Two shifted pair-sum reciprocals total at most one half at unit product

Proved

WorkbookSource.plus_16145

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

lean-workbooksource-checked

Let $a,b,c,d>0$ and $abcd=1.$ Prove that

\frac{1}{a+b+2}+\frac{1}{c+d+2} \le \frac{1}{2}

Source: InternLM Lean-Workbook, record lean_workbook_plus_16145 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_16145 (a b c d : ℝ) (hab : 0 < a) (hbc : 0 < b) (hcd : 0 < c) (hda : 0 < d) (habcd : a * b * c * d = 1) : 1 / (a + b + 2) + 1 / (c + d + 2) ≤ 1 / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_16145; Apache-2.0

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