A mixed quadratic-cubic lower bound

Proved

WorkbookSource.plus_75744

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

lean-workbooksource-checked

If $a,b,c > 0$ and $ab + bc + ca = 3$ , then

$2(a^2 + b^2 + c^2) + 3abc\geq a + b + c + 6$

Source: InternLM Lean-Workbook, record lean_workbook_plus_75744 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75744 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a * b + b * c + c * a = 3) : 2 * (a ^ 2 + b ^ 2 + c ^ 2) + 3 * a * b * c ≥ a + b + c + 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_75744; Apache-2.0

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