YES After renaming modulo { a->0, b->1, c->2 }, it remains to prove termination of the 3-rule system { 0 1 2 -> 0 0 1 , 0 1 2 -> 1 2 1 2 , 0 1 2 -> 2 1 2 0 } The system was reversed. After renaming modulo { 2->0, 1->1, 0->2 }, it remains to prove termination of the 3-rule system { 0 1 2 -> 1 2 2 , 0 1 2 -> 0 1 0 1 , 0 1 2 -> 2 0 1 0 } Applying sparse 2-tiling [Hofbauer/Geser/Waldmann, FSCD 2019]. After renaming modulo { (0,0)->0, (0,1)->1, (1,2)->2, (2,0)->3, (2,2)->4, (2,1)->5, (2,4)->6, (1,0)->7, (1,1)->8, (3,0)->9, (3,1)->10, (1,4)->11, (0,2)->12, (0,4)->13, (3,2)->14 }, it remains to prove termination of the 48-rule system { 0 1 2 3 -> 1 2 4 3 , 0 1 2 5 -> 1 2 4 5 , 0 1 2 4 -> 1 2 4 4 , 0 1 2 6 -> 1 2 4 6 , 7 1 2 3 -> 8 2 4 3 , 7 1 2 5 -> 8 2 4 5 , 7 1 2 4 -> 8 2 4 4 , 7 1 2 6 -> 8 2 4 6 , 3 1 2 3 -> 5 2 4 3 , 3 1 2 5 -> 5 2 4 5 , 3 1 2 4 -> 5 2 4 4 , 3 1 2 6 -> 5 2 4 6 , 9 1 2 3 -> 10 2 4 3 , 9 1 2 5 -> 10 2 4 5 , 9 1 2 4 -> 10 2 4 4 , 9 1 2 6 -> 10 2 4 6 , 0 1 2 3 -> 0 1 7 1 7 , 0 1 2 5 -> 0 1 7 1 8 , 0 1 2 4 -> 0 1 7 1 2 , 0 1 2 6 -> 0 1 7 1 11 , 7 1 2 3 -> 7 1 7 1 7 , 7 1 2 5 -> 7 1 7 1 8 , 7 1 2 4 -> 7 1 7 1 2 , 7 1 2 6 -> 7 1 7 1 11 , 3 1 2 3 -> 3 1 7 1 7 , 3 1 2 5 -> 3 1 7 1 8 , 3 1 2 4 -> 3 1 7 1 2 , 3 1 2 6 -> 3 1 7 1 11 , 9 1 2 3 -> 9 1 7 1 7 , 9 1 2 5 -> 9 1 7 1 8 , 9 1 2 4 -> 9 1 7 1 2 , 9 1 2 6 -> 9 1 7 1 11 , 0 1 2 3 -> 12 3 1 7 0 , 0 1 2 5 -> 12 3 1 7 1 , 0 1 2 4 -> 12 3 1 7 12 , 0 1 2 6 -> 12 3 1 7 13 , 7 1 2 3 -> 2 3 1 7 0 , 7 1 2 5 -> 2 3 1 7 1 , 7 1 2 4 -> 2 3 1 7 12 , 7 1 2 6 -> 2 3 1 7 13 , 3 1 2 3 -> 4 3 1 7 0 , 3 1 2 5 -> 4 3 1 7 1 , 3 1 2 4 -> 4 3 1 7 12 , 3 1 2 6 -> 4 3 1 7 13 , 9 1 2 3 -> 14 3 1 7 0 , 9 1 2 5 -> 14 3 1 7 1 , 9 1 2 4 -> 14 3 1 7 12 , 9 1 2 6 -> 14 3 1 7 13 } 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 | \ / 6 is interpreted by / \ | 1 1 | | 0 1 | \ / 7 is interpreted by / \ | 1 0 | | 0 1 | \ / 8 is interpreted by / \ | 1 0 | | 0 1 | \ / 9 is interpreted by / \ | 1 1 | | 0 1 | \ / 10 is interpreted by / \ | 1 0 | | 0 1 | \ / 11 is interpreted by / \ | 1 0 | | 0 1 | \ / 12 is interpreted by / \ | 1 0 | | 0 1 | \ / 13 is interpreted by / \ | 1 0 | | 0 1 | \ / 14 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, 12->10 }, it remains to prove termination of the 19-rule system { 0 1 2 3 -> 1 2 4 3 , 0 1 2 5 -> 1 2 4 5 , 0 1 2 4 -> 1 2 4 4 , 0 1 2 6 -> 1 2 4 6 , 7 1 2 3 -> 8 2 4 3 , 7 1 2 5 -> 8 2 4 5 , 7 1 2 4 -> 8 2 4 4 , 7 1 2 6 -> 8 2 4 6 , 3 1 2 3 -> 5 2 4 3 , 3 1 2 5 -> 5 2 4 5 , 3 1 2 4 -> 5 2 4 4 , 3 1 2 6 -> 5 2 4 6 , 0 1 2 4 -> 0 1 7 1 2 , 7 1 2 4 -> 7 1 7 1 2 , 3 1 2 4 -> 3 1 7 1 2 , 9 1 2 4 -> 9 1 7 1 2 , 7 1 2 3 -> 2 3 1 7 0 , 7 1 2 5 -> 2 3 1 7 1 , 7 1 2 4 -> 2 3 1 7 10 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 3 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 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 0 0 0 0 | \ / 6 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 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10 }, it remains to prove termination of the 