0.56/0.86 YES 0.56/0.86 0.56/0.86 Problem 1: 0.56/0.86 0.56/0.86 (VAR v_NonEmpty:S x1:S) 0.56/0.86 (RULES 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ) 0.56/0.86 0.56/0.86 Problem 1: 0.56/0.86 0.56/0.86 Dependency Pairs Processor: 0.56/0.86 -> Pairs: 0.56/0.86 F(s(x1:S)) -> G(p(s(s(x1:S)))) 0.56/0.86 F(s(x1:S)) -> P(s(g(p(s(s(x1:S)))))) 0.56/0.86 F(s(x1:S)) -> P(s(s(x1:S))) 0.56/0.86 G(s(x1:S)) -> J(s(p(s(p(s(x1:S)))))) 0.56/0.86 G(s(x1:S)) -> P(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 G(s(x1:S)) -> P(s(p(s(x1:S)))) 0.56/0.86 G(s(x1:S)) -> P(s(s(s(j(s(p(s(p(s(x1:S)))))))))) 0.56/0.86 G(s(x1:S)) -> P(s(x1:S)) 0.56/0.86 HALF(0(x1:S)) -> HALF(p(s(p(s(x1:S))))) 0.56/0.86 HALF(0(x1:S)) -> P(s(p(s(x1:S)))) 0.56/0.86 HALF(0(x1:S)) -> P(s(x1:S)) 0.56/0.86 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.56/0.86 HALF(s(s(x1:S))) -> P(p(s(s(x1:S)))) 0.56/0.86 HALF(s(s(x1:S))) -> P(s(s(x1:S))) 0.56/0.86 J(s(x1:S)) -> F(p(s(p(p(s(x1:S)))))) 0.56/0.86 J(s(x1:S)) -> P(p(s(x1:S))) 0.56/0.86 J(s(x1:S)) -> P(s(f(p(s(p(p(s(x1:S)))))))) 0.56/0.86 J(s(x1:S)) -> P(s(p(p(s(x1:S))))) 0.56/0.86 J(s(x1:S)) -> P(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 J(s(x1:S)) -> P(s(x1:S)) 0.56/0.86 P(p(s(x1:S))) -> P(x1:S) 0.56/0.86 P(0(x1:S)) -> P(x1:S) 0.56/0.86 RD(0(x1:S)) -> RD(x1:S) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 0.56/0.86 Problem 1: 0.56/0.86 0.56/0.86 SCC Processor: 0.56/0.86 -> Pairs: 0.56/0.86 F(s(x1:S)) -> G(p(s(s(x1:S)))) 0.56/0.86 F(s(x1:S)) -> P(s(g(p(s(s(x1:S)))))) 0.56/0.86 F(s(x1:S)) -> P(s(s(x1:S))) 0.56/0.86 G(s(x1:S)) -> J(s(p(s(p(s(x1:S)))))) 0.56/0.86 G(s(x1:S)) -> P(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 G(s(x1:S)) -> P(s(p(s(x1:S)))) 0.56/0.86 G(s(x1:S)) -> P(s(s(s(j(s(p(s(p(s(x1:S)))))))))) 0.56/0.86 G(s(x1:S)) -> P(s(x1:S)) 0.56/0.86 HALF(0(x1:S)) -> HALF(p(s(p(s(x1:S))))) 0.56/0.86 HALF(0(x1:S)) -> P(s(p(s(x1:S)))) 0.56/0.86 HALF(0(x1:S)) -> P(s(x1:S)) 0.56/0.86 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.56/0.86 HALF(s(s(x1:S))) -> P(p(s(s(x1:S)))) 0.56/0.86 HALF(s(s(x1:S))) -> P(s(s(x1:S))) 0.56/0.86 J(s(x1:S)) -> F(p(s(p(p(s(x1:S)))))) 0.56/0.86 J(s(x1:S)) -> P(p(s(x1:S))) 0.56/0.86 J(s(x1:S)) -> P(s(f(p(s(p(p(s(x1:S)))))))) 0.56/0.86 J(s(x1:S)) -> P(s(p(p(s(x1:S))))) 0.56/0.86 J(s(x1:S)) -> P(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 J(s(x1:S)) -> P(s(x1:S)) 0.56/0.86 P(p(s(x1:S))) -> P(x1:S) 0.56/0.86 P(0(x1:S)) -> P(x1:S) 0.56/0.86 RD(0(x1:S)) -> RD(x1:S) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->Strongly Connected Components: 0.56/0.86 ->->Cycle: 0.56/0.86 ->->-> Pairs: 0.56/0.86 RD(0(x1:S)) -> RD(x1:S) 0.56/0.86 ->->-> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->->Cycle: 0.56/0.86 ->->-> Pairs: 0.56/0.86 P(p(s(x1:S))) -> P(x1:S) 0.56/0.86 P(0(x1:S)) -> P(x1:S) 0.56/0.86 ->->-> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->->Cycle: 0.56/0.86 ->->-> Pairs: 0.56/0.86 HALF(0(x1:S)) -> HALF(p(s(p(s(x1:S))))) 0.56/0.86 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.56/0.86 ->->-> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->->Cycle: 0.56/0.86 ->->-> Pairs: 0.56/0.86 F(s(x1:S)) -> G(p(s(s(x1:S)))) 0.56/0.86 G(s(x1:S)) -> J(s(p(s(p(s(x1:S)))))) 0.56/0.86 J(s(x1:S)) -> F(p(s(p(p(s(x1:S)))))) 0.56/0.86 ->->-> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 0.56/0.86 0.56/0.86 The problem is decomposed in 4 subproblems. 