A cubic and pairwise-product lower bound

Proved

WorkbookSource.plus_51848

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

lean-workbooksource-checked

Let a,b,c>0 such that $a+b+c=3$ .Prove that $a^3+b^3+c^3+ab+ac+bc$ ≥ $6$

Source: InternLM Lean-Workbook, record lean_workbook_plus_51848 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_51848 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a ^ 3 + b ^ 3 + c ^ 3 + a * b + b * c + c * a ≥ 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_51848; Apache-2.0

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