A product of elementary symmetric sums bounds a five-degree term

Proved

WorkbookSource.plus_23756

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

lean-workbooksource-checked

Prove that for non-negative numbers $a, b, c, d$ , the following inequality holds: $(ab+bc+cd+ad+ac+bd)(acd+abd+abc+bcd) \geq 6(a+b+c+d)abcd$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_23756 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_23756 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) : (a * b + b * c + c * d + d * a + a * c + b * d) * (a * c * d + b * c * d + b * a * c + a * b * d) ≥ 6 * (a + b + c + d) * a * b * c * d   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_23756; Apache-2.0

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