A symmetric quartic sum is at most two at fixed total two

Proved

WorkbookSource.plus_37409

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

lean-workbooksource-checked

Let $a,b,c\geq 0$ such that $a+b+c=2$ . Prove that: $\sum ab(a^{2}+b^{2})\leq 2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_37409 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_37409 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 2) : a * b * (a ^ 2 + b ^ 2) + b * c * (b ^ 2 + c ^ 2) + c * a * (c ^ 2 + a ^ 2) ≤ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_37409; Apache-2.0

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