A four-variable weighted cyclic ratio sum is at least one

Proved

WorkbookSource.base_16402

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

lean-workbooksource-checked

Prove that if ${x_1,x_2,x_3,x_4}\\in(0;+\\infty)$ , then

$\\frac{x_1}{3x_2+x_3}+\\frac{x_2}{3x_3+x_4}+\\frac{x_3}{3x_4+x_1}+\\frac{x_4}{3x_1+x_2}\\ge1$

Source: InternLM Lean-Workbook, record lean_workbook_16402 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16402 (x1 x2 x3 x4 : ℝ) (hx1 : 0 < x1) (hx2 : 0 < x2) (hx3 : 0 < x3) (hx4 : 0 < x4) : (x1 / (3 * x2 + x3) + x2 / (3 * x3 + x4) + x3 / (3 * x4 + x1) + x4 / (3 * x1 + x2)) ≥ 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16402; Apache-2.0

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