173.64/44.31 YES 173.80/44.32 173.80/44.32 Problem: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 173.80/44.32 Proof: 173.80/44.32 DP Processor: 173.80/44.32 DPs: 173.80/44.32 a#(a(a(x1))) -> b#(x1) 173.80/44.32 b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 TDG Processor: 173.80/44.32 DPs: 173.80/44.32 a#(a(a(x1))) -> b#(x1) 173.80/44.32 b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 graph: 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1) 173.80/44.32 b#(b(x1)) -> a#(x1) -> a#(a(a(x1))) -> b#(x1) 173.80/44.32 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1)) 173.80/44.32 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(x1) 173.80/44.32 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(b(x1)) 173.80/44.32 Arctic Interpretation Processor: 173.80/44.32 dimension: 2 173.80/44.32 usable rules: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 interpretation: 173.80/44.32 [b#](x0) = [0 -&]x0 + [0], 173.80/44.32 173.80/44.32 [a#](x0) = [0 0]x0 + [0], 173.80/44.32 173.80/44.32 [2 0] [0 ] 173.80/44.32 [b](x0) = [2 0]x0 + [-&], 173.80/44.32 173.80/44.32 [-& 0 ] [0] 173.80/44.32 [a](x0) = [2 0 ]x0 + [0] 173.80/44.32 orientation: 173.80/44.32 a#(a(a(x1))) = [2 2]x1 + [2] >= [0 -&]x1 + [0] = b#(x1) 173.80/44.32 173.80/44.32 b#(b(x1)) = [2 0]x1 + [0] >= [2 0]x1 + [0] = a#(b(x1)) 173.80/44.32 173.80/44.32 b#(b(x1)) = [2 0]x1 + [0] >= [2 0]x1 + [0] = b#(a(b(x1))) 173.80/44.32 173.80/44.32 b#(b(x1)) = [2 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1) 173.80/44.32 173.80/44.32 b#(b(x1)) = [2 0]x1 + [0] >= [2 0]x1 + [0] = a#(a(x1)) 173.80/44.32 173.80/44.32 [2 2] [2] [2 0] [0 ] 173.80/44.32 a(a(a(x1))) = [4 2]x1 + [2] >= [2 0]x1 + [-&] = b(x1) 173.80/44.32 173.80/44.32 [4 2] [2] [4 2] [2] 173.80/44.32 b(b(x1)) = [4 2]x1 + [2] >= [4 2]x1 + [2] = b(a(b(x1))) 173.80/44.32 173.80/44.32 [4 2] [2] [2 0] [0] 173.80/44.32 b(b(x1)) = [4 2]x1 + [2] >= [2 2]x1 + [2] = a(a(x1)) 173.80/44.32 problem: 173.80/44.32 DPs: 173.80/44.32 b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 Restore Modifier: 173.80/44.32 DPs: 173.80/44.32 b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 EDG Processor: 173.80/44.32 DPs: 173.80/44.32 b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> a#(a(x1)) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 graph: 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1)) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1) 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1)) 173.80/44.32 SCC Processor: 173.80/44.32 #sccs: 1 173.80/44.32 #rules: 1 173.80/44.32 #arcs: 4/16 173.80/44.32 DPs: 173.80/44.32 b#(b(x1)) -> b#(a(b(x1))) 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 Arctic Interpretation Processor: 173.80/44.32 dimension: 2 173.80/44.32 usable rules: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 interpretation: 173.80/44.32 [b#](x0) = [-& 0 ]x0 + [0], 173.80/44.32 173.80/44.32 [-& 2 ] [0] 173.80/44.32 [b](x0) = [0 3 ]x0 + [2], 173.80/44.32 173.80/44.32 [1 2 ] [0] 173.80/44.32 [a](x0) = [0 -&]x0 + [0] 173.80/44.32 orientation: 173.80/44.32 b#(b(x1)) = [0 3]x1 + [2] >= [-& 2 ]x1 + [0] = b#(a(b(x1))) 173.80/44.32 173.80/44.32 [3 4] [3] [-& 2 ] [0] 173.80/44.32 a(a(a(x1))) = [2 3]x1 + [2] >= [0 3 ]x1 + [2] = b(x1) 173.80/44.32 173.80/44.32 [2 5] [4] [-& 4 ] [2] 173.80/44.32 b(b(x1)) = [3 6]x1 + [5] >= [2 5 ]x1 + [4] = b(a(b(x1))) 173.80/44.32 173.80/44.32 [2 5] [4] [2 3] [2] 173.80/44.32 b(b(x1)) = [3 6]x1 + [5] >= [1 2]x1 + [0] = a(a(x1)) 173.80/44.32 problem: 173.80/44.32 DPs: 173.80/44.32 173.80/44.32 TRS: 173.80/44.32 a(a(a(x1))) -> b(x1) 173.80/44.32 b(b(x1)) -> b(a(b(x1))) 173.80/44.32 b(b(x1)) -> a(a(x1)) 173.80/44.32 Qed 173.80/44.33 EOF