A quadratic lower bound under a determinant relation

Proved

WorkbookSource.base_26207

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

lean-workbooksource-checked

Let $a,b,c,d$ be reals such that $ad-3bc=1 .$ Prove that

a^2+ b^2+3c^2+d^2+ac+ bd\geq 1

Source: InternLM Lean-Workbook, record lean_workbook_26207 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26207 (a b c d : ℝ) (h : a * d - 3 * b * c = 1) :
  a^2 + b^2 + 3 * c^2 + d^2 + a * c + b * d ≥ 1  :=  by sorry

Source