A squared-distance lower bound under two quadratic constraints

Proved

WorkbookSource.base_7402

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

lean-workbooksource-checked

Let $a, b, c, d$ be real numbers such that $a^2+b^2 = 1$ and $cd = 2$ . Prove that

(a - d)^2 + (b - c)^2\geq 1

Source: InternLM Lean-Workbook, record lean_workbook_7402 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7402 (a b c d : ℝ) (ha : a^2 + b^2 = 1) (hb : c * d = 2) : (a - d)^2 + (b - c)^2 ≥ 1  :=  by sorry

Source