A four-variable squared-total inequality with a quartic ratio correction

Proved

WorkbookSource.base_44682

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

lean-workbooksource-checked

Prove that for positive numbers $a, b, c, d$ , the following inequality holds: $\frac{1}{4}(a+b+c+d)^2 \geq ac + bd + \frac{abc^2 + cbd^2 + a^2cd + b^2ad}{ac + bd}$ .

Source: InternLM Lean-Workbook, record lean_workbook_44682 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_44682 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (1 / 4) * (a + b + c + d) ^ 2 ≥ a * c + b * d + (a * b * c ^ 2 + b * c * d ^ 2 + a ^ 2 * c * d + b ^ 2 * a * d) / (a * c + b * d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_44682; Apache-2.0

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