A quadratic excess bounds a product of shifted variables

Proved

WorkbookSource.plus_36410

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

lean-workbooksource-checked

given $a,b,c,d \ge 0$ such that $a+b+c+d=4$ .Prove that $a^2+b^2+c^2+d^2-4 \ge 4(a-1)(b-1)(c-1)(d-1)$

Source: InternLM Lean-Workbook, record lean_workbook_plus_36410 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_36410 (a b c d : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (hd : d ≥ 0) (hab : a + b + c + d = 4) : a^2 + b^2 + c^2 + d^2 - 4 ≥ 4 * (a - 1) * (b - 1) * (c - 1) * (d - 1)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_36410; Apache-2.0

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