A quadratic and cubic lower bound at sum three

Proved

WorkbookSource.base_2562

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

lean-workbooksource-checked

Let $x;y;z \geq 0$ such that $x+y+z=3$ Prove: $x^2+y^2+z^2+3xyz \geq \dfrac{9}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_2562 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2562 (x y z : ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hz : z ≥ 0) (h : x + y + z = 3) : x ^ 2 + y ^ 2 + z ^ 2 + 3 * x * y * z ≥ 9 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2562; Apache-2.0

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