A four-variable product and pairwise-sum bound

Proved

WorkbookSource.plus_39693

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

lean-workbooksource-checked

Let $a,b,c,d\geq 0$ ,prove that: $(a+c+d)(d+a+b)(a+b+c)(b+c+d)-\\frac{27}{32}(a+b+c+d)^2(ac+bd)-\\frac{27}{32}(a+b+c+d)^2(cd+ab)-\\frac{27}{32}(a+b+c+d)^2(bc+ad)\\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_plus_39693 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_39693 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) : (a + c + d) * (d + a + b) * (a + b + c) * (b + c + d) - (27 / 32) * (a + b + c + d) ^ 2 * (a * c + b * d) - (27 / 32) * (a + b + c + d) ^ 2 * (c * d + a * b) - (27 / 32) * (a + b + c + d) ^ 2 * (b * c + a * d) ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_39693; Apache-2.0

View graph