Prove2Me

A four-variable cubic ratio inequality with a product of signed triple sums

Proved
WorkbookSource.base_258

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

lean-workbooksource-checked

For a,b,c,d>0a, b, c, d > 0 real numbers, prove: ab+ac+ad+bc+bd+cd+a3+b3+c3+d3a+b+c+d≥a2+b2+c2+d2+3(a+b+c−d)(a+b+d−c)(a+c+d−b)(b+c+d−a)(a+b+c+d)2  ;ab+ac+ad+bc+bd+cd +\frac{a^3+b^3+c^3+d^3}{a+b+c+d} \ge a^2+b^2+c^2+d^2+\frac{3(a+b+c-d)(a+b+d-c)(a+c+d-b)(b+c+d-a)}{(a+b+c+d)^2} \ \ ;

Source: InternLM Lean-Workbook, record lean_workbook_258 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_258 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : a * b + a * c + a * d + b * c + b * d + c * d + (a ^ 3 + b ^ 3 + c ^ 3 + d ^ 3) / (a + b + c + d) ≥ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 + 3 * (a + b + c - d) * (a + b + d - c) * (a + c + d - b) * (b + c + d - a) / (a + b + c + d) ^ 2  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_258; Apache-2.0

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