A cubic-pairwise lower bound at unit sum

Proved

WorkbookSource.base_26775

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

lean-workbooksource-checked

Let $a,b,c > 0, a+b+c = 1$ . Prove that $15(a^{3}+b^{3}+c^{3}+ab+bc+ca)+9abc \ge 7$ .

Source: InternLM Lean-Workbook, record lean_workbook_26775 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26775 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (habc : a + b + c = 1) :  15 * (a ^ 3 + b ^ 3 + c ^ 3 + a * b + b * c + c * a) + 9 * a * b * c ≥ 7  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26775; Apache-2.0

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