YES After renaming modulo { c->0, a->1, b->2 }, it remains to prove termination of the 3-rule system { 0 0 -> 1 1 1 2 2 2 , 2 2 2 1 -> 2 2 2 2 2 2 2 2 , 2 2 0 0 -> 0 0 0 1 1 1 1 } The system was reversed. After renaming modulo { 0->0, 2->1, 1->2 }, it remains to prove termination of the 3-rule system { 0 0 -> 1 1 1 2 2 2 , 2 1 1 1 -> 1 1 1 1 1 1 1 1 , 0 0 1 1 -> 2 2 2 2 0 0 0 } Applying sparse 2-tiling [Hofbauer/Geser/Waldmann, FSCD 2019]. After renaming modulo { (0,0)->0, (0,1)->1, (1,1)->2, (1,2)->3, (2,2)->4, (2,0)->5, (2,1)->6, (0,2)->7, (0,4)->8, (2,4)->9, (1,4)->10, (3,2)->11, (3,1)->12 }, it remains to prove termination of the 26-rule system { 0 0 0 -> 1 2 2 3 4 4 5 , 0 0 1 -> 1 2 2 3 4 4 6 , 0 0 7 -> 1 2 2 3 4 4 4 , 0 0 8 -> 1 2 2 3 4 4 9 , 5 0 0 -> 6 2 2 3 4 4 5 , 5 0 1 -> 6 2 2 3 4 4 6 , 5 0 7 -> 6 2 2 3 4 4 4 , 5 0 8 -> 6 2 2 3 4 4 9 , 7 6 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 7 6 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 7 6 2 2 10 -> 1 2 2 2 2 2 2 2 10 , 3 6 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 6 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 3 6 2 2 10 -> 2 2 2 2 2 2 2 2 10 , 4 6 2 2 2 -> 6 2 2 2 2 2 2 2 2 , 4 6 2 2 3 -> 6 2 2 2 2 2 2 2 3 , 4 6 2 2 10 -> 6 2 2 2 2 2 2 2 10 , 11 6 2 2 2 -> 12 2 2 2 2 2 2 2 2 , 11 6 2 2 3 -> 12 2 2 2 2 2 2 2 3 , 11 6 2 2 10 -> 12 2 2 2 2 2 2 2 10 , 0 0 1 2 2 -> 7 4 4 4 5 0 0 1 , 0 0 1 2 3 -> 7 4 4 4 5 0 0 7 , 0 0 1 2 10 -> 7 4 4 4 5 0 0 8 , 5 0 1 2 2 -> 4 4 4 4 5 0 0 1 , 5 0 1 2 3 -> 4 4 4 4 5 0 0 7 , 5 0 1 2 10 -> 4 4 4 4 5 0 0 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 0 | | 0 1 | \ / 2 is interpreted by / \ | 1 0 | | 0 1 | \ / 3 is interpreted by / \ | 1 0 | | 0 1 | \ / 4 is interpreted by / \ | 1 0 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 0 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 0 | | 0 1 | \ / 9 is interpreted by / \ | 1 0 | | 0 1 | \ / 10 is interpreted by / \ | 1 1 | | 0 1 | \ / 11 is interpreted by / \ | 1 1 | | 0 1 | \ / 12 is interpreted by / \ | 1 0 | | 0 1 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10 }, it remains to prove termination of the 21-rule system { 0 0 0 -> 1 2 2 3 4 4 5 , 0 0 1 -> 1 2 2 3 4 4 6 , 0 0 7 -> 1 2 2 3 4 4 4 , 0 0 8 -> 1 2 2 3 4 4 9 , 5 0 0 -> 6 2 2 3 4 4 5 , 5 0 1 -> 6 2 2 3 4 4 6 , 5 0 7 -> 6 2 2 3 4 4 4 , 5 0 8 -> 6 2 2 3 4 4 9 , 7 6 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 7 6 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 7 6 2 2 10 -> 1 2 2 2 2 2 2 2 10 , 3 6 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 6 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 3 6 2 2 10 -> 2 2 2 2 2 2 2 2 10 , 4 6 2 2 2 -> 6 2 2 2 2 2 2 2 2 , 4 6 2 2 3 -> 6 2 2 2 2 2 2 2 3 , 4 6 2 2 10 -> 6 2 2 2 2 2 2 2 10 , 0 0 1 2 2 -> 7 4 4 4 5 0 0 1 , 0 0 1 2 3 -> 7 4 4 4 5 0 0 7 , 5 0 1 2 2 -> 4 4 4 4 5 0 0 1 , 5 0 1 2 3 -> 4 4 4 4 5 0 0 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 0 | | 0 1 | \ / 2 is interpreted by / \ | 1 0 | | 0 1 | \ / 3 is interpreted by / \ | 1 0 | | 0 1 | \ / 4 is interpreted by / \ | 1 0 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 0 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 1 | | 0 1 | \ / 9 is interpreted by / \ | 1 0 | | 0 1 | \ / 10 is interpreted by / \ | 1 1 | | 0 1 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 10->8 }, it remains to prove termination of the 19-rule system { 0 0 0 -> 1 2 2 3 4 4 5 , 0 0 1 -> 1 2 2 3 4 4 6 , 0 0 7 -> 1 2 2 3 4 4 4 , 5 0 0 -> 6 2 2 3 4 4 5 , 5 0 1 -> 6 2 2 3 4 4 6 , 5 0 7 -> 6 2 2 3 4 4 4 , 7 6 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 7 6 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 7 6 2 2 8 -> 1 2 2 2 2 2 2 2 8 , 3 6 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 6 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 3 6 2 2 8 -> 2 2 2 2 2 2 2 2 8 , 4 6 2 2 2 -> 6 2 2 2 2 2 2 2 2 , 4 6 2 2 3 -> 6 2 2 2 2 2 2 2 3 , 4 6 2 2 8 -> 6 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 7 4 4 4 5 0 0 1 , 0 0 1 2 3 -> 7 4 4 4 5 0 0 7 , 5 0 1 2 2 -> 4 4 4 4 5 0 0 1 , 5 0 1 2 3 -> 4 4 4 4 5 0 0 7 } 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 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 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 0 0 0 | \ / 6 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 7 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 8 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, 2->2, 3->3, 4->4, 6->5, 7->6, 5->7, 8->8 }, it remains to prove termination of the 18-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 7 0 1 -> 5 2 2 3 4 4 5 , 7 0 6 -> 5 2 2 3 4 4 4 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 6 5 2 2 8 -> 1 2 2 2 2 2 2 2 8 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 5 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 