18-rule system { 0 1 2 3 -> 1 2 4 3 , 0 1 2 5 -> 1 2 4 5 , 0 1 2 4 -> 1 2 4 4 , 0 1 2 6 -> 1 2 4 6 , 7 1 2 3 -> 8 2 4 3 , 7 1 2 5 -> 8 2 4 5 , 7 1 2 4 -> 8 2 4 4 , 7 1 2 6 -> 8 2 4 6 , 3 1 2 3 -> 5 2 4 3 , 3 1 2 5 -> 5 2 4 5 , 3 1 2 4 -> 5 2 4 4 , 0 1 2 4 -> 0 1 7 1 2 , 7 1 2 4 -> 7 1 7 1 2 , 3 1 2 4 -> 3 1 7 1 2 , 9 1 2 4 -> 9 1 7 1 2 , 7 1 2 3 -> 2 3 1 7 0 , 7 1 2 5 -> 2 3 1 7 1 , 7 1 2 4 -> 2 3 1 7 10 } Applying the dependency pairs transformation. After renaming modulo { (0,true)->0, (1,false)->1, (2,false)->2, (3,false)->3, (3,true)->4, (7,true)->5, (4,false)->6, (7,false)->7, (9,true)->8, (0,false)->9, (5,false)->10, (10,false)->11, (6,false)->12, (8,false)->13, (9,false)->14 }, it remains to prove termination of the 36-rule system { 0 1 2 3 -> 4 , 5 1 2 3 -> 4 , 4 1 2 3 -> 4 , 0 1 2 6 -> 0 1 7 1 2 , 0 1 2 6 -> 5 1 2 , 5 1 2 6 -> 5 1 7 1 2 , 5 1 2 6 -> 5 1 2 , 4 1 2 6 -> 4 1 7 1 2 , 4 1 2 6 -> 5 1 2 , 8 1 2 6 -> 8 1 7 1 2 , 8 1 2 6 -> 5 1 2 , 5 1 2 3 -> 4 1 7 9 , 5 1 2 3 -> 5 9 , 5 1 2 3 -> 0 , 5 1 2 10 -> 4 1 7 1 , 5 1 2 10 -> 5 1 , 5 1 2 6 -> 4 1 7 11 , 5 1 2 6 -> 5 11 , 9 1 2 3 ->= 1 2 6 3 , 9 1 2 10 ->= 1 2 6 10 , 9 1 2 6 ->= 1 2 6 6 , 9 1 2 12 ->= 1 2 6 12 , 7 1 2 3 ->= 13 2 6 3 , 7 1 2 10 ->= 13 2 6 10 , 7 1 2 6 ->= 13 2 6 6 , 7 1 2 12 ->= 13 2 6 12 , 3 1 2 3 ->= 10 2 6 3 , 3 1 2 10 ->= 10 2 6 10 , 3 1 2 6 ->= 10 2 6 6 , 9 1 2 6 ->= 9 1 7 1 2 , 7 1 2 6 ->= 7 1 7 1 2 , 3 1 2 6 ->= 3 1 7 1 2 , 14 1 2 6 ->= 14 1 7 1 2 , 7 1 2 3 ->= 2 3 1 7 9 , 7 1 2 10 ->= 2 3 1 7 1 , 7 1 2 6 ->= 2 3 1 7 11 } 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 | \ / 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 0 | | 0 1 | \ / 11 is interpreted by / \ | 1 0 | | 0 1 | \ / 12 is interpreted by / \ | 1 1 | | 0 1 | \ / 13 is interpreted by / \ | 1 0 | | 0 1 | \ / 14 is interpreted by / \ | 1 1 | | 0 1 | \ / After renaming modulo { 0->0, 1->1, 2->2, 6->3, 7->4, 5->5, 4->6, 8->7, 9->8, 3->9, 10->10, 12->11, 13->12, 14->13, 11->14 }, it remains to prove termination of the 25-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 5 1 2 , 5 1 2 3 -> 5 1 4 1 2 , 5 1 2 3 -> 5 1 2 , 6 1 2 3 -> 6 1 4 1 2 , 6 1 2 3 -> 5 1 2 , 7 1 2 3 -> 7 1 4 1 2 , 8 1 2 9 ->= 1 2 3 9 , 8 1 2 10 ->= 1 2 3 10 , 8 1 2 3 ->= 1 2 3 3 , 8 1 2 11 ->= 1 2 3 11 , 4 1 2 9 ->= 12 2 3 9 , 4 1 2 10 ->= 12 2 3 10 , 4 1 2 3 ->= 12 2 3 3 , 4 1 2 11 ->= 12 2 3 11 , 9 1 2 9 ->= 10 2 3 9 , 9 1 2 10 ->= 10 2 3 10 , 9 1 2 3 ->= 10 2 3 3 , 8 1 2 3 ->= 8 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 9 1 2 3 ->= 9 1 4 1 2 , 13 1 2 3 ->= 13 1 4 1 2 , 4 1 2 9 ->= 2 9 1 4 8 , 4 1 2 10 ->= 2 9 1 4 1 , 4 1 2 3 ->= 2 9 1 4 14 } 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 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 | \ / 6 is interpreted by / \ | 1 1 | | 0 1 | \ / 7 is interpreted by / \ | 1 1 | | 0 1 | \ / 8 is interpreted by / \ | 1 0 | | 0 1 | \ / 9 is interpreted by / \ | 1 0 | | 0 1 | \ / 10 is interpreted by / \ | 1 0 | | 0 1 | \ / 11 is interpreted by / \ | 1 1 | | 0 1 | \ / 12 is interpreted