A four-variable quartic difference inequality

Proved

WorkbookSource.base_56570

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

lean-workbooksource-checked

Let $a,b,c,d$ be positive numbers such that $a+b+c+d=2$ . Prove that $a^4+b^4+c^4+d^4-4abcd \ge 2(a-b)(c-d)$ .

Source: InternLM Lean-Workbook, record lean_workbook_56570 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_56570 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (hab : a + b + c + d = 2) : a^4 + b^4 + c^4 + d^4 - 4 * a * b * c * d ≥ 2 * (a - b) * (c - d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_56570; Apache-2.0

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