173.13/44.59 YES 173.20/44.59 173.20/44.59 Problem: 173.20/44.59 a(x1) -> x1 173.20/44.59 a(b(x1)) -> b(b(a(c(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> b(a(x1)) 173.20/44.59 173.20/44.59 Proof: 173.20/44.59 String Reversal Processor: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 DP Processor: 173.20/44.59 DPs: 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> a#(b(b(x1))) 173.20/44.59 b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 c#(c(x1)) -> a#(b(x1)) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 TDG Processor: 173.20/44.59 DPs: 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> a#(b(b(x1))) 173.20/44.59 b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 c#(c(x1)) -> a#(b(x1)) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 graph: 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> c#(a(b(b(x1)))) -> c#(c(x1)) -> a#(b(x1)) 173.20/44.59 b#(a(x1)) -> c#(a(b(b(x1)))) -> c#(c(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> a#(b(b(x1))) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(b(x1))) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 173.20/44.59 SCC Processor: 173.20/44.59 #sccs: 1 173.20/44.59 #rules: 4 173.20/44.59 #arcs: 14/36 173.20/44.59 DPs: 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> c#(a(b(b(x1)))) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 Arctic Interpretation Processor: 173.20/44.59 dimension: 2 173.20/44.59 usable rules: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 interpretation: 173.20/44.59 [c#](x0) = [-& 1 ]x0 + [0], 173.20/44.59 173.20/44.59 [b#](x0) = [0 2]x0 + [1], 173.20/44.59 173.20/44.59 [0 2] [0] 173.20/44.59 [c](x0) = [0 2]x0 + [0], 173.20/44.59 173.20/44.59 [0 -&] 173.20/44.59 [b](x0) = [0 0 ]x0, 173.20/44.59 173.20/44.59 [2 2 ] [0 ] 173.20/44.59 [a](x0) = [-& 0 ]x0 + [-&] 173.20/44.59 orientation: 173.20/44.59 c#(c(x1)) = [1 3]x1 + [1] >= [0 2]x1 + [1] = b#(x1) 173.20/44.59 173.20/44.59 b#(a(x1)) = [2 2]x1 + [1] >= [0 2]x1 + [1] = b#(x1) 173.20/44.59 173.20/44.59 b#(a(x1)) = [2 2]x1 + [1] >= [2 2]x1 + [1] = b#(b(x1)) 173.20/44.59 173.20/44.59 b#(a(x1)) = [2 2]x1 + [1] >= [1 1]x1 + [0] = c#(a(b(b(x1)))) 173.20/44.59 173.20/44.59 [2 2 ] [0 ] 173.20/44.59 a(x1) = [-& 0 ]x1 + [-&] >= x1 = x1 173.20/44.59 173.20/44.59 [2 2] [0] [2 2] [0] 173.20/44.59 b(a(x1)) = [2 2]x1 + [0] >= [2 2]x1 + [0] = c(a(b(b(x1)))) 173.20/44.59 173.20/44.59 [0 -&] 173.20/44.59 b(x1) = [0 0 ]x1 >= x1 = x1 173.20/44.59 173.20/44.59 [2 4] [2] [2 2] [0 ] 173.20/44.59 c(c(x1)) = [2 4]x1 + [2] >= [0 0]x1 + [-&] = a(b(x1)) 173.20/44.59 problem: 173.20/44.59 DPs: 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 Restore Modifier: 173.20/44.59 DPs: 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 EDG Processor: 173.20/44.59 DPs: 173.20/44.59 c#(c(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) 173.20/44.59 TRS: 173.20/44.59 a(x1) -> x1 173.20/44.59 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.59 b(x1) -> x1 173.20/44.59 c(c(x1)) -> a(b(x1)) 173.20/44.59 graph: 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 173.20/44.59 c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) 173.20/44.59 b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(x1)) 173.20/44.60 SCC Processor: 173.20/44.60 #sccs: 1 173.20/44.60 #rules: 2 173.20/44.60 #arcs: 6/9 173.20/44.60 DPs: 173.20/44.60 b#(a(x1)) -> b#(b(x1)) 173.20/44.60 b#(a(x1)) -> b#(x1) 173.20/44.60 TRS: 173.20/44.60 a(x1) -> x1 173.20/44.60 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.60 b(x1) -> x1 173.20/44.60 c(c(x1)) -> a(b(x1)) 173.20/44.60 Arctic Interpretation Processor: 173.20/44.60 dimension: 2 173.20/44.60 usable rules: 173.20/44.60 a(x1) -> x1 173.20/44.60 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.60 b(x1) -> x1 173.20/44.60 c(c(x1)) -> a(b(x1)) 173.20/44.60 interpretation: 173.20/44.60 [b#](x0) = [0 0]x0, 173.20/44.60 173.20/44.60 [2 0 ] [1 ] 173.20/44.60 [c](x0) = [2 -&]x0 + [-&], 173.20/44.60 173.20/44.60 [0 0] [0 ] 173.20/44.60 [b](x0) = [0 0]x0 + [-&], 173.20/44.60 173.20/44.60 [0 0] [0] 173.20/44.60 [a](x0) = [2 2]x0 + [2] 173.20/44.60 orientation: 173.20/44.60 b#(a(x1)) = [2 2]x1 + [2] >= [0 0]x1 + [0] = b#(b(x1)) 173.20/44.60 173.20/44.60 b#(a(x1)) = [2 2]x1 + [2] >= [0 0]x1 = b#(x1) 173.20/44.60 173.20/44.60 [0 0] [0] 173.20/44.60 a(x1) = [2 2]x1 + [2] >= x1 = x1 173.20/44.60 173.20/44.60 [2 2] [2] [2 2] [2] 173.20/44.60 b(a(x1)) = [2 2]x1 + [2] >= [2 2]x1 + [2] = c(a(b(b(x1)))) 173.20/44.60 173.20/44.60 [0 0] [0 ] 173.20/44.60 b(x1) = [0 0]x1 + [-&] >= x1 = x1 173.20/44.60 173.20/44.60 [4 2] [3] [0 0] [0] 173.20/44.60 c(c(x1)) = [4 2]x1 + [3] >= [2 2]x1 + [2] = a(b(x1)) 173.20/44.60 problem: 173.20/44.60 DPs: 173.20/44.60 173.20/44.60 TRS: 173.20/44.60 a(x1) -> x1 173.20/44.60 b(a(x1)) -> c(a(b(b(x1)))) 173.20/44.60 b(x1) -> x1 173.20/44.60 c(c(x1)) -> a(b(x1)) 173.20/44.60 Qed 173.20/44.60 EOF