Prove2Me

A pair-sum reciprocal product inequality at fixed total four

Proved
WorkbookSource.base_22

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

lean-workbooksource-checked

Let a,b,c,d>0,a+b+c+d=4a,b,c,d>0,a+b+c+d=4 ,prove that: 1(a+b)(c+d)+1(b+c)(d+a)+1(c+a)(b+d)≥34\frac{1}{(a+b)(c+d)}+\frac{1}{(b+c)(d+a)}+\frac{1}{(c+a)(b+d)} \geq \frac{3}{4}

Source: InternLM Lean-Workbook, record lean_workbook_22 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_22 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (habc : a + b + c + d = 4) : 1 / (a + b) / (c + d) + 1 / (b + c) / (d + a) + 1 / (c + a) / (b + d) ≥ 3 / 4  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_22; Apache-2.0

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