A four-variable product bound for paired products

Proved

WorkbookSource.plus_79000

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

lean-workbooksource-checked

Let $a,b,c,d\geq 0$ ,prove that: $(a+c+d)(d+a+b)(a+b+c)(b+c+d)-\\frac{27}{4}(cd+ab)(bc+ad)-\\frac{27}{4}(ac+bd)(bc+ad)-\\frac{27}{4}(cd+ab)(ac+bd)\\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_plus_79000 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_79000 (a b c d : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (hd : d ≥ 0) : (a + c + d) * (d + a + b) * (a + b + c) * (b + c + d) - 27 / 4 * (c * d + a * b) * (b * c + a * d) - 27 / 4 * (a * c + b * d) * (b * c + a * d) - 27 / 4 * (c * d + a * b) * (a * c + b * d) ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_79000; Apache-2.0

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