3 5 2 2 8 -> 2 2 2 2 2 2 2 2 8 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 , 7 0 1 2 2 -> 4 4 4 4 7 0 0 1 , 7 0 1 2 3 -> 4 4 4 4 7 0 0 6 } 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 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 | \ / 1 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 | \ / 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 1 | | 0 0 0 0 0 0 | \ / 3 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 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 0 | | 0 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 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 0 0 0 0 0 | \ / 6 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 | \ / 7 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 | \ / 8 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 1 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, it remains to prove termination of the 17-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 7 0 1 -> 5 2 2 3 4 4 5 , 7 0 6 -> 5 2 2 3 4 4 4 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 6 5 2 2 8 -> 1 2 2 2 2 2 2 2 8 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 5 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 , 7 0 1 2 2 -> 4 4 4 4 7 0 0 1 , 7 0 1 2 3 -> 4 4 4 4 7 0 0 6 } 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 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 0 1 0 0 | | 0 0 1 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 1 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 | \ / 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 0 0 | | 0 0 0 0 0 1 | | 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 1 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 0 | | 0 0 0 0 0 0 | \ / 5 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 | \ / 6 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 | \ / 7 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 0 0 0 0 0 | \ / 8 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 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, it remains to prove termination of the 16-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 7 0 1 -> 5 2 2 3 4 4 5 , 7 0 6 -> 5 2 2 3 4 4 4 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 6 5 2 2 8 -> 1 2 2 2 2 2 2 2 8 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 5 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 , 7 0 1 2 2 -> 4 4 4 4 7 0 0 1 } 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 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 | \ / 1 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 | \ / 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 1 | | 0 0 0 0 0 0 | \ / 3 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 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 0 | | 0 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 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 0 0 0 0 0 | \ / 6 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 0 0 0 0 0 | \ / 7 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 | \ / 8 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 1 0 0 0 0 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, it remains to prove termination of the 15-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 7 0 1 -> 5 2 2 3 4 4 5 , 7 0 6 -> 5 2 2 3 4 4 4 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 3 5 2 2 3 -> 2 2 2 2 2 2 2 2 3 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 , 7 0 1 2 2 -> 4 4 4 4 7 0 0 1 } 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 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 | \ / 1 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 | \ / 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 1 | | 0 0 0 0 0 0 | \ / 3 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 | \ / 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 0 | | 0 0 0 0 0 0 | \ / 5 is interpreted by / \ | 1 0 0 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 0 0 0 0 0 | \ / 6 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 | \ / 7 is interpreted by / \ | 1 0 0 0 0 0 | | 0 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 0 0 0 0 | \ / 8 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 | \ / After renaming modulo { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, it remains to prove termination of the 14-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 7 0 1 -> 5 2 2 3 4 4 5 , 7 0 6 -> 5 2 2 3 4 4 4 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 , 7 0 1 2 2 -> 4 4 4 4 7 0 0 1 } 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 0 0 | | 0 1 0 0 | | 0 0 0 1 | | 0 0 1 0 | \ / 1 is interpreted by / \ | 1 0 0 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 | \ / 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 0 0 