YES After renaming modulo { 0->0, 1->1, 2->2 }, it remains to prove termination of the 18-rule system { 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 , 0 1 2 1 -> 1 2 1 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 } The system was reversed. After renaming modulo { 1->0, 2->1, 0->2 }, it remains to prove termination of the 18-rule system { 0 1 0 2 -> 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 , 0 1 0 2 -> 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 1 0 2 0 0 1 0 } Applying sparse 2-tiling [Hofbauer/Geser/Waldmann, FSCD 2019]. After renaming modulo { (0,0)->0, (0,1)->1, (1,0)->2, (0,2)->3, (2,0)->4, (2,1)->5, (1,1)->6 }, it remains to prove termination of the 108-rule system { 0 1 2 3 4 -> 1 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 4 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 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 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 1 | | 0 0 0 0 0 0 | \ / 4 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 | \ / 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 { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6 }, it remains to prove termination of the 90-rule system { 0 1 2 3 4 -> 1 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 0 1 2 3 5 -> 1 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 2 1 2 3 4 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 0 , 2 1 2 3 5 -> 6 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 1 , 4 1 2 3 5 -> 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 4 0 1 2 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 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 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 1 | | 0 0 0 0 0 0 | \ / 4 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 | \ / 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 { 0->0, 1->1, 2->2, 3->3, 5->4, 4->5, 6->6 }, it remains to prove termination of the 72-rule system { 0 1 2 3 4 -> 1 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 5 1 2 3 4 -> 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 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 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 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 1 | | 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 1 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 | \ / 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 { 0->0, 1->1, 2->2, 3->3, 4->4, 5->5, 6->6 }, it remains to prove termination of the 54-rule system { 0 1 2 3 4 -> 1 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 0 1 2 3 4 -> 1 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 1 , 2 1 2 3 5 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 0 , 2 1 2 3 4 -> 6 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 5 0 1 2 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 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 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 1 | | 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 1 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 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 { 2->0, 1->1, 3->2, 5->3, 6->4, 4->5, 0->6 }, it remains to prove termination of the 36-rule system { 0 1 0 2 3 -> 4 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 0 6 , 0 1 0 2 5 -> 4 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 5 0 2 3 6 1 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 1 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 | \ / 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 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 1 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 { 0->0, 1->1, 2->2, 5->3, 4->4, 3->5, 6->6 }, it remains to prove termination of the 18-rule system { 0 1 0 2 3 -> 4 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 0 1 , 0 1 0 2 3 -> 4 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 5 6 1 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 1 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 | \ / 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 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 1 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 { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.