A refined cubic inequality at fixed sum three

Proved

WorkbookSource.plus_54627

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

lean-workbooksource-checked

Prove that $a^2+b^2+c^2+3abc\ge 2(ab+bc+ca)+\frac{1}{6}\cdot abc(1-abc)$ given $a,b,c>0$ and $a+b+c=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_54627 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_54627 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : a^2 + b^2 + c^2 + 3 * a * b * c ≥ 2 * (a * b + b * c + c * a) + (1 / 6) * a * b * c * (1 - a * b * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_54627; Apache-2.0

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