YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ) Problem 1: Dependency Pairs Processor: -> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(a12(x1:S)) A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a34(a34(x1:S)))) -> A34(a12(a12(x1:S))) A12(a12(a34(a34(x1:S)))) -> A34(a34(a12(a12(x1:S)))) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A45(a12(a12(x1:S))) A12(a12(a45(a45(x1:S)))) -> A45(a45(a12(a12(x1:S)))) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A56(a12(a12(x1:S))) A12(a12(a56(a56(x1:S)))) -> A56(a56(a12(a12(x1:S)))) A13(a13(x1:S)) -> A12(a12(a23(a23(a12(a12(x1:S)))))) A13(a13(x1:S)) -> A12(a12(x1:S)) A13(a13(x1:S)) -> A12(a23(a23(a12(a12(x1:S))))) A13(a13(x1:S)) -> A12(x1:S) A13(a13(x1:S)) -> A23(a12(a12(x1:S))) A13(a13(x1:S)) -> A23(a23(a12(a12(x1:S)))) A14(a14(x1:S)) -> A12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A14(a14(x1:S)) -> A12(a12(x1:S)) A14(a14(x1:S)) -> A12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S))))))))) A14(a14(x1:S)) -> A12(x1:S) A14(a14(x1:S)) -> A23(a12(a12(x1:S))) A14(a14(x1:S)) -> A23(a23(a12(a12(x1:S)))) A14(a14(x1:S)) -> A23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))) A14(a14(x1:S)) -> A23(a34(a34(a23(a23(a12(a12(x1:S))))))) A14(a14(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A14(a14(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A15(a15(x1:S)) -> A12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) A15(a15(x1:S)) -> A12(a12(x1:S)) A15(a15(x1:S)) -> A12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))) A15(a15(x1:S)) -> A12(x1:S) A15(a15(x1:S)) -> A23(a12(a12(x1:S))) A15(a15(x1:S)) -> A23(a23(a12(a12(x1:S)))) A15(a15(x1:S)) -> A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))) A15(a15(x1:S)) -> A23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))) A15(a15(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A15(a15(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A15(a15(x1:S)) -> A34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A15(a15(x1:S)) -> A34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))) A15(a15(x1:S)) -> A45(a34(a34(a23(a23(a12(a12(x1:S))))))) A15(a15(x1:S)) -> A45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))) A16(a16(x1:S)) -> A12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) A16(a16(x1:S)) -> A12(a12(x1:S)) A16(a16(x1:S)) -> A12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))))))) A16(a16(x1:S)) -> A12(x1:S) A16(a16(x1:S)) -> A23(a12(a12(x1:S))) A16(a16(x1:S)) -> A23(a23(a12(a12(x1:S)))) A16(a16(x1:S)) -> A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))) A16(a16(x1:S)) -> A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))))) A16(a16(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A16(a16(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A16(a16(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) A16(a16(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))) A16(a16(x1:S)) -> A45(a34(a34(a23(a23(a12(a12(x1:S))))))) A16(a16(x1:S)) -> A45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))) A16(a16(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))) A16(a16(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))) A16(a16(x1:S)) -> A56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))) A16(a16(x1:S)) -> A56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A23(a23(a45(a45(x1:S)))) -> A23(a23(x1:S)) A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a45(a45(x1:S)))) -> A45(a23(a23(x1:S))) A23(a23(a45(a45(x1:S)))) -> A45(a45(a23(a23(x1:S)))) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A56(a23(a23(x1:S))) A23(a23(a56(a56(x1:S)))) -> A56(a56(a23(a23(x1:S)))) A24(a24(x1:S)) -> A23(a23(a34(a34(a23(a23(x1:S)))