A squared total and normalized triple-product inequality in four variables

Proved

WorkbookSource.base_36046

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

lean-workbooksource-checked

For $a, b, c, d > 0$ real numbers prove that: $\frac{9}{16} (a+b+c+d)^2 + \frac{3(abc+bcd+cda+dab)}{a+b+c+d} \ge 2(ab+ac+ad+bc+bd+cd) \ \ ; (2)$

Source: InternLM Lean-Workbook, record lean_workbook_36046 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36046 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (9 / 16) * (a + b + c + d) ^ 2 + (3 * (a * b * c + b * c * d + c * d * a + d * a * b)) / (a + b + c + d) ≥ 2 * (a * b + a * c + a * d + b * c + b * d + c * d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36046; Apache-2.0

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