A four-variable quartic inequality with four linear factors

Proved

WorkbookSource.base_36270

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

lean-workbooksource-checked

Let $a, b, c, d$ be real positive numbers. Prove: $2(a^4+b^4+c^4+d^4) + (a+b+c-d)(a+b+d-c)(a+c+d-b)(b+c+d-a) \ge 8abcd+ (a+b+c+d)(abc+bcd+cda+dab) \ \ ;$

Source: InternLM Lean-Workbook, record lean_workbook_36270 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36270 (a b c d : ℝ) : 2 * (a ^ 4 + b ^ 4 + c ^ 4 + d ^ 4) + (a + b + c - d) * (a + b + d - c) * (a + c + d - b) * (b + c + d - a) ≥ 8 * a * b * c * d + (a + b + c + d) * (a * b * c + b * c * d + c * d * a + d * a * b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36270; Apache-2.0

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