YES After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 46-rule system { 0 0 1 -> 0 1 2 0 , 0 0 1 -> 0 3 1 0 , 0 0 1 -> 1 0 4 0 , 0 0 1 -> 0 1 3 0 2 , 0 0 1 -> 0 1 3 0 4 , 0 0 1 -> 0 2 0 1 2 , 0 0 1 -> 0 3 0 1 2 , 0 0 1 -> 0 3 0 3 1 , 0 0 1 -> 0 4 0 4 1 , 0 0 1 -> 1 2 0 2 0 , 0 0 1 -> 1 2 2 0 0 , 0 0 1 -> 0 0 2 2 1 2 , 0 0 1 -> 0 1 2 4 2 0 , 0 0 1 -> 1 2 0 3 0 4 , 0 1 1 -> 0 2 1 1 , 0 1 1 -> 0 3 1 1 , 0 1 1 -> 1 1 3 0 4 , 0 1 1 -> 1 2 0 2 1 , 0 1 1 -> 1 0 3 1 2 4 , 0 1 1 -> 1 0 4 2 1 2 , 0 1 1 -> 1 1 2 4 3 0 , 0 1 1 -> 1 2 1 0 4 4 , 0 1 1 -> 1 2 2 1 3 0 , 0 5 1 -> 0 3 1 5 , 0 5 1 -> 0 4 5 1 , 0 5 1 -> 0 2 3 1 5 , 0 5 1 -> 0 3 1 5 2 , 0 5 1 -> 0 3 1 2 5 2 , 5 0 1 -> 5 1 2 4 0 , 5 0 1 -> 5 0 2 1 2 4 , 5 0 1 -> 5 1 2 3 0 4 , 0 0 1 5 -> 0 4 1 0 5 , 0 0 2 1 -> 2 0 3 0 2 1 , 0 0 2 1 -> 2 3 0 2 0 1 , 0 1 0 1 -> 1 0 2 0 1 , 0 1 1 1 -> 1 1 3 1 0 , 5 0 1 1 -> 1 5 1 2 0 , 5 3 0 1 -> 5 1 2 3 0 , 5 3 1 5 -> 5 3 1 2 5 , 5 3 2 1 -> 1 2 3 5 2 , 5 4 0 1 -> 1 2 5 0 4 , 0 0 5 1 5 -> 1 2 5 5 0 0 , 0 5 3 0 1 -> 1 0 5 3 0 4 , 0 5 3 4 1 -> 1 0 3 5 4 5 , 0 5 4 0 1 -> 0 1 3 0 4 5 , 5 4 2 1 1 -> 5 4 1 2 1 2 } The system was reversed. After renaming modulo { 1->0, 0->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 46-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 5 1 -> 2 5 2 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 5 0 1 1 -> 5 1 0 4 1 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 0 1 5 -> 1 2 0 5 0 , 0 1 3 5 -> 1 3 2 0 5 , 5 0 3 5 -> 5 2 0 3 5 , 0 2 3 5 -> 2 5 3 2 0 , 0 1 4 5 -> 4 1 5 2 0 , 5 0 5 1 1 -> 1 1 5 5 2 0 , 0 1 3 5 1 -> 4 1 3 5 1 0 , 0 4 3 5 1 -> 5 4 5 3 1 0 , 0 1 4 5 1 -> 5 4 1 3 0 1 , 0 0 2 4 5 -> 2 0 2 0 4 5 } Applying sparse 2-untiling [Hofbauer/Geser/Waldmann, FSCD 2019]. After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 45-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 5 1 -> 2 5 2 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 5 0 1 1 -> 5 1 0 4 1 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 0 1 5 -> 1 2 0 5 0 , 0 1 3 5 -> 1 3 2 0 5 , 5 0 3 5 -> 5 2 0 3 5 , 0 2 3 5 -> 2 5 3 2 0 , 0 1 4 5 -> 4 1 5 2 0 , 5 0 5 1 1 -> 1 1 5 5 2 0 , 0 1 3 5 1 -> 4 1 3 5 1 0 , 0 1 4 5 1 -> 5 4 1 3 0 1 , 0 0 2 4 5 -> 2 0 2 0 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | | 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 1 1 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 44-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 5 1 -> 2 5 2 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 5 0 1 1 -> 5 1 0 4 1 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 0 1 5 -> 1 2 0 5 0 , 0 1 3 5 -> 1 3 2 0 5 , 0 2 3 5 -> 2 5 3 2 0 , 0 1 4 5 -> 4 1 5 2 0 , 5 0 5 1 1 -> 1 1 5 5 2 0 , 0 1 3 5 1 -> 4 1 3 5 1 0 , 0 1 4 5 1 -> 5 4 1 3 0 1 , 0 0 2 4 5 -> 2 0 2 0 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 is interpreted by / \ | 2 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 1 0 1 0 | | 0 1 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | | 0 1 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 1 1 1 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | | 0 0 0 0 1 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 38-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 0 1 5 -> 1 2 0 5 0 , 0 1 3 5 -> 1 3 2 0 5 , 0 1 3 5 1 -> 4 1 3 5 1 0 , 0 0 2 4 5 -> 2 0 2 0 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 is interpreted by / \ | 1 0 1 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | \ / 1 is interpreted by / \ | 1 0 0 1 0 | | 0 1 0 0 0 | | 0 0 1 0 0 | | 0 0 0 0 1 | | 0 0 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 1 1 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 37-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 1 3 5 -> 1 3 2 0 5 , 0 1 3 5 1 -> 4 1 3 5 1 0 , 0 0 2 4 5 -> 2 