A reverse product inequality under a disk constraint

Proved

WorkbookSource.base_5933

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

lean-workbooksource-checked

Prove that: $(ac+bd-1)^{2}\geq (a^{2}+b^{2}-1)(c^{2}+d^{2}-1)$ where $a;b$ are real numbers satisfying $a^{2}+b^{2}\leq 1$

Source: InternLM Lean-Workbook, record lean_workbook_5933 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5933 (a b c d : ℝ) (h : a^2 + b^2 ≤ 1) :
  (a * c + b * d - 1)^2 ≥ (a^2 + b^2 - 1) * (c^2 + d^2 - 1)  :=  by sorry

Source