A product inequality and its difference factor

Proved

WorkbookSource.problem_33749

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

For real numbers $a,b,c,d$ ,

(a+b)(c+d)\le 2(ac+bd)\quad\Longleftrightarrow\quad(a-b)(c-d)\ge0.

Source: InternLM Lean-Workbook, record lean_workbook_33749 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_33749  (a b c d : ℝ) :
  (a + b) * (c + d) ≤ 2 * (a * c + b * d) ↔ (a - b) * (c - d) ≥ 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33749; Apache-2.0

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