58.82/59.64 YES 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 (VAR v_NonEmpty:S x1:S) 58.82/59.64 (RULES 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ) 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 Dependency Pairs Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(b(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(b(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 ->->Cycle: 58.82/59.64 ->->-> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(b(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 ->->-> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 Reduction Pair Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(b(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 -> Usable rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Interpretation type: 58.82/59.64 Linear 58.82/59.64 ->Coefficients: 58.82/59.64 Natural Numbers 58.82/59.64 ->Dimension: 58.82/59.64 2 58.82/59.64 ->Bound: 58.82/59.64 1 58.82/59.64 ->Interpretation: 58.82/59.64 58.82/59.64 [a](X) = [1 0;1 0].X 58.82/59.64 [b](X) = [0 1;1 0].X + [0;1] 58.82/59.64 [c](X) = [0 1;0 0].X 58.82/59.64 [A](X) = [1 0;1 0].X + [0;1] 58.82/59.64 [B](X) = [0 1;0 1].X + [0;1] 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 ->->Cycle: 58.82/59.64 ->->-> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 ->->-> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 Reduction Pair Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> A(x1:S) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 -> Usable rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Interpretation type: 58.82/59.64 Linear 58.82/59.64 ->Coefficients: 58.82/59.64 Natural Numbers 58.82/59.64 ->Dimension: 58.82/59.64 2 58.82/59.64 ->Bound: 58.82/59.64 1 58.82/59.64 ->Interpretation: 58.82/59.64 58.82/59.64 [a](X) = [0 1;0 1].X 58.82/59.64 [b](X) = [0 1;1 0].X + [1;0] 58.82/59.64 [c](X) = [0 0;1 0].X 58.82/59.64 [A](X) = [0 1;0 1].X + [1;1] 58.82/59.64 [B](X) = [1 0;1 0].X + [1;1] 58.82/59.64 58.82/59.64 Problem 1: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 A(a(x1:S)) -> B(c(a(x1:S))) 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 ->->Cycle: 58.82/59.64 ->->-> Pairs: 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 ->->-> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->->Cycle: 58.82/59.64 ->->-> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 ->->-> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 58.82/59.64 58.82/59.64 The problem is decomposed in 2 subproblems. 58.82/59.64 58.82/59.64 Problem 1.1: 58.82/59.64 58.82/59.64 Reduction Pair Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(c(b(x1:S))) -> B(a(b(a(x1:S)))) 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 -> Usable rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Interpretation type: 58.82/59.64 Linear 58.82/59.64 ->Coefficients: 58.82/59.64 Natural Numbers 58.82/59.64 ->Dimension: 58.82/59.64 2 58.82/59.64 ->Bound: 58.82/59.64 1 58.82/59.64 ->Interpretation: 58.82/59.64 58.82/59.64 [a](X) = [1 0;1 0].X 58.82/59.64 [b](X) = [0 1;1 0].X + [0;1] 58.82/59.64 [c](X) = [0 1;0 0].X 58.82/59.64 [A](X) = [1 0;0 0].X + [1;0] 58.82/59.64 [B](X) = [0 1;0 0].X + [1;0] 58.82/59.64 58.82/59.64 Problem 1.1: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 ->->Cycle: 58.82/59.64 ->->-> Pairs: 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 ->->-> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 58.82/59.64 Problem 1.1: 58.82/59.64 58.82/59.64 Reduction Pair Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(c(b(x1:S))) -> B(a(x1:S)) 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 -> Usable rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Interpretation type: 58.82/59.64 Linear 58.82/59.64 ->Coefficients: 58.82/59.64 Natural Numbers 58.82/59.64 ->Dimension: 58.82/59.64 2 58.82/59.64 ->Bound: 58.82/59.64 1 58.82/59.64 ->Interpretation: 58.82/59.64 58.82/59.64 [a](X) = [0 1;0 1].X 58.82/59.64 [b](X) = [0 1;1 0].X + [1;0] 58.82/59.64 [c](X) = [0 0;1 0].X 58.82/59.64 [A](X) = [0 1;0 1].X + [1;1] 58.82/59.64 [B](X) = [1 0;1 0].X + [1;1] 58.82/59.64 58.82/59.64 Problem 1.1: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 B(x1:S) -> A(c(x1:S)) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 There is no strongly connected component 58.82/59.64 58.82/59.64 The problem is finite. 58.82/59.64 58.82/59.64 Problem 1.2: 58.82/59.64 58.82/59.64 Reduction Pair Processor: 58.82/59.64 -> Pairs: 58.82/59.64 A(a(x1:S)) -> A(b(c(a(x1:S)))) 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 -> Usable rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Interpretation type: 58.82/59.64 Linear 58.82/59.64 ->Coefficients: 58.82/59.64 Natural Numbers 58.82/59.64 ->Dimension: 58.82/59.64 1 58.82/59.64 ->Bound: 58.82/59.64 2 58.82/59.64 ->Interpretation: 58.82/59.64 58.82/59.64 [a](X) = 2.X + 2 58.82/59.64 [b](X) = 1 58.82/59.64 [c](X) = 0 58.82/59.64 [A](X) = X 58.82/59.64 58.82/59.64 Problem 1.2: 58.82/59.64 58.82/59.64 SCC Processor: 58.82/59.64 -> Pairs: 58.82/59.64 Empty 58.82/59.64 -> Rules: 58.82/59.64 a(a(x1:S)) -> a(b(c(a(x1:S)))) 58.82/59.64 a(c(b(x1:S))) -> b(a(b(a(x1:S)))) 58.82/59.64 b(x1:S) -> c(a(c(x1:S))) 58.82/59.64 ->Strongly Connected Components: 58.82/59.64 There is no strongly connected component 58.82/59.64 58.82/59.64 The problem is finite. 58.82/59.64 EOF