A complementary product bound at unit sum

Proved

WorkbookSource.base_38516

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

lean-workbooksource-checked

Prove $(1-a)(1-b)(1-c) \geq 8abc$ given $a+b+c=1$ and $a, b, c \in \mathbb{R}^{+}$ .

Source: InternLM Lean-Workbook, record lean_workbook_38516 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_38516 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : (1 - a) * (1 - b) * (1 - c) ≥ 8 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_38516; Apache-2.0

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