lean-workbooksource-checked

Given that $a$ , $b$ , $c$ , and $d$ are positive numbers, prove that $\frac{1}{ab}+\frac{1}{cd}\geq\frac{8}{(a+b)(c+d)}$ .

Source: InternLM Lean-Workbook, record lean_workbook_20401 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_20401 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (1 / (a * b) + 1 / (c * d)) ≥ 8 / (a + b) / (c + d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_20401; Apache-2.0

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