by / \ | 1 0 | | 0 1 | \ / 13 is interpreted by / \ | 1 1 | | 0 1 | \ / 14 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, 11->11, 12->12, 13->13, 14->14 }, it remains to prove termination of the 23-rule system { 0 1 2 3 -> 0 1 4 1 2 , 5 1 2 3 -> 5 1 4 1 2 , 5 1 2 3 -> 5 1 2 , 6 1 2 3 -> 6 1 4 1 2 , 7 1 2 3 -> 7 1 4 1 2 , 8 1 2 9 ->= 1 2 3 9 , 8 1 2 10 ->= 1 2 3 10 , 8 1 2 3 ->= 1 2 3 3 , 8 1 2 11 ->= 1 2 3 11 , 4 1 2 9 ->= 12 2 3 9 , 4 1 2 10 ->= 12 2 3 10 , 4 1 2 3 ->= 12 2 3 3 , 4 1 2 11 ->= 12 2 3 11 , 9 1 2 9 ->= 10 2 3 9 , 9 1 2 10 ->= 10 2 3 10 , 9 1 2 3 ->= 10 2 3 3 , 8 1 2 3 ->= 8 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 9 1 2 3 ->= 9 1 4 1 2 , 13 1 2 3 ->= 13 1 4 1 2 , 4 1 2 9 ->= 2 9 1 4 8 , 4 1 2 10 ->= 2 9 1 4 1 , 4 1 2 3 ->= 2 9 1 4 14 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 0 0 0 | | 0 1 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 1 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 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 8 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 0 | \ / 9 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 0 | \ / 10 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 | \ / 11 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 | \ / 12 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 | \ / 13 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 | \ / 14 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, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10, 12->11, 11->12, 13->13, 14->14 }, it remains to prove termination of the 22-rule system { 0 1 2 3 -> 0 1 4 1 2 , 5 1 2 3 -> 5 1 4 1 2 , 5 1 2 3 -> 5 1 2 , 6 1 2 3 -> 6 1 4 1 2 , 7 1 2 3 -> 7 1 4 1 2 , 8 1 2 9 ->= 1 2 3 9 , 8 1 2 10 ->= 1 2 3 10 , 8 1 2 3 ->= 1 2 3 3 , 4 1 2 9 ->= 11 2 3 9 , 4 1 2 10 ->= 11 2 3 10 , 4 1 2 3 ->= 11 2 3 3 , 4 1 2 12 ->= 11 2 3 12 , 9 1 2 9 ->= 10 2 3 9 , 9 1 2 10 ->= 10 2 3 10 , 9 1 2 3 ->= 10 2 3 3 , 8 1 2 3 ->= 8 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 9 1 2 3 ->= 9 1 4 1 2 , 13 1 2 3 ->= 13 1 4 1 2 , 4 1 2 9 ->= 2 9 1 4 8 , 4 1 2 10 ->= 2 9 1 4 1 , 4 1 2 3 ->= 2 9 1 4 14 } 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 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 0 0 0 0 | \ / 6 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 | \ / 7 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 | \ / 8 is interpreted by / \ | 1 0 0 0 0 | | 0 1 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 9 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 | \ / 10 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 | \ / 11 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 | \ / 12 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 | \ / 13 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 | \ / 14 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 { 5->0, 1->1, 2->2, 3->3, 4->4, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12, 14->13 }, it remains to prove termination of the 21-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 