A quadratic lower bound for three positive variables

Proved

WorkbookSource.plus_68450

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

lean-workbooksource-checked

Let $a,b,c > 0$ , prove that $3a^2 + 2b^2 + c^2 + 4ab + 2c(a + b) -2(6a + 5b + 3c) + 14 \geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_plus_68450 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_68450 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 3 * a ^ 2 + 2 * b ^ 2 + c ^ 2 + 4 * a * b + 2 * c * (a + b) - 2 * (6 * a + 5 * b + 3 * c) + 14 ≥ 0   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_68450; Apache-2.0

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