0 | \ / 6 is interpreted by / \ | 1 0 0 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 1 0 0 | \ / 7 is interpreted by / \ | 1 0 1 0 | | 0 1 0 0 | | 0 0 0 0 | | 0 0 0 0 | \ / 8 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, 2->2, 3->3, 4->4, 5->5, 6->6, 7->7, 8->8 }, it remains to prove termination of the 11-rule system { 0 0 1 -> 1 2 2 3 4 4 5 , 0 0 6 -> 1 2 2 3 4 4 4 , 7 0 0 -> 5 2 2 3 4 4 7 , 6 5 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 6 5 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 3 5 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 4 5 2 2 2 -> 5 2 2 2 2 2 2 2 2 , 4 5 2 2 3 -> 5 2 2 2 2 2 2 2 3 , 4 5 2 2 8 -> 5 2 2 2 2 2 2 2 8 , 0 0 1 2 2 -> 6 4 4 4 7 0 0 1 , 0 0 1 2 3 -> 6 4 4 4 7 0 0 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 1 | | 0 1 | \ / 1 is interpreted by / \ | 1 0 | | 0 1 | \ / 2 is interpreted by / \ | 1 0 | | 0 1 | \ / 3 is interpreted by / \ | 1 0 | | 0 1 | \ / 4 is interpreted by / \ | 1 0 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 0 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 1 | | 0 1 | \ / After renaming modulo { 6->0, 5->1, 2->2, 1->3, 3->4, 4->5, 8->6, 0->7, 7->8 }, it remains to prove termination of the 8-rule system { 0 1 2 2 2 -> 3 2 2 2 2 2 2 2 2 , 0 1 2 2 4 -> 3 2 2 2 2 2 2 2 4 , 4 1 2 2 2 -> 2 2 2 2 2 2 2 2 2 , 5 1 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 5 1 2 2 4 -> 1 2 2 2 2 2 2 2 4 , 5 1 2 2 6 -> 1 2 2 2 2 2 2 2 6 , 7 7 3 2 2 -> 0 5 5 5 8 7 7 3 , 7 7 3 2 4 -> 0 5 5 5 8 7 7 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 1 | | 0 1 | \ / 2 is interpreted by / \ | 1 0 | | 0 1 | \ / 3 is interpreted by / \ | 1 0 | | 0 1 | \ / 4 is interpreted by / \ | 1 1 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / 6 is interpreted by / \ | 1 1 | | 0 1 | \ / 7 is interpreted by / \ | 1 1 | | 0 1 | \ / 8 is interpreted by / \ | 1 0 | | 0 1 | \ / After renaming modulo { 5->0, 1->1, 2->2, 4->3, 6->4, 7->5, 3->6, 0->7, 8->8 }, it remains to prove termination of the 4-rule system { 0 1 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 0 1 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 0 1 2 2 4 -> 1 2 2 2 2 2 2 2 4 , 5 5 6 2 2 -> 7 0 0 0 8 5 5 6 } 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 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 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 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 1 | | 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 1 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 0 | | 0 0 0 0 0 0 | \ / 5 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 | \ / 6 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 | \ / 7 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 | \ / 8 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 | \ / After renaming modulo { 0->0, 1->1, 2->2, 4->3, 5->4, 6->5, 7->6, 8->7 }, it remains to prove termination of the 3-rule system { 0 1 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 0 1 2 2 3 -> 1 2 2 2 2 2 2 2 3 , 4 4 5 2 2 -> 6 0 0 0 7 4 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 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 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 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 1 | | 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 1 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 0 | | 0 0 0 0 0 0 | \ / 5 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 | \ / 6 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 | \ / 7 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 | \ / After renaming modulo { 0->0, 1->1, 2->2, 4->3, 5->4, 6->5, 7->6 }, it remains to prove termination of the 2-rule system { 0 1 2 2 2 -> 1 2 2 2 2 2 2 2 2 , 3 3 4 2 2 -> 5 0 0 0 6 3 3 4 } 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 1 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | \ / 1 is interpreted by / \ | 1 0 0 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 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 1 | | 0 1 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 0 | | 0 0 0 0 0 0 | \ / 5 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 | \ / 6 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 | \ / After renaming modulo { 3->0, 4->1, 2->2, 5->3, 0->4, 6->5 }, it remains to prove termination of the 1-rule system { 0 0 1 2 2 -> 3 4 4 4 5 0 0 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 is interpreted by / \ | 1 0 | | 0 1 | \ / 1 is interpreted by / \ | 1 0 | | 0 1 | \ / 2 is interpreted by / \ | 1 1 | | 0 1 | \ / 3 is interpreted by / \ | 1 0 | | 0 1 | \ / 4 is interpreted by / \ | 1 0 | | 0 1 | \ / 5 is interpreted by / \ | 1 0 | | 0 1 | \ / After renaming modulo { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.