A squared-distance inequality under two product constraints

Proved

WorkbookSource.plus_9080

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

lean-workbooksource-checked

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

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

Source: InternLM Lean-Workbook, record lean_workbook_plus_9080 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_9080 (a b c d : ℝ) (h : a * b = 1) (h' : c ^ 2 + d ^ 2 = 1) :  (a - c) ^ 2 + (b - d) ^ 2 + 2 * a * d + 2 * b * c ≥ 1   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_9080; Apache-2.0

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