168.27/43.05 YES 168.27/43.06 168.27/43.06 Problem: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 a(c(x1)) -> c(c(c(a(b(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 168.27/43.06 Proof: 168.27/43.06 String Reversal Processor: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 DP Processor: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> a#(c(c(c(x1)))) 168.27/43.06 c#(a(x1)) -> b#(a(c(c(c(x1))))) 168.27/43.06 b#(b(x1)) -> a#(x1) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 TDG Processor: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> a#(c(c(c(x1)))) 168.27/43.06 c#(a(x1)) -> b#(a(c(c(c(x1))))) 168.27/43.06 b#(b(x1)) -> a#(x1) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 graph: 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) -> c#(a(x1)) -> b#(a(c(c(c(x1))))) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) -> c#(a(x1)) -> a#(c(c(c(x1)))) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) -> c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) -> c#(a(x1)) -> c#(c(x1)) 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) -> c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> b#(a(c(c(c(x1))))) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> a#(c(c(c(x1)))) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(c(x1)) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) -> c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(a(c(c(c(x1))))) 168.27/43.06 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> a#(c(c(c(x1)))) 168.27/43.06 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(c(x1)) 168.27/43.06 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> b#(a(c(c(c(x1))))) -> b#(b(x1)) -> a#(x1) 168.27/43.06 c#(a(x1)) -> a#(c(c(c(x1)))) -> a#(x1) -> b#(x1) 168.27/43.06 b#(b(x1)) -> a#(x1) -> a#(x1) -> b#(x1) 168.27/43.06 a#(x1) -> b#(x1) -> b#(b(x1)) -> a#(x1) 168.27/43.06 SCC Processor: 168.27/43.06 #sccs: 2 168.27/43.06 #rules: 5 168.27/43.06 #arcs: 19/49 168.27/43.06 DPs: 168.27/43.06 c#(a(x1)) -> c#(c(c(x1))) 168.27/43.06 c#(a(x1)) -> c#(x1) 168.27/43.06 c#(a(x1)) -> c#(c(x1)) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 Arctic Interpretation Processor: 168.27/43.06 dimension: 2 168.27/43.06 usable rules: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 interpretation: 168.27/43.06 [c#](x0) = [-& 1 ]x0 + [0], 168.27/43.06 168.27/43.06 [0 1 ] [0 ] 168.27/43.06 [c](x0) = [-& 0 ]x0 + [-&], 168.27/43.06 168.27/43.06 [0 1] [-&] 168.27/43.06 [b](x0) = [0 0]x0 + [0 ], 168.27/43.06 168.27/43.06 [0 1] [0] 168.27/43.06 [a](x0) = [0 1]x0 + [0] 168.27/43.06 orientation: 168.27/43.06 c#(a(x1)) = [1 2]x1 + [1] >= [-& 1 ]x1 + [0] = c#(c(c(x1))) 168.27/43.06 168.27/43.06 c#(a(x1)) = [1 2]x1 + [1] >= [-& 1 ]x1 + [0] = c#(x1) 168.27/43.06 168.27/43.06 c#(a(x1)) = [1 2]x1 + [1] >= [-& 1 ]x1 + [0] = c#(c(x1)) 168.27/43.06 168.27/43.06 [0 1] [0] [0 1] [-&] 168.27/43.06 a(x1) = [0 1]x1 + [0] >= [0 0]x1 + [0 ] = b(x1) 168.27/43.06 168.27/43.06 [1 2] [1] [1 2] [1] 168.27/43.06 c(a(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = b(a(c(c(c(x1))))) 168.27/43.06 168.27/43.06 [1 1] [1] [0 1] [0] 168.27/43.06 b(b(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = a(x1) 168.27/43.06 problem: 168.27/43.06 DPs: 168.27/43.06 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 Qed 168.27/43.06 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 b#(b(x1)) -> a#(x1) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 Usable Rule Processor: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 b#(b(x1)) -> a#(x1) 168.27/43.06 TRS: 168.27/43.06 168.27/43.06 Arctic Interpretation Processor: 168.27/43.06 dimension: 4 168.27/43.06 usable rules: 168.27/43.06 168.27/43.06 interpretation: 168.27/43.06 [b#](x0) = [0 -& -& -&]x0, 168.27/43.06 168.27/43.06 [a#](x0) = [0 -& -& -&]x0, 168.27/43.06 168.27/43.06 [1 0 0 1] [0] 168.27/43.06 [0 0 0 0] [0] 168.27/43.06 [b](x0) = [0 0 0 0]x0 + [0] 168.27/43.06 [0 0 0 0] [0] 168.27/43.06 orientation: 168.27/43.06 a#(x1) = [0 -& -& -&]x1 >= [0 -& -& -&]x1 = b#(x1) 168.27/43.06 168.27/43.06 b#(b(x1)) = [1 0 0 1]x1 + [0] >= [0 -& -& -&]x1 = a#(x1) 168.27/43.06 problem: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 TRS: 168.27/43.06 168.27/43.06 Restore Modifier: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 EDG Processor: 168.27/43.06 DPs: 168.27/43.06 a#(x1) -> b#(x1) 168.27/43.06 TRS: 168.27/43.06 a(x1) -> b(x1) 168.27/43.06 c(a(x1)) -> b(a(c(c(c(x1))))) 168.27/43.06 b(b(x1)) -> a(x1) 168.27/43.06 graph: 168.27/43.06 168.27/43.06 SCC Processor: 168.27/43.06 #sccs: 0 168.27/43.06 #rules: 0 168.27/43.06 #arcs: 0/1 168.27/43.06 168.60/43.07 EOF