An elementary symmetric inequality on the four-variable simplex

Proved

WorkbookSource.base_33790

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

lean-workbooksource-checked

Let $a,b,c,d$ be nonnegative real numbers such that $a+b+c+d=1$ . Prove that $ab+ac+ad+bc+bd+cd+48abcd\ge 9(abc+abd+acd+bcd).$

Source: InternLM Lean-Workbook, record lean_workbook_33790 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_33790 (a b c d : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) (habc : a + b + c + d = 1) : a * b + a * c + a * d + b * c + b * d + c * d + 48 * a * b * c * d ≥ 9 * (a * b * c + a * b * d + a * c * d + b * c * d)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_33790; Apache-2.0

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