A cubic upper bound under a sum and norm relation

Proved

WorkbookSource.plus_10275

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

lean-workbooksource-checked

If $a,b,c>0$ and $a+b+c=a^2+b^2+c^2$ , prove that $a^3+b^3+c^3+3(ab+bc+ca)\leq 12$

Source: InternLM Lean-Workbook, record lean_workbook_plus_10275 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_10275 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = a^2 + b^2 + c^2) : a^3 + b^3 + c^3 + 3 * (a * b + b * c + c * a) ≤ 12   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_10275; Apache-2.0

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