A distance bound between constrained pairs

Proved

WorkbookSource.base_1776

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

lean-workbooksource-checked

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

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

Source: InternLM Lean-Workbook, record lean_workbook_1776 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1776 (a b c d : ℝ) (h1 : a^2 + b^2 = 1/2) (h2 : c * d = 1) : (a - d)^2 + (b - c)^2 ≥ 1/2  :=  by sorry

Source