A cubic correction to a squared-norm bound

Proved

WorkbookSource.base_5012

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

lean-workbooksource-checked

Let $a,b,c > 0$ be such that $a + b + c = 1$ . Prove the followings: (1) $18abc + a^2 + b^2 + c^2 \le 1$

Source: InternLM Lean-Workbook, record lean_workbook_5012 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5012 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : 18 * a * b * c + a ^ 2 + b ^ 2 + c ^ 2 ≤ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5012; Apache-2.0

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