1.09/1.31 YES 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 (VAR v_NonEmpty:S x1:S) 1.09/1.31 (RULES 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Dependency Pairs Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(b(x1:S)) 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> C(c(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> C(x1:S) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(b(x1:S)) 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> C(c(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> C(x1:S) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(b(x1:S)) 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(b(x1:S)) 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 Natural Numbers 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X 1.09/1.31 [b](X) = X 1.09/1.31 [c](X) = X + 1 1.09/1.31 [d](X) = X + 1 1.09/1.31 [A](X) = 2.X + 2 1.09/1.31 [B](X) = 2.X + 2 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(d(d(x1:S)))) -> B(x1:S) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 Natural Numbers 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X 1.09/1.31 [b](X) = X 1.09/1.31 [c](X) = 2.X + 2 1.09/1.31 [d](X) = 2.X + 2 1.09/1.31 [A](X) = 2.X + 2 1.09/1.31 [B](X) = 2.X + 2 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 A(a(x1:S)) -> B(x1:S) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 Natural Numbers 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 1 1.09/1.31 [b](X) = 2 1.09/1.31 [c](X) = 0 1.09/1.31 [d](X) = 0 1.09/1.31 [A](X) = 2.X + 2 1.09/1.31 [B](X) = X + 2 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 Natural Numbers 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 1 1.09/1.31 [b](X) = 2 1.09/1.31 [c](X) = 0 1.09/1.31 [d](X) = 0 1.09/1.31 [A](X) = 2.X + 2 1.09/1.31 [B](X) = 2.X 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 All rationals 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 2 1.09/1.31 [b](X) = X + 1/2 1.09/1.31 [c](X) = 2.X + 2 1.09/1.31 [d](X) = 2.X + 2 1.09/1.31 [A](X) = X + 1 1.09/1.31 [B](X) = X 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(b(x1:S))))) 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 All rationals 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 2 1.09/1.31 [b](X) = X + 1/2 1.09/1.31 [c](X) = 2.X + 2 1.09/1.31 [d](X) = 2.X + 2 1.09/1.31 [A](X) = 2.X + 1 1.09/1.31 [B](X) = 2.X 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(b(x1:S)))) 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 All rationals 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 2 1.09/1.31 [b](X) = X + 1/2 1.09/1.31 [c](X) = 2.X 1.09/1.31 [d](X) = 2.X 1.09/1.31 [A](X) = 1/2.X + 1 1.09/1.31 [B](X) = 1/2.X 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(b(x1:S))) 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 All rationals 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 2 1.09/1.31 [b](X) = X + 1/2 1.09/1.31 [c](X) = 2.X + 2 1.09/1.31 [d](X) = 2.X + 2 1.09/1.31 [A](X) = 2.X + 1 1.09/1.31 [B](X) = 2.X 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 ->->Cycle: 1.09/1.31 ->->-> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 ->->-> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 Reduction Pair Processor: 1.09/1.31 -> Pairs: 1.09/1.31 A(a(x1:S)) -> B(b(x1:S)) 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 -> Usable rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Interpretation type: 1.09/1.31 Linear 1.09/1.31 ->Coefficients: 1.09/1.31 All rationals 1.09/1.31 ->Dimension: 1.09/1.31 1 1.09/1.31 ->Bound: 1.09/1.31 2 1.09/1.31 ->Interpretation: 1.09/1.31 1.09/1.31 [a](X) = X + 2 1.09/1.31 [b](X) = X + 1/2 1.09/1.31 [c](X) = 2.X + 1 1.09/1.31 [d](X) = 2.X + 1 1.09/1.31 [A](X) = 2.X + 2 1.09/1.31 [B](X) = 2.X + 1 1.09/1.31 1.09/1.31 Problem 1: 1.09/1.31 1.09/1.31 SCC Processor: 1.09/1.31 -> Pairs: 1.09/1.31 B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) 1.09/1.31 -> Rules: 1.09/1.31 a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) 1.09/1.31 a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) 1.09/1.31 b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) 1.09/1.31 c(c(x1:S)) -> d(d(x1:S)) 1.09/1.31 ->Strongly Connected Components: 1.09/1.31 There is no strongly connected component 1.09/1.31 1.09/1.31 The problem is finite. 1.09/1.31 EOF