Prove2Me

A pair and triple-product ratio inequality bounds nine times the product

Proved
WorkbookSource.plus_70121

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

lean-workbooksource-checked

If a,b,c,da,b,c,d are positive real numbers, then

(ab+bc+cd+da+ac+bd)(abc+bcd+cda+dab)a+b+c+d+\frac{(ab+bc+cd+da+ac+bd)(abc+bcd+cda+dab)}{a+b+c+d}+ +3(abc+bcd+cda+dab)2(a+b+c+d)2≥9abcd.+\frac{3(abc+bcd+cda+dab)^2}{(a+b+c+d)^2}\ge 9 abcd.

It's just ∑sym(a3b2c+a2b2c2−a3bcd−a2b2cd)≥0,\sum\limits_{sym}(a^3b^2c+a^2b^2c^2-a^3bcd-a^2b^2cd)\geq0, which is true by Muirhead.

Source: InternLM Lean-Workbook, record lean_workbook_plus_70121 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_70121 {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 * b * c + b * c * d + c * d * a + d * a * b) / (a + b + c + d) + 3 * (a * b * c + b * c * d + c * d * a + d * a * b) ^ 2 / (a + b + c + d) ^ 2 ≥ 9 * a * b * c * d   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_70121; Apache-2.0

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