A lower bound under two linked quadratic relations

Proved

WorkbookSource.plus_75566

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

lean-workbooksource-checked

Let $a,b,c$ are real numbers such that $a^2+2b=7$ and $b^2+4c=-7.$ Prove that

c^2+6a\geq -14

Source: InternLM Lean-Workbook, record lean_workbook_plus_75566 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75566 (a b c : ℝ) (ha : a^2 + 2 * b = 7) (hb : b^2 + 4 * c = -7) : c^2 + 6 * a ≥ -14   :=  by sorry

Source