A cubic sum with a triple-product correction bounds the linear sum

Proved

WorkbookSource.base_38506

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

lean-workbooksource-checked

Let $a, b,c\geq 0.$ Prove that

a^3+b^3+c^3+abc+1 \geq a+b+c

Source: InternLM Lean-Workbook, record lean_workbook_38506 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_38506 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) : a^3 + b^3 + c^3 + a * b * c + 1 ≥ a + b + c  :=  by sorry

Source