176.26/45.80 YES 176.26/45.81 176.26/45.81 Problem: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 176.26/45.81 Proof: 176.26/45.81 DP Processor: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 c#(b(b(x1))) -> a#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 TDG Processor: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 c#(b(b(x1))) -> a#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 graph: 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> c#(x1) -> c#(b(b(x1))) -> a#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) -> a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) -> a#(a(x1)) -> c#(x1) 176.26/45.81 EDG Processor: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 c#(b(b(x1))) -> a#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 graph: 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> c#(x1) 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 c#(b(b(x1))) -> a#(x1) -> a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 a#(a(x1)) -> c#(x1) -> c#(b(b(x1))) -> a#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(b(a(c(x1)))) 176.26/45.81 SCC Processor: 176.26/45.81 #sccs: 1 176.26/45.81 #rules: 3 176.26/45.81 #arcs: 7/16 176.26/45.81 DPs: 176.26/45.81 c#(b(b(x1))) -> a#(x1) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 Arctic Interpretation Processor: 176.26/45.81 dimension: 2 176.26/45.81 usable rules: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 interpretation: 176.26/45.81 [c#](x0) = [0 2]x0 + [0], 176.26/45.81 176.26/45.81 [a#](x0) = [-& 1 ]x0 + [0], 176.26/45.81 176.26/45.81 [0 0] [0] 176.26/45.81 [c](x0) = [0 1]x0 + [0], 176.26/45.81 176.26/45.81 [-& 0 ] [0] 176.26/45.81 [b](x0) = [0 -&]x0 + [0], 176.26/45.81 176.26/45.81 [-& 0 ] [0] 176.26/45.81 [a](x0) = [0 1 ]x0 + [0] 176.26/45.81 orientation: 176.26/45.81 c#(b(b(x1))) = [0 2]x1 + [2] >= [-& 1 ]x1 + [0] = a#(x1) 176.26/45.81 176.26/45.81 a#(a(x1)) = [1 2]x1 + [1] >= [1 2]x1 + [1] = a#(c(x1)) 176.26/45.81 176.26/45.81 a#(a(x1)) = [1 2]x1 + [1] >= [0 2]x1 + [0] = c#(x1) 176.26/45.81 176.26/45.81 [-& 0 ] [0] [-& 0 ] [0] 176.26/45.81 a(x1) = [0 1 ]x1 + [0] >= [0 -&]x1 + [0] = b(x1) 176.26/45.81 176.26/45.81 [0 1] [0] [0 1] [0] 176.26/45.81 a(a(x1)) = [1 2]x1 + [1] >= [1 2]x1 + [1] = a(b(a(c(x1)))) 176.26/45.81 176.26/45.81 [0 0] [0] [-& 0 ] [0] 176.26/45.81 c(b(b(x1))) = [0 1]x1 + [1] >= [0 1 ]x1 + [0] = a(x1) 176.26/45.81 problem: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 Restore Modifier: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 EDG Processor: 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> c#(x1) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 graph: 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> a#(c(x1)) 176.26/45.81 a#(a(x1)) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1) 176.26/45.81 SCC Processor: 176.26/45.81 #sccs: 1 176.26/45.81 #rules: 1 176.26/45.81 #arcs: 2/4 176.26/45.81 DPs: 176.26/45.81 a#(a(x1)) -> a#(c(x1)) 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 Arctic Interpretation Processor: 176.26/45.81 dimension: 2 176.26/45.81 usable rules: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 interpretation: 176.26/45.81 [a#](x0) = [0 2]x0, 176.26/45.81 176.26/45.81 [0 1 ] 176.26/45.81 [c](x0) = [-3 0 ]x0, 176.26/45.81 176.26/45.81 [-2 3 ] [0] 176.26/45.81 [b](x0) = [0 -4]x0 + [0], 176.26/45.81 176.26/45.81 [0 3] [0] 176.26/45.81 [a](x0) = [0 3]x0 + [0] 176.26/45.81 orientation: 176.26/45.81 a#(a(x1)) = [2 5]x1 + [2] >= [0 2]x1 = a#(c(x1)) 176.26/45.81 176.26/45.81 [0 3] [0] [-2 3 ] [0] 176.26/45.81 a(x1) = [0 3]x1 + [0] >= [0 -4]x1 + [0] = b(x1) 176.26/45.81 176.26/45.81 [3 6] [3] [3 6] [3] 176.26/45.81 a(a(x1)) = [3 6]x1 + [3] >= [3 6]x1 + [3] = a(b(a(c(x1)))) 176.26/45.81 176.26/45.81 [3 4] [3] [0 3] [0] 176.26/45.81 c(b(b(x1))) = [0 3]x1 + [0] >= [0 3]x1 + [0] = a(x1) 176.26/45.81 problem: 176.26/45.81 DPs: 176.26/45.81 176.26/45.81 TRS: 176.26/45.81 a(x1) -> b(x1) 176.26/45.81 a(a(x1)) -> a(b(a(c(x1)))) 176.26/45.81 c(b(b(x1))) -> a(x1) 176.26/45.81 Qed 176.26/45.81 EOF