A mixed distance and product lower bound

Proved

WorkbookSource.plus_21888

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+ad+bc\geq \frac{27}{20}

Source: InternLM Lean-Workbook, record lean_workbook_plus_21888 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_21888 (a b c d : ℝ) (h : a * b = 1) (h' : c ^ 2 + d ^ 2 = 1) : (a - c) ^ 2 + (b - d) ^ 2 + a * d + b * c ≥ 27 / 20   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_21888; Apache-2.0

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