An asymmetrically shifted product bound at fixed sum one

Proved

WorkbookSource.base_24516

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

lean-workbooksource-checked

Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, prove that $(x + 1)(y + 2)(z + 3) \leq 12$ for non-negative real numbers $x, y, z$ with $x + y + z = 1$ .

Source: InternLM Lean-Workbook, record lean_workbook_24516 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_24516 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) (h : x + y + z = 1) : (x + 1) * (y + 2) * (z + 3) ≤ 12  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_24516; Apache-2.0

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