Tiltan and graphs with no infinite paths

We prove the consistency of tiltan with the positive relation ${\omega }^{\ast }\cdot {\omega }_1\to ({\omega }^{\ast }\cdot {\omega }_1,\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}{)}^2$.