A quadratic norm bound under two linked relations

Proved

WorkbookSource.plus_4648

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

lean-workbooksource-checked

Let $a,b,c$ are real numbers such that $a^2+2b=-2$ and $b^2+4c=2.$ Prove that

a^2+8c^2\geq \frac{1}{2}

Source: InternLM Lean-Workbook, record lean_workbook_plus_4648 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_4648 (a b c : ℝ) (ha2 : a^2 + 2 * b = -2) (hb2 : b^2 + 4 * c = 2) : a^2 + 8 * c^2 ≥ 1 / 2   :=  by sorry

Source