A cubic lower bound with a quadratic correction

Proved

WorkbookSource.base_19984

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

lean-workbooksource-checked

Prove that for positives $x$ , $y$ , and $z$ such that $x+y+z=3$ , the inequality $2(x^3+y^3+z^3)\ge x^2+y^2+z^2+3$ holds.

Source: InternLM Lean-Workbook, record lean_workbook_19984 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_19984 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : 2 * (x ^ 3 + y ^ 3 + z ^ 3) ≥ x ^ 2 + y ^ 2 + z ^ 2 + 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_19984; Apache-2.0

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