0.00/0.10 YES 0.00/0.10 0.00/0.10 Problem 1: 0.00/0.10 0.00/0.10 (VAR v_NonEmpty:S x1:S) 0.00/0.10 (RULES 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ) 0.00/0.10 0.00/0.10 Problem 1: 0.00/0.10 0.00/0.10 Dependency Pairs Processor: 0.00/0.10 -> Pairs: 0.00/0.10 P(p(s(x1:S))) -> P(x1:S) 0.00/0.10 Q(0(x1:S)) -> P(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 Q(0(x1:S)) -> P(s(s(0(s(s(s(s(x1:S)))))))) 0.00/0.10 Q(s(x1:S)) -> P(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 Q(s(x1:S)) -> P(p(s(s(x1:S)))) 0.00/0.10 Q(s(x1:S)) -> P(s(s(s(s(s(s(r(p(p(s(s(x1:S)))))))))))) 0.00/0.10 Q(s(x1:S)) -> P(s(s(x1:S))) 0.00/0.10 Q(s(x1:S)) -> R(p(p(s(s(x1:S))))) 0.00/0.10 R(0(x1:S)) -> P(p(p(s(s(s(x1:S)))))) 0.00/0.10 R(0(x1:S)) -> P(p(s(s(s(x1:S))))) 0.00/0.10 R(0(x1:S)) -> P(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 R(0(x1:S)) -> P(s(0(p(p(p(s(s(s(x1:S))))))))) 0.00/0.10 R(0(x1:S)) -> P(s(s(s(x1:S)))) 0.00/0.10 R(s(x1:S)) -> P(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 R(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.10 R(s(x1:S)) -> P(s(s(q(p(s(p(s(x1:S)))))))) 0.00/0.10 R(s(x1:S)) -> P(s(x1:S)) 0.00/0.10 R(s(x1:S)) -> Q(p(s(p(s(x1:S))))) 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 0.00/0.10 Problem 1: 0.00/0.10 0.00/0.10 SCC Processor: 0.00/0.10 -> Pairs: 0.00/0.10 P(p(s(x1:S))) -> P(x1:S) 0.00/0.10 Q(0(x1:S)) -> P(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 Q(0(x1:S)) -> P(s(s(0(s(s(s(s(x1:S)))))))) 0.00/0.10 Q(s(x1:S)) -> P(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 Q(s(x1:S)) -> P(p(s(s(x1:S)))) 0.00/0.10 Q(s(x1:S)) -> P(s(s(s(s(s(s(r(p(p(s(s(x1:S)))))))))))) 0.00/0.10 Q(s(x1:S)) -> P(s(s(x1:S))) 0.00/0.10 Q(s(x1:S)) -> R(p(p(s(s(x1:S))))) 0.00/0.10 R(0(x1:S)) -> P(p(p(s(s(s(x1:S)))))) 0.00/0.10 R(0(x1:S)) -> P(p(s(s(s(x1:S))))) 0.00/0.10 R(0(x1:S)) -> P(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 R(0(x1:S)) -> P(s(0(p(p(p(s(s(s(x1:S))))))))) 0.00/0.10 R(0(x1:S)) -> P(s(s(s(x1:S)))) 0.00/0.10 R(s(x1:S)) -> P(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 R(s(x1:S)) -> P(s(p(s(x1:S)))) 0.00/0.10 R(s(x1:S)) -> P(s(s(q(p(s(p(s(x1:S)))))))) 0.00/0.10 R(s(x1:S)) -> P(s(x1:S)) 0.00/0.10 R(s(x1:S)) -> Q(p(s(p(s(x1:S))))) 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ->Strongly Connected Components: 0.00/0.10 ->->Cycle: 0.00/0.10 ->->-> Pairs: 0.00/0.10 P(p(s(x1:S))) -> P(x1:S) 0.00/0.10 ->->-> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ->->Cycle: 0.00/0.10 ->->-> Pairs: 0.00/0.10 Q(s(x1:S)) -> R(p(p(s(s(x1:S))))) 0.00/0.10 R(s(x1:S)) -> Q(p(s(p(s(x1:S))))) 0.00/0.10 ->->-> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 0.00/0.10 0.00/0.10 The problem is decomposed in 2 subproblems. 0.00/0.10 0.00/0.10 Problem 1.1: 0.00/0.10 0.00/0.10 Subterm Processor: 0.00/0.10 -> Pairs: 0.00/0.10 P(p(s(x1:S))) -> P(x1:S) 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ->Projection: 0.00/0.10 pi(P) = 1 0.00/0.10 0.00/0.10 Problem 1.1: 0.00/0.10 0.00/0.10 SCC Processor: 0.00/0.10 -> Pairs: 0.00/0.10 Empty 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ->Strongly Connected Components: 0.00/0.10 There is no strongly connected component 0.00/0.10 0.00/0.10 The problem is finite. 0.00/0.10 0.00/0.10 Problem 1.2: 0.00/0.10 0.00/0.10 Reduction Pair Processor: 0.00/0.10 -> Pairs: 0.00/0.10 Q(s(x1:S)) -> R(p(p(s(s(x1:S))))) 0.00/0.10 R(s(x1:S)) -> Q(p(s(p(s(x1:S))))) 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 -> Usable rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 ->Interpretation type: 0.00/0.10 Linear 0.00/0.10 ->Coefficients: 0.00/0.10 All rationals 0.00/0.10 ->Dimension: 0.00/0.10 1 0.00/0.10 ->Bound: 0.00/0.10 2 0.00/0.10 ->Interpretation: 0.00/0.10 0.00/0.10 [p](X) = 1/2.X 0.00/0.10 [0](X) = 0 0.00/0.10 [s](X) = 2.X + 1 0.00/0.10 [Q](X) = 1/2.X + 1/2 0.00/0.10 [R](X) = X 0.00/0.10 0.00/0.10 Problem 1.2: 0.00/0.10 0.00/0.10 SCC Processor: 0.00/0.10 -> Pairs: 0.00/0.10 R(s(x1:S)) -> Q(p(s(p(s(x1:S))))) 0.00/0.10 -> Rules: 0.00/0.10 p(p(s(x1:S))) -> p(x1:S) 0.00/0.10 p(0(x1:S)) -> 0(s(s(s(x1:S)))) 0.00/0.10 p(s(x1:S)) -> x1:S 0.00/0.10 q(0(x1:S)) -> p(p(s(s(0(s(s(s(s(x1:S))))))))) 0.00/0.10 q(s(x1:S)) -> p(p(s(s(s(s(s(s(r(p(p(s(s(x1:S))))))))))))) 0.00/0.10 r(0(x1:S)) -> p(s(p(s(0(p(p(p(s(s(s(x1:S))))))))))) 0.00/0.10 r(s(x1:S)) -> p(s(p(s(s(q(p(s(p(s(x1:S)))))))))) 0.00/0.10 ->Strongly Connected Components: 0.00/0.10 There is no strongly connected component 0.00/0.10 0.00/0.10 The problem is finite. 0.00/0.10 EOF