A cubic lower bound under half-unit caps

Proved

WorkbookSource.plus_51590

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

lean-workbooksource-checked

Let $a,b,c\leq \frac{1}{2}$ be nonnegative real numbers such that: $a+b+c=1$ . Prove that: $a^{2}+b^{2}+c^{2}+9abc\geq 2(ab+bc+ca)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_51590 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_51590 (a b c : ℝ) (ha : a ≤ 1 / 2) (hb : b ≤ 1 / 2) (hc : c ≤ 1 / 2) (habc : a + b + c = 1) : a^2 + b^2 + c^2 + 9 * a * b * c ≥ 2 * (a * b + b * c + c * a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_51590; Apache-2.0

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