A six-term pair-sum reciprocal lower bound

Proved

WorkbookSource.base_1057

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

lean-workbooksource-checked

Let $a,b,c,d>0$ ,prove ${\frac {1}{a+b}}+{\frac {1}{b+c}}+{\frac {1}{c+d}}+{\frac {1}{d+a}}+{\frac {1}{c+a}}+{\frac {1}{b+d}}\geq \frac{9}{2}\,{\frac {a+b+c+d}{ab+bc+cd+ad+ac+bd}}$

Source: InternLM Lean-Workbook, record lean_workbook_1057 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1057 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : 1 / (a + b) + 1 / (b + c) + 1 / (c + d) + 1 / (d + a) + 1 / (c + a) + 1 / (b + d) ≥ 9 / 2 * (a + b + c + d) / (a * b + b * c + c * d + d * a + a * c + b * d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1057; Apache-2.0

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