0.56/0.86 0.56/0.86 Problem 1.1: 0.56/0.86 0.56/0.86 Subterm Processor: 0.56/0.86 -> Pairs: 0.56/0.86 RD(0(x1:S)) -> RD(x1:S) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->Projection: 0.56/0.86 pi(RD) = 1 0.56/0.86 0.56/0.86 Problem 1.1: 0.56/0.86 0.56/0.86 SCC Processor: 0.56/0.86 -> Pairs: 0.56/0.86 Empty 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->Strongly Connected Components: 0.56/0.86 There is no strongly connected component 0.56/0.86 0.56/0.86 The problem is finite. 0.56/0.86 0.56/0.86 Problem 1.2: 0.56/0.86 0.56/0.86 Subterm Processor: 0.56/0.86 -> Pairs: 0.56/0.86 P(p(s(x1:S))) -> P(x1:S) 0.56/0.86 P(0(x1:S)) -> P(x1:S) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->Projection: 0.56/0.86 pi(P) = 1 0.56/0.86 0.56/0.86 Problem 1.2: 0.56/0.86 0.56/0.86 SCC Processor: 0.56/0.86 -> Pairs: 0.56/0.86 Empty 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 ->Strongly Connected Components: 0.56/0.86 There is no strongly connected component 0.56/0.86 0.56/0.86 The problem is finite. 0.56/0.86 0.56/0.86 Problem 1.3: 0.56/0.86 0.56/0.86 Reduction Pair Processor: 0.56/0.86 -> Pairs: 0.56/0.86 HALF(0(x1:S)) -> HALF(p(s(p(s(x1:S))))) 0.56/0.86 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.56/0.86 -> Usable rules: 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 ->Interpretation type: 0.56/0.86 Linear 0.56/0.86 ->Coefficients: 0.56/0.86 Natural Numbers 0.56/0.86 ->Dimension: 0.56/0.86 1 0.56/0.86 ->Bound: 0.56/0.86 2 0.56/0.86 ->Interpretation: 0.56/0.86 0.56/0.86 [p](X) = X 0.56/0.86 [0](X) = X + 1 0.56/0.86 [s](X) = X 0.56/0.86 [HALF](X) = X 0.56/0.86 0.56/0.86 Problem 1.3: 0.56/0.86 0.56/0.86 SCC Processor: 0.56/0.86 -> Pairs: 0.56/0.86 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.56/0.86 -> Rules: 0.56/0.86 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.56/0.86 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.56/0.86 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.56/0.86 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.56/0.86 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.56/0.86 p(p(s(x1:S))) -> p(x1:S) 0.56/0.86 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.56/0.86 p(s(x1:S)) -> x1:S 0.56/0.86 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 ->Strongly Connected Components: 0.67/0.87 ->->Cycle: 0.67/0.87 ->->-> Pairs: 0.67/0.87 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.67/0.87 ->->-> Rules: 0.67/0.87 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.67/0.87 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.67/0.87 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.67/0.87 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.67/0.87 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 0.67/0.87 Problem 1.3: 0.67/0.87 0.67/0.87 Reduction Pair Processor: 0.67/0.87 -> Pairs: 0.67/0.87 HALF(s(s(x1:S))) -> HALF(p(p(s(s(x1:S))))) 0.67/0.87 -> Rules: 0.67/0.87 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.67/0.87 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.67/0.87 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.67/0.87 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.67/0.87 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 -> Usable rules: 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 ->Interpretation type: 0.67/0.87 Linear 0.67/0.87 ->Coefficients: 0.67/0.87 All rationals 0.67/0.87 ->Dimension: 0.67/0.87 1 0.67/0.87 ->Bound: 0.67/0.87 2 0.67/0.87 ->Interpretation: 0.67/0.87 0.67/0.87 [p](X) = 1/2.X + 1 0.67/0.87 [0](X) = 2 0.67/0.87 [s](X) = 2.X + 2 0.67/0.87 [HALF](X) = 2.X 0.67/0.87 0.67/0.87 Problem 1.3: 0.67/0.87 0.67/0.87 SCC Processor: 0.67/0.87 -> Pairs: 0.67/0.87 Empty 0.67/0.87 -> Rules: 0.67/0.87 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.67/0.87 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.67/0.87 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.67/0.87 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.67/0.87 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 ->Strongly Connected Components: 0.67/0.87 There is no strongly connected component 0.67/0.87 0.67/0.87 The problem is finite. 0.67/0.87 0.67/0.87 Problem 1.4: 0.67/0.87 0.67/0.87 Reduction Pair Processor: 0.67/0.87 -> Pairs: 0.67/0.87 F(s(x1:S)) -> G(p(s(s(x1:S)))) 0.67/0.87 G(s(x1:S)) -> J(s(p(s(p(s(x1:S)))))) 0.67/0.87 J(s(x1:S)) -> F(p(s(p(p(s(x1:S)))))) 0.67/0.87 -> Rules: 0.67/0.87 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.67/0.87 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.67/0.87 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.67/0.87 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.67/0.87 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 -> Usable rules: 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 ->Interpretation type: 0.67/0.87 Linear 0.67/0.87 ->Coefficients: 0.67/0.87 All rationals 0.67/0.87 ->Dimension: 0.67/0.87 1 0.67/0.87 ->Bound: 0.67/0.87 4 0.67/0.87 ->Interpretation: 0.67/0.87 0.67/0.87 [p](X) = 1/3.X 0.67/0.87 [0](X) = 0 0.67/0.87 [s](X) = 3.X + 3/2 0.67/0.87 [F](X) = 4.X + 1/4 0.67/0.87 [G](X) = X + 4 0.67/0.87 [J](X) = 2/3.X + 2 0.67/0.87 0.67/0.87 Problem 1.4: 0.67/0.87 0.67/0.87 SCC Processor: 0.67/0.87 -> Pairs: 0.67/0.87 G(s(x1:S)) -> J(s(p(s(p(s(x1:S)))))) 0.67/0.87 J(s(x1:S)) -> F(p(s(p(p(s(x1:S)))))) 0.67/0.87 -> Rules: 0.67/0.87 f(s(x1:S)) -> p(s(g(p(s(s(x1:S)))))) 0.67/0.87 g(s(x1:S)) -> p(p(s(s(s(j(s(p(s(p(s(x1:S))))))))))) 0.67/0.87 half(0(x1:S)) -> 0(s(s(half(p(s(p(s(x1:S)))))))) 0.67/0.87 half(s(s(x1:S))) -> s(half(p(p(s(s(x1:S)))))) 0.67/0.87 j(s(x1:S)) -> p(s(s(p(s(f(p(s(p(p(s(x1:S))))))))))) 0.67/0.87 p(p(s(x1:S))) -> p(x1:S) 0.67/0.87 p(0(x1:S)) -> 0(s(s(p(x1:S)))) 0.67/0.87 p(s(x1:S)) -> x1:S 0.67/0.87 rd(0(x1:S)) -> 0(s(0(0(0(0(s(0(rd(x1:S))))))))) 0.67/0.87 ->Strongly Connected Components: 0.67/0.87 There is no strongly connected component 0.67/0.87 0.67/0.87 The problem is finite. 0.67/0.88 EOF