A cubic and linear lower bound on a sphere

Proved

WorkbookSource.base_52236

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

lean-workbooksource-checked

Let $x,y,z >0$ and $x^2+y^2+z^2=3$ Prove that

x^3+y^3+z^3+3(x+y+z) \geq 9+3xyz

Source: InternLM Lean-Workbook, record lean_workbook_52236 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52236 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x^2 + y^2 + z^2 = 3) :  x^3 + y^3 + z^3 + 3 * (x + y + z) ≥ 9 + 3 * x * y * z  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52236; Apache-2.0

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