A cubic sum bound on the unit simplex

Proved

WorkbookSource.plus_19763

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

lean-workbooksource-checked

Let $a,b,c\ge 0$ such that $a + b + c = 1$ . Prove that $a^3 + b^3 + c^3\le 3(1 + abc)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_19763 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_19763 (a b c : ℝ) (ha : a ≥ 0 ∧ b ≥ 0 ∧ c ≥ 0 ∧ a + b + c = 1) :
  a^3 + b^3 + c^3 ≤ 3 * (1 + a * b * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_19763; Apache-2.0

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