A determinant square bounded by a product of sums and squared differences

Proved

WorkbookSource.plus_4889

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

lean-workbooksource-checked

Let $a,b,c,d \geq 0$ . Prove that

(ad-bc)^2 \leq (a+b)(c+d)((a-c)^2+(b-d)^2).

Source: InternLM Lean-Workbook, record lean_workbook_plus_4889 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_4889 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) : (a * d - b * c) ^ 2 ≤ (a + b) * (c + d) * ((a - c) ^ 2 + (b - d) ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_4889; Apache-2.0

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