170.44/45.03 YES 170.44/45.04 170.44/45.04 Problem: 170.44/45.04 a(a(b(x1))) -> b(a(x1)) 170.44/45.04 b(a(a(x1))) -> a(a(a(b(x1)))) 170.44/45.04 a(c(x1)) -> c(b(x1)) 170.44/45.04 170.44/45.04 Proof: 170.44/45.04 String Reversal Processor: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 DP Processor: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 c#(a(x1)) -> c#(x1) 170.44/45.04 c#(a(x1)) -> b#(c(x1)) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 TDG Processor: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 c#(a(x1)) -> c#(x1) 170.44/45.04 c#(a(x1)) -> b#(c(x1)) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 graph: 170.44/45.04 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> b#(c(x1)) 170.44/45.04 c#(a(x1)) -> c#(x1) -> c#(a(x1)) -> c#(x1) 170.44/45.04 c#(a(x1)) -> b#(c(x1)) -> b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 c#(a(x1)) -> b#(c(x1)) -> b#(a(a(x1))) -> b#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) -> a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) -> a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(x1) -> a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> a#(x1) 170.44/45.04 b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) 170.44/45.04 SCC Processor: 170.44/45.04 #sccs: 2 170.44/45.04 #rules: 7 170.44/45.04 #arcs: 24/64 170.44/45.04 DPs: 170.44/45.04 c#(a(x1)) -> c#(x1) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 Usable Rule Processor: 170.44/45.04 DPs: 170.44/45.04 c#(a(x1)) -> c#(x1) 170.44/45.04 TRS: 170.44/45.04 170.44/45.04 Arctic Interpretation Processor: 170.44/45.04 dimension: 4 170.44/45.04 usable rules: 170.44/45.04 170.44/45.04 interpretation: 170.44/45.04 [c#](x0) = [0 -& -& -&]x0, 170.44/45.04 170.44/45.04 [1 0 0 0] [0] 170.44/45.04 [0 0 0 0] [0] 170.44/45.04 [a](x0) = [1 0 0 0]x0 + [0] 170.44/45.04 [0 0 0 0] [0] 170.44/45.04 orientation: 170.44/45.04 c#(a(x1)) = [1 0 0 0]x1 + [0] >= [0 -& -& -&]x1 = c#(x1) 170.44/45.04 problem: 170.44/45.04 DPs: 170.44/45.04 170.44/45.04 TRS: 170.44/45.04 170.44/45.04 Qed 170.44/45.04 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 Usable Rule Processor: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(x1) 170.44/45.04 a#(a(b(x1))) -> a#(a(x1)) 170.44/45.04 a#(a(b(x1))) -> a#(a(a(x1))) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 Arctic Interpretation Processor: 170.44/45.04 dimension: 1 170.44/45.04 usable rules: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 interpretation: 170.44/45.04 [a#](x0) = 12x0, 170.44/45.04 170.44/45.04 [b#](x0) = 15x0, 170.44/45.04 170.44/45.04 [a](x0) = x0, 170.44/45.04 170.44/45.04 [b](x0) = 3x0 170.44/45.04 orientation: 170.44/45.04 b#(a(a(x1))) = 15x1 >= 15x1 = b#(x1) 170.44/45.04 170.44/45.04 b#(a(a(x1))) = 15x1 >= 15x1 = a#(b(x1)) 170.44/45.04 170.44/45.04 a#(a(b(x1))) = 15x1 >= 12x1 = a#(x1) 170.44/45.04 170.44/45.04 a#(a(b(x1))) = 15x1 >= 12x1 = a#(a(x1)) 170.44/45.04 170.44/45.04 a#(a(b(x1))) = 15x1 >= 12x1 = a#(a(a(x1))) 170.44/45.04 170.44/45.04 a#(a(b(x1))) = 15x1 >= 15x1 = b#(a(a(a(x1)))) 170.44/45.04 170.44/45.04 b(a(a(x1))) = 3x1 >= 3x1 = a(b(x1)) 170.44/45.04 170.44/45.04 a(a(b(x1))) = 3x1 >= 3x1 = b(a(a(a(x1)))) 170.44/45.04 problem: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 Restore Modifier: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 EDG Processor: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 c(a(x1)) -> b(c(x1)) 170.44/45.04 graph: 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> b#(x1) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) -> b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) -> a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 Usable Rule Processor: 170.44/45.04 DPs: 170.44/45.04 b#(a(a(x1))) -> b#(x1) 170.44/45.04 b#(a(a(x1))) -> a#(b(x1)) 170.44/45.04 a#(a(b(x1))) -> b#(a(a(a(x1)))) 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 Matrix Interpretation Processor: dim=1 170.44/45.04 170.44/45.04 usable rules: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 interpretation: 170.44/45.04 [a#](x0) = x0 + 5, 170.44/45.04 170.44/45.04 [b#](x0) = 1/2x0 + 3, 170.44/45.04 170.44/45.04 [a](x0) = x0 + 4, 170.44/45.04 170.44/45.04 [b](x0) = 1/2x0 + 3/2 170.44/45.04 orientation: 170.44/45.04 b#(a(a(x1))) = 1/2x1 + 7 >= 1/2x1 + 3 = b#(x1) 170.44/45.04 170.44/45.04 b#(a(a(x1))) = 1/2x1 + 7 >= 1/2x1 + 13/2 = a#(b(x1)) 170.44/45.04 170.44/45.04 a#(a(b(x1))) = 1/2x1 + 21/2 >= 1/2x1 + 9 = b#(a(a(a(x1)))) 170.44/45.04 170.44/45.04 b(a(a(x1))) = 1/2x1 + 11/2 >= 1/2x1 + 11/2 = a(b(x1)) 170.44/45.04 170.44/45.04 a(a(b(x1))) = 1/2x1 + 19/2 >= 1/2x1 + 15/2 = b(a(a(a(x1)))) 170.44/45.04 problem: 170.44/45.04 DPs: 170.44/45.04 170.44/45.04 TRS: 170.44/45.04 b(a(a(x1))) -> a(b(x1)) 170.44/45.04 a(a(b(x1))) -> b(a(a(a(x1)))) 170.44/45.04 Qed 170.52/45.05 EOF