180.35/46.09 YES 180.73/46.10 180.73/46.10 Problem: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 180.73/46.10 Proof: 180.73/46.10 DP Processor: 180.73/46.10 DPs: 180.73/46.10 a#(a(a(x1))) -> b#(x1) 180.73/46.10 b#(b(x1)) -> a#(x1) 180.73/46.10 b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> b#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 TDG Processor: 180.73/46.10 DPs: 180.73/46.10 a#(a(a(x1))) -> b#(x1) 180.73/46.10 b#(b(x1)) -> a#(x1) 180.73/46.10 b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> b#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 graph: 180.73/46.10 b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> b#(a(x1)) 180.73/46.10 b#(b(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) 180.73/46.10 b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 b#(b(x1)) -> a#(x1) -> a#(a(x1)) -> b#(a(x1)) 180.73/46.10 b#(b(x1)) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) 180.73/46.10 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(x1) 180.73/46.10 a#(a(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> b#(a(x1)) -> b#(b(x1)) -> a#(x1) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(x1) 180.73/46.10 Arctic Interpretation Processor: 180.73/46.10 dimension: 2 180.73/46.10 usable rules: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 interpretation: 180.73/46.10 [b#](x0) = [0 1]x0, 180.73/46.10 180.73/46.10 [a#](x0) = [1 0]x0, 180.73/46.10 180.73/46.10 [0 1] 180.73/46.10 [b](x0) = [1 0]x0, 180.73/46.10 180.73/46.10 [1 0 ] 180.73/46.10 [a](x0) = [0 -&]x0 180.73/46.10 orientation: 180.73/46.10 a#(a(a(x1))) = [3 2]x1 >= [0 1]x1 = b#(x1) 180.73/46.10 180.73/46.10 b#(b(x1)) = [2 1]x1 >= [1 0]x1 = a#(x1) 180.73/46.10 180.73/46.10 b#(b(x1)) = [2 1]x1 >= [2 1]x1 = a#(a(x1)) 180.73/46.10 180.73/46.10 a#(a(x1)) = [2 1]x1 >= [1 0]x1 = b#(a(x1)) 180.73/46.10 180.73/46.10 a#(a(x1)) = [2 1]x1 >= [2 1]x1 = a#(b(a(x1))) 180.73/46.10 180.73/46.10 [3 2] [0 1] 180.73/46.10 a(a(a(x1))) = [2 1]x1 >= [1 0]x1 = b(x1) 180.73/46.10 180.73/46.10 [2 1] [2 1] 180.73/46.10 b(b(x1)) = [1 2]x1 >= [1 0]x1 = a(a(x1)) 180.73/46.10 180.73/46.10 [2 1] [2 1] 180.73/46.10 a(a(x1)) = [1 0]x1 >= [1 0]x1 = a(b(a(x1))) 180.73/46.10 problem: 180.73/46.10 DPs: 180.73/46.10 b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 Restore Modifier: 180.73/46.10 DPs: 180.73/46.10 b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 EDG Processor: 180.73/46.10 DPs: 180.73/46.10 b#(b(x1)) -> a#(a(x1)) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 graph: 180.73/46.10 b#(b(x1)) -> a#(a(x1)) -> a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 SCC Processor: 180.73/46.10 #sccs: 1 180.73/46.10 #rules: 1 180.73/46.10 #arcs: 2/4 180.73/46.10 DPs: 180.73/46.10 a#(a(x1)) -> a#(b(a(x1))) 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 Arctic Interpretation Processor: 180.73/46.10 dimension: 2 180.73/46.10 usable rules: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 interpretation: 180.73/46.10 [a#](x0) = [-& 3 ]x0 + [0], 180.73/46.10 180.73/46.10 [0 0 ] [2] 180.73/46.10 [b](x0) = [2 -3]x0 + [0], 180.73/46.10 180.73/46.10 [-& -2] [0] 180.73/46.10 [a](x0) = [1 1 ]x0 + [3] 180.73/46.10 orientation: 180.73/46.10 a#(a(x1)) = [4 4]x1 + [6] >= [1 3]x1 + [5] = a#(b(a(x1))) 180.73/46.10 180.73/46.10 [0 0] [2] [0 0 ] [2] 180.73/46.10 a(a(a(x1))) = [3 3]x1 + [5] >= [2 -3]x1 + [0] = b(x1) 180.73/46.10 180.73/46.10 [2 0] [2] [-1 -1] [1] 180.73/46.10 b(b(x1)) = [2 2]x1 + [4] >= [2 2 ]x1 + [4] = a(a(x1)) 180.73/46.10 180.73/46.10 [-1 -1] [1] [-4 -2] [0] 180.73/46.10 a(a(x1)) = [2 2 ]x1 + [4] >= [2 2 ]x1 + [4] = a(b(a(x1))) 180.73/46.10 problem: 180.73/46.10 DPs: 180.73/46.10 180.73/46.10 TRS: 180.73/46.10 a(a(a(x1))) -> b(x1) 180.73/46.10 b(b(x1)) -> a(a(x1)) 180.73/46.10 a(a(x1)) -> a(b(a(x1))) 180.73/46.10 Qed 180.73/46.11 EOF