A quadratic and triple-product bound at unit sum

Proved

WorkbookSource.base_15098

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

lean-workbooksource-checked

Prove that for $a, b, c > 0$ and $a + b + c = 1$ , $a^2 + b^2 + c^2 + 3abc \geq \frac{4}{9}$ using AM-GM.

Source: InternLM Lean-Workbook, record lean_workbook_15098 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15098 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : a^2 + b^2 + c^2 + 3 * a * b * c ≥ 4 / 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15098; Apache-2.0

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