0 2 0 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 6: 0 is interpreted by / \ | 1 0 1 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 1 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 1 0 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 1 1 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 1 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 1 | | 0 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 1 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 1 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 36-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 1 5 -> 1 4 2 0 5 , 0 1 5 -> 4 2 0 2 1 5 , 0 1 5 -> 4 1 3 2 0 5 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 1 3 5 -> 1 3 2 0 5 , 0 1 3 5 1 -> 4 1 3 5 1 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 1 | \ / 1 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 1 | | 0 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 1 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 33-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 , 0 1 3 5 -> 1 3 2 0 5 , 0 1 3 5 1 -> 4 1 3 5 1 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 is interpreted by / \ | 1 0 1 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | \ / 1 is interpreted by / \ | 1 0 1 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 1 | | 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5 }, it remains to prove termination of the 31-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 5 1 -> 5 0 3 1 , 0 5 1 -> 0 5 4 1 , 0 5 1 -> 5 0 3 2 1 , 0 5 1 -> 2 5 0 3 1 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 1 | \ / 1 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 1 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 1 | \ / 3 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 1 | | 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4 }, it remains to prove termination of the 27-rule system { 0 1 1 -> 1 2 0 1 , 0 1 1 -> 1 0 3 1 , 0 1 1 -> 1 4 1 0 , 0 1 1 -> 2 1 3 0 1 , 0 1 1 -> 4 1 3 0 1 , 0 1 1 -> 2 0 1 2 1 , 0 1 1 -> 2 0 1 3 1 , 0 1 1 -> 0 3 1 3 1 , 0 1 1 -> 0 4 1 4 1 , 0 1 1 -> 1 2 1 2 0 , 0 1 1 -> 1 1 2 2 0 , 0 1 1 -> 2 0 2 2 1 1 , 0 1 1 -> 1 2 4 2 0 1 , 0 1 1 -> 4 1 3 1 2 0 , 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 0 2 1 2 0 , 0 0 1 -> 4 2 0 3 1 0 , 0 0 1 -> 2 0 2 4 1 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 , 0 0 1 -> 1 3 0 2 2 0 , 0 2 1 1 -> 0 2 1 3 1 2 , 0 2 1 1 -> 0 1 2 1 3 2 , 0 1 0 1 -> 0 1 2 1 0 , 0 0 0 1 -> 1 0 3 0 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 5: 0 is interpreted by / \ | 1 0 1 0 0 | | 0 1 0 0 0 | | 0 1 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 1 | \ / 1 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 1 0 1 | | 0 0 0 0 1 | | 0 1 0 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 1 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4 }, it remains to prove termination of the 5-rule system { 0 0 1 -> 0 0 2 1 , 0 0 1 -> 0 0 3 1 , 0 0 1 -> 4 1 3 0 0 , 0 0 1 -> 1 3 4 2 0 0 , 0 0 1 -> 4 4 1 0 2 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 1 | | 0 1 0 0 | \ / 1 is interpreted by / \ | 1 0 1 1 | | 0 1 0 0 | | 0 0 0 0 | | 0 1 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 4 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 4->2, 2->3 }, it remains to prove termination of the 1-rule system { 0 0 1 -> 2 2 1 0 3 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 4: 0 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 1 | | 0 0 0 0 | \ / 1 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 1 0 0 | \ / 2 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 3 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / After renaming modulo { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.