Prove2Me

An alternating reciprocal sum bounds a ratio of cyclic pair and triple products

Proved
WorkbookSource.base_31687

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

lean-workbooksource-checked

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

1c+a+1b+d≥814(a+b)(b+c)(c+d)(d+a)(a+b+c+d)(a+c+d)(d+a+b)(a+b+c)(b+c+d)\frac{1}{c+a}+\frac{1}{b+d}\geq \frac{81}{4}\frac{(a+b)(b+c)(c+d)(d+a)}{(a+b+c+d)(a+c+d)(d+a+b)(a+b+c)(b+c+d)}

Source: InternLM Lean-Workbook, record lean_workbook_31687 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_31687 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (1 / (c + a) + 1 / (b + d)) ≥ (81 / 4) * (a + b) * (b + c) * (c + d) * (d + a) / ((a + b + c + d) * (a + c + d) * (d + a + b) * (a + b + c) * (b + c + d))  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_31687; Apache-2.0

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