A cubic inequality for three nonnegative variables

Proved

WorkbookSource.problem_10096

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

If $x, y, z \ge 0$ , prove: $\frac{3}{2} (x+y+z)[3(x+y+z)^2+xy+yz+zx] \ge (3x+y+z)(3y+x+z)(3z+x+y)$

Source: InternLM Lean-Workbook, record lean_workbook_10096 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_10096 (x y z : ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hz : z ≥ 0) : (3 / 2) * (x + y + z) * (3 * (x + y + z) ^ 2 + x * y + x * z + y * z) ≥ (3 * x + y + z) * (3 * y + x + z) * (3 * z + x + y)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10096; Apache-2.0

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