1.89/1.96 YES 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 (VAR v_NonEmpty:S x1:S) 1.89/1.96 (RULES 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Dependency Pairs Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> A(x1:S) 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> A(x1:S) 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> A(x1:S) 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> A(x1:S) 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 1 1.89/1.96 [b](X) = 2.X + 1 1.89/1.96 [A](X) = 2.X + 2 1.89/1.96 [B](X) = 2.X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(a(x1:S)) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = X + 2 1.89/1.96 [B](X) = X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(a(x1:S))) 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = X + 1 1.89/1.96 [b](X) = X + 1 1.89/1.96 [A](X) = X + 2 1.89/1.96 [B](X) = X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = 2.X + 2 1.89/1.96 [B](X) = 2.X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = 2.X + 1 1.89/1.96 [B](X) = 2.X + 1 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = 2.X + 2 1.89/1.96 [B](X) = 2.X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(a(b(x1:S))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = 2.X + 2 1.89/1.96 [B](X) = 2.X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> A(b(x1:S)) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = X + 1 1.89/1.96 [b](X) = X + 1 1.89/1.96 [A](X) = X + 2 1.89/1.96 [B](X) = X + 2 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(x1:S) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 Natural Numbers 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 2 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 2.X + 2 1.89/1.96 [b](X) = 2.X + 2 1.89/1.96 [A](X) = 2.X 1.89/1.96 [B](X) = 2.X 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 A(a(b(b(x1:S)))) -> B(b(b(a(x1:S)))) 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 All rationals 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 4 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 4/3.X + 1/4 1.89/1.96 [b](X) = 4/3.X + 1/2 1.89/1.96 [A](X) = 2.X + 3/4 1.89/1.96 [B](X) = 2.X + 1 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 B(a(b(a(x1:S)))) -> A(a(a(b(x1:S)))) 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 ->->Cycle: 1.89/1.96 ->->-> Pairs: 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 ->->-> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 Reduction Pair Processor: 1.89/1.96 -> Pairs: 1.89/1.96 B(b(b(a(x1:S)))) -> B(a(a(b(x1:S)))) 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 -> Usable rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Interpretation type: 1.89/1.96 Linear 1.89/1.96 ->Coefficients: 1.89/1.96 All rationals 1.89/1.96 ->Dimension: 1.89/1.96 1 1.89/1.96 ->Bound: 1.89/1.96 4 1.89/1.96 ->Interpretation: 1.89/1.96 1.89/1.96 [a](X) = 4/3.X 1.89/1.96 [b](X) = 4/3.X + 3 1.89/1.96 [B](X) = 1/2.X 1.89/1.96 1.89/1.96 Problem 1: 1.89/1.96 1.89/1.96 SCC Processor: 1.89/1.96 -> Pairs: 1.89/1.96 Empty 1.89/1.96 -> Rules: 1.89/1.96 a(a(b(b(x1:S)))) -> b(b(b(a(x1:S)))) 1.89/1.96 b(a(b(a(x1:S)))) -> a(a(a(b(x1:S)))) 1.89/1.96 b(b(b(a(x1:S)))) -> b(a(a(b(x1:S)))) 1.89/1.96 ->Strongly Connected Components: 1.89/1.96 There is no strongly connected component 1.89/1.96 1.89/1.96 The problem is finite. 1.89/1.96 EOF