A triple-product sum has a weighted pair-product bound at fixed total four

Proved

WorkbookSource.plus_28550

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

lean-workbooksource-checked

Prove that for $a>0,b>0,c>0,d>0$ and $a+b+c+d=4$ is $abc+abd+bcd+acd\le ac+bd+\frac{ad+ab+bc+cd}{2}$ . Sonnhard.

Source: InternLM Lean-Workbook, record lean_workbook_plus_28550 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_28550 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (hab : a + b + c + d = 4) : a * b * c + a * b * d + b * c * d + a * c * d ≤ a * c + b * d + (a * d + a * b + b * c + c * d) / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_28550; Apache-2.0

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