169.74/43.26 YES 169.74/43.27 169.74/43.27 Problem: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 b(c(x1)) -> c(c(b(a(x1)))) 169.74/43.27 169.74/43.27 Proof: 169.74/43.27 String Reversal Processor: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 DP Processor: 169.74/43.27 DPs: 169.74/43.27 a#(a(x1)) -> b#(x1) 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 c#(b(x1)) -> c#(x1) 169.74/43.27 c#(b(x1)) -> c#(c(x1)) 169.74/43.27 c#(b(x1)) -> b#(c(c(x1))) 169.74/43.27 c#(b(x1)) -> a#(b(c(c(x1)))) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 TDG Processor: 169.74/43.27 DPs: 169.74/43.27 a#(a(x1)) -> b#(x1) 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 c#(b(x1)) -> c#(x1) 169.74/43.27 c#(b(x1)) -> c#(c(x1)) 169.74/43.27 c#(b(x1)) -> b#(c(c(x1))) 169.74/43.27 c#(b(x1)) -> a#(b(c(c(x1)))) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 graph: 169.74/43.27 c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> a#(b(c(c(x1)))) 169.74/43.27 c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> b#(c(c(x1))) 169.74/43.27 c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) 169.74/43.27 c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(x1) 169.74/43.27 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> a#(b(c(c(x1)))) 169.74/43.27 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> b#(c(c(x1))) 169.74/43.27 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(c(x1)) 169.74/43.27 c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1) 169.74/43.27 c#(b(x1)) -> b#(c(c(x1))) -> b#(x1) -> a#(x1) 169.74/43.27 c#(b(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> b#(x1) 169.74/43.27 b#(x1) -> a#(x1) -> a#(a(x1)) -> b#(x1) 169.74/43.27 a#(a(x1)) -> b#(x1) -> b#(x1) -> a#(x1) 169.74/43.27 SCC Processor: 169.74/43.27 #sccs: 2 169.74/43.27 #rules: 4 169.74/43.27 #arcs: 12/36 169.74/43.27 DPs: 169.74/43.27 c#(b(x1)) -> c#(c(x1)) 169.74/43.27 c#(b(x1)) -> c#(x1) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 Arctic Interpretation Processor: 169.74/43.27 dimension: 2 169.74/43.27 usable rules: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 interpretation: 169.74/43.27 [c#](x0) = [0 -&]x0 + [0], 169.74/43.27 169.74/43.27 [0 0 ] [-&] 169.74/43.27 [c](x0) = [0 -&]x0 + [0 ], 169.74/43.27 169.74/43.27 [2 2] [3] 169.74/43.27 [b](x0) = [0 0]x0 + [1], 169.74/43.27 169.74/43.27 [0 2] [0] 169.74/43.27 [a](x0) = [0 0]x0 + [1] 169.74/43.27 orientation: 169.74/43.27 c#(b(x1)) = [2 2]x1 + [3] >= [0 0]x1 + [0] = c#(c(x1)) 169.74/43.27 169.74/43.27 c#(b(x1)) = [2 2]x1 + [3] >= [0 -&]x1 + [0] = c#(x1) 169.74/43.27 169.74/43.27 [0 2] [0] 169.74/43.27 a(x1) = [0 0]x1 + [1] >= x1 = x1 169.74/43.27 169.74/43.27 [2 2] [3] [2 2] [3] 169.74/43.27 a(a(x1)) = [0 2]x1 + [1] >= [0 0]x1 + [1] = b(x1) 169.74/43.27 169.74/43.27 [2 2] [3] [0 2] [0] 169.74/43.27 b(x1) = [0 0]x1 + [1] >= [0 0]x1 + [1] = a(x1) 169.74/43.27 169.74/43.27 [2 2] [3] [2 2] [3] 169.74/43.27 c(b(x1)) = [2 2]x1 + [3] >= [2 2]x1 + [3] = a(b(c(c(x1)))) 169.74/43.27 problem: 169.74/43.27 DPs: 169.74/43.27 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 Qed 169.74/43.27 169.74/43.27 DPs: 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 a#(a(x1)) -> b#(x1) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 Usable Rule Processor: 169.74/43.27 DPs: 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 a#(a(x1)) -> b#(x1) 169.74/43.27 TRS: 169.74/43.27 169.74/43.27 Arctic Interpretation Processor: 169.74/43.27 dimension: 4 169.74/43.27 usable rules: 169.74/43.27 169.74/43.27 interpretation: 169.74/43.27 [b#](x0) = [0 -& -& -&]x0, 169.74/43.27 169.74/43.27 [a#](x0) = [0 -& -& -&]x0, 169.74/43.27 169.74/43.27 [1 0 0 1] [0] 169.74/43.27 [0 0 0 0] [0] 169.74/43.27 [a](x0) = [0 0 0 0]x0 + [0] 169.74/43.27 [0 0 0 0] [0] 169.74/43.27 orientation: 169.74/43.27 b#(x1) = [0 -& -& -&]x1 >= [0 -& -& -&]x1 = a#(x1) 169.74/43.27 169.74/43.27 a#(a(x1)) = [1 0 0 1]x1 + [0] >= [0 -& -& -&]x1 = b#(x1) 169.74/43.27 problem: 169.74/43.27 DPs: 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 TRS: 169.74/43.27 169.74/43.27 Restore Modifier: 169.74/43.27 DPs: 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 EDG Processor: 169.74/43.27 DPs: 169.74/43.27 b#(x1) -> a#(x1) 169.74/43.27 TRS: 169.74/43.27 a(x1) -> x1 169.74/43.27 a(a(x1)) -> b(x1) 169.74/43.27 b(x1) -> a(x1) 169.74/43.27 c(b(x1)) -> a(b(c(c(x1)))) 169.74/43.27 graph: 169.74/43.27 169.74/43.27 SCC Processor: 169.74/43.27 #sccs: 0 169.74/43.27 #rules: 0 169.74/43.27 #arcs: 0/1 169.74/43.27 169.74/43.27 EOF