A lower bound for three quadratic factors under normalization

Proved

WorkbookSource.base_3378

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

lean-workbooksource-checked

Let $a,b,c\ge 0$ and $a+b+c=1$ . Prove that $(8-27ab)(8-27bc)(8-27ca) \geq 80$

Source: InternLM Lean-Workbook, record lean_workbook_3378 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_3378 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 1) : (8 - 27 * a * b) * (8 - 27 * b * c) * (8 - 27 * c * a) ≥ 80  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_3378; Apache-2.0

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