3 -> 5 1 4 1 2 , 6 1 2 3 -> 6 1 4 1 2 , 7 1 2 8 ->= 1 2 3 8 , 7 1 2 9 ->= 1 2 3 9 , 7 1 2 3 ->= 1 2 3 3 , 4 1 2 8 ->= 10 2 3 8 , 4 1 2 9 ->= 10 2 3 9 , 4 1 2 3 ->= 10 2 3 3 , 4 1 2 11 ->= 10 2 3 11 , 8 1 2 8 ->= 9 2 3 8 , 8 1 2 9 ->= 9 2 3 9 , 8 1 2 3 ->= 9 2 3 3 , 7 1 2 3 ->= 7 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 8 1 2 3 ->= 8 1 4 1 2 , 12 1 2 3 ->= 12 1 4 1 2 , 4 1 2 8 ->= 2 8 1 4 7 , 4 1 2 9 ->= 2 8 1 4 1 , 4 1 2 3 ->= 2 8 1 4 13 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 0 0 | | 0 1 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 1 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 1 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 8 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 | \ / 9 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 | \ / 10 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 | \ / 11 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 | \ / 12 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 | \ / 13 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, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11, 13->12 }, it remains to prove termination of the 20-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 3 -> 5 1 4 1 2 , 6 1 2 7 ->= 1 2 3 7 , 6 1 2 8 ->= 1 2 3 8 , 6 1 2 3 ->= 1 2 3 3 , 4 1 2 7 ->= 9 2 3 7 , 4 1 2 8 ->= 9 2 3 8 , 4 1 2 3 ->= 9 2 3 3 , 4 1 2 10 ->= 9 2 3 10 , 7 1 2 7 ->= 8 2 3 7 , 7 1 2 8 ->= 8 2 3 8 , 7 1 2 3 ->= 8 2 3 3 , 6 1 2 3 ->= 6 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 7 1 2 3 ->= 7 1 4 1 2 , 11 1 2 3 ->= 11 1 4 1 2 , 4 1 2 7 ->= 2 7 1 4 6 , 4 1 2 8 ->= 2 7 1 4 1 , 4 1 2 3 ->= 2 7 1 4 12 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 0 0 | | 0 1 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 1 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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 | \ / 11 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 | \ / 12 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, 6->5, 7->6, 8->7, 9->8, 10->9, 11->10, 12->11 }, it remains to prove termination of the 19-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 6 ->= 1 2 3 6 , 5 1 2 7 ->= 1 2 3 7 , 5 1 2 3 ->= 1 2 3 3 , 4 1 2 6 ->= 8 2 3 6 , 4 1 2 7 ->= 8 2 3 7 , 4 1 2 3 ->= 8 2 3 3 , 4 1 2 9 ->= 8 2 3 9 , 6 1 2 6 ->= 7 2 3 6 , 6 1 2 7 ->= 7 2 3 7 , 6 1 2 3 ->= 7 2 3 3 , 5 1 2 3 ->= 5 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 6 1 2 3 ->= 6 1 4 1 2 , 10 1 2 3 ->= 10 1 4 1 2 , 4 1 2 6 ->= 2 6 1 4 5 , 4 1 2 7 ->= 2 6 1 4 1 , 4 1 2 3 ->= 2 6 1 4 11 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 6 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 0 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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 | \ / 11 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, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10, 11->11 }, it remains to prove termination of the 17-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 