A cyclic weighted-square lower bound

Proved

WorkbookSource.plus_2356

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

lean-workbooksource-checked

If $a,b,c$ are positive numbers and $a+b+c=1$ , prove: $\sum_{cyc}a(2a+b+1)^2\ge\ 4$

Source: InternLM Lean-Workbook, record lean_workbook_plus_2356 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_2356 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 1) : a * (2 * a + b + 1) ^ 2 + b * (2 * b + c + 1) ^ 2 + c * (2 * c + a + 1) ^ 2 ≥ 4   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_2356; Apache-2.0

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