Tiltan and graphs with no infinite paths

We prove the consistency of tiltan with the positive relation ω∗·ω1→(ω∗·ω1,infinite path)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^*\cdot \omega _1 \rightarrow (\omega ^*\cdot \omega _1,\text {infinite path})^2$$\end{document}.