6 ->= 1 2 3 6 , 5 1 2 7 ->= 1 2 3 7 , 5 1 2 3 ->= 1 2 3 3 , 4 1 2 6 ->= 8 2 3 6 , 4 1 2 7 ->= 8 2 3 7 , 4 1 2 3 ->= 8 2 3 3 , 4 1 2 9 ->= 8 2 3 9 , 6 1 2 6 ->= 7 2 3 6 , 6 1 2 7 ->= 7 2 3 7 , 5 1 2 3 ->= 5 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 10 1 2 3 ->= 10 1 4 1 2 , 4 1 2 6 ->= 2 6 1 4 5 , 4 1 2 7 ->= 2 6 1 4 1 , 4 1 2 3 ->= 2 6 1 4 11 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 6 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 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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 0 | \ / 11 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, 5->5, 6->6, 7->7, 8->8, 9->9, 11->10 }, it remains to prove termination of the 16-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 6 ->= 1 2 3 6 , 5 1 2 7 ->= 1 2 3 7 , 5 1 2 3 ->= 1 2 3 3 , 4 1 2 6 ->= 8 2 3 6 , 4 1 2 7 ->= 8 2 3 7 , 4 1 2 3 ->= 8 2 3 3 , 4 1 2 9 ->= 8 2 3 9 , 6 1 2 6 ->= 7 2 3 6 , 6 1 2 7 ->= 7 2 3 7 , 5 1 2 3 ->= 5 1 4 1 2 , 4 1 2 3 ->= 4 1 4 1 2 , 4 1 2 6 ->= 2 6 1 4 5 , 4 1 2 7 ->= 2 6 1 4 1 , 4 1 2 3 ->= 2 6 1 4 10 } 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 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 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 0 0 | | 0 1 0 0 0 | | 0 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 6 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 0 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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, 5->5, 6->6, 7->7, 8->8, 9->9, 10->10 }, it remains to prove termination of the 14-rule system { 0 1 2 3 -> 0 1 4 1 2 , 0 1 2 3 -> 0 1 2 , 5 1 2 6 ->= 1 2 3 6 , 5 1 2 7 ->= 1 2 3 7 , 4 1 2 6 ->= 8 2 3 6 , 4 1 2 7 ->= 8 2 3 7 , 4 1 2 3 ->= 8 2 3 3 , 4 1 2 9 ->= 8 2 3 9 , 6 1 2 6 ->= 7 2 3 6 , 6 1 2 7 ->= 7 2 3 7 , 4 1 2 3 ->= 4 1 4 1 2 , 4 1 2 6 ->= 2 6 1 4 5 , 4 1 2 7 ->= 2 6 1 4 1 , 4 1 2 3 ->= 2 6 1 4 10 } 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 0 | \ / 1 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 | \ / 2 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 | \ / 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 1 0 0 1 | \ / 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 1 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 0 | \ / 6 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 | \ / 7 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 | \ / 8 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 | \ / 9 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 | \ / 10 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 { 5->0, 1->1, 2->2, 6->3, 3->4, 7->5, 4->6, 8->7, 9->8, 10->9 }, it remains to prove termination of the 12-rule system { 0 1 2 3 ->= 1 2 4 3 , 0 1 2 5 ->= 1 2 4 5 , 6 1 2 3 ->= 7 2 4 3 , 6 1 2 5 ->= 7 2 4 5 , 6 1 2 4 ->= 7 2 4 4 , 6 1 2 8 ->= 7 2 4 8 , 3 1 2 3 ->= 5 2 4 3 , 3 1 2 5 ->= 5 2 4 5 , 6 1 2 4 ->= 6 1 6 1 2 , 6 1 2 3 ->= 2 3 1 6 0 , 6 1 2 5 ->= 2 3 1 6 1 , 6 1 2 4 ->= 2 3 1 6 9 } The system is trivially terminating.