A mixed cubic bound at sum three

Proved

WorkbookSource.base_14228

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

lean-workbooksource-checked

Give $a,b,c$ be non-negative real numbers with sum 3.Prove that $\frac {ab + bc + ca}3 + 3\ge abc + a^2\frac {b + c}2 + b^2\frac {c + a}2 + c^2\frac {a + b}2$

Source: InternLM Lean-Workbook, record lean_workbook_14228 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_14228 (a b c: ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a + b + c = 3):  (a * b + b * c + c * a) / 3 + 3 ≥ a * b * c + a ^ 2 * (b + c) / 2 + b ^ 2 * (c + a) / 2 + c ^ 2 * (a + b) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14228; Apache-2.0

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