A mixed cubic bound at unit sum

Proved

WorkbookSource.base_16538

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

lean-workbooksource-checked

Let $a,b,c > 0$ be such that $a + b + c = 1$ . Prove the followings: (4) $a^2(b + c) + b^2(c + a) + c^2(a + b) \ge \frac {1}{3} (1 - a^2 - b^2 - c^2)$

Source: InternLM Lean-Workbook, record lean_workbook_16538 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16538 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : a^2 * (b + c) + b^2 * (c + a) + c^2 * (a + b) ≥ 1 / 3 * (1 - a^2 - b^2 - c^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16538; Apache-2.0

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