A shifted quadratic ratio with a product reciprocal lower bound

Proved

WorkbookSource.base_22582

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

lean-workbooksource-checked

Let $a,b>0$ . Prove that

\frac{ a^2+ b^2+1}{(a+1)(b+1)}+\frac{2ab}{ ab( ab+3)} \geq \frac{5}{4}

Source: InternLM Lean-Workbook, record lean_workbook_22582 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_22582 (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : (a^2 + b^2 + 1) / (a + 1) / (b + 1) + 2 * a * b / (a * b * (a * b + 3)) ≥ 5 / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_22582; Apache-2.0

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