A cubic sum bound on the unit sphere

Proved

WorkbookSource.base_49027

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

lean-workbooksource-checked

Let $a,b,c$ are real numbers such that $a^2+b^2+c^2=1$ . Prove that $a^3+b^3+c^3 \le 1+3abc$ .

Source: InternLM Lean-Workbook, record lean_workbook_49027 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_49027 (a b c : ℝ) (h : a^2 + b^2 + c^2 = 1) : a^3 + b^3 + c^3 ≤ 1 + 3 * a * b * c  :=  by sorry

Source