A squared-norm bound at unit positive sum

Proved

WorkbookSource.base_53090

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

lean-workbooksource-checked

If $a,b,c>0$ and $a+b+c=1$ , prove that $a^2+b^2+c^2+1\ge 4(ab+bc+ca)$ .

Source: InternLM Lean-Workbook, record lean_workbook_53090 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53090 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : a^2 + b^2 + c^2 + 1 ≥ 4 * (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53090; Apache-2.0

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