A norm bound under two linked quadratic constraints

Proved

WorkbookSource.plus_79287

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

lean-workbooksource-checked

Let $a,b,c$ are real numbers such that $a^2+2b=1$ and $b^2+4c=-1.$ Prove that

a^2+b^2+c^2\geq \frac{89}{256}

Source: InternLM Lean-Workbook, record lean_workbook_plus_79287 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_79287 (a b c : ℝ) (ha : a^2 + 2 * b = 1) (hb : b^2 + 4 * c = -1) : 89 / 256 ≤ a^2 + b^2 + c^2   :=  by sorry

Source