))) A24(a24(x1:S)) -> A23(a23(x1:S)) A24(a24(x1:S)) -> A23(a34(a34(a23(a23(x1:S))))) A24(a24(x1:S)) -> A23(x1:S) A24(a24(x1:S)) -> A34(a23(a23(x1:S))) A24(a24(x1:S)) -> A34(a34(a23(a23(x1:S)))) A25(a25(x1:S)) -> A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) A25(a25(x1:S)) -> A23(a23(x1:S)) A25(a25(x1:S)) -> A23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S))))))))) A25(a25(x1:S)) -> A23(x1:S) A25(a25(x1:S)) -> A34(a23(a23(x1:S))) A25(a25(x1:S)) -> A34(a34(a23(a23(x1:S)))) A25(a25(x1:S)) -> A34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))) A25(a25(x1:S)) -> A34(a45(a45(a34(a34(a23(a23(x1:S))))))) A25(a25(x1:S)) -> A45(a34(a34(a23(a23(x1:S))))) A25(a25(x1:S)) -> A45(a45(a34(a34(a23(a23(x1:S)))))) A26(a26(x1:S)) -> A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) A26(a26(x1:S)) -> A23(a23(x1:S)) A26(a26(x1:S)) -> A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))))))) A26(a26(x1:S)) -> A23(x1:S) A26(a26(x1:S)) -> A34(a23(a23(x1:S))) A26(a26(x1:S)) -> A34(a34(a23(a23(x1:S)))) A26(a26(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))) A26(a26(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))))) A26(a26(x1:S)) -> A45(a34(a34(a23(a23(x1:S))))) A26(a26(x1:S)) -> A45(a45(a34(a34(a23(a23(x1:S)))))) A26(a26(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))) A26(a26(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))) A26(a26(x1:S)) -> A56(a45(a45(a34(a34(a23(a23(x1:S))))))) A26(a26(x1:S)) -> A56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))) A34(a34(a56(a56(x1:S)))) -> A34(a34(x1:S)) A34(a34(a56(a56(x1:S)))) -> A34(x1:S) A34(a34(a56(a56(x1:S)))) -> A56(a34(a34(x1:S))) A34(a34(a56(a56(x1:S)))) -> A56(a56(a34(a34(x1:S)))) A35(a35(x1:S)) -> A34(a34(a45(a45(a34(a34(x1:S)))))) A35(a35(x1:S)) -> A34(a34(x1:S)) A35(a35(x1:S)) -> A34(a45(a45(a34(a34(x1:S))))) A35(a35(x1:S)) -> A34(x1:S) A35(a35(x1:S)) -> A45(a34(a34(x1:S))) A35(a35(x1:S)) -> A45(a45(a34(a34(x1:S)))) A36(a36(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) A36(a36(x1:S)) -> A34(a34(x1:S)) A36(a36(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S))))))))) A36(a36(x1:S)) -> A34(x1:S) A36(a36(x1:S)) -> A45(a34(a34(x1:S))) A36(a36(x1:S)) -> A45(a45(a34(a34(x1:S)))) A36(a36(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))) A36(a36(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(x1:S))))))) A36(a36(x1:S)) -> A56(a45(a45(a34(a34(x1:S))))) A36(a36(x1:S)) -> A56(a56(a45(a45(a34(a34(x1:S)))))) A46(a46(x1:S)) -> A45(a45(a56(a56(a45(a45(x1:S)))))) A46(a46(x1:S)) -> A45(a45(x1:S)) A46(a46(x1:S)) -> A45(a56(a56(a45(a45(x1:S))))) A46(a46(x1:S)) -> A45(x1:S) A46(a46(x1:S)) -> A56(a45(a45(x1:S))) A46(a46(x1:S)) -> A56(a56(a45(a45(x1:S)))) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1: SCC Processor: -> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(a12(x1:S)) A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a34(a34(x1:S)))) -> A34(a12(a12(x1:S))) A12(a12(a34(a34(x1:S)))) -> A34(a34(a12(a12(x1:S)))) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A45(a12(a12(x1:S))) A12(a12(a45(a45(x1:S)))) -> A45(a45(a12(a12(x1:S)))) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A56(a12(a12(x1:S))) A12(a12(a56(a56(x1:S)))) -> A56(a56(a12(a12(x1:S)))) A13(a13(x1:S)) -> A12(a12(a23(a23(a12(a12(x1:S)))))) A13(a13(x1:S)) -> A12(a12(x1:S)) A13(a13(x1:S)) -> A12(a23(a23(a12(a12(x1:S))))) A13(a13(x1:S)) -> A12(x1:S) A13(a13(x1:S)) -> A23(a12(a12(x1:S))) A13(a13(x1:S)) -> A23(a23(a12(a12(x1:S)))) A14(a14(x1:S)) -> A12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A14(a14(x1:S)) -> A12(a12(x1:S)) A14(a14(x1:S)) -> A12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S))))))))) A14(a14(x1:S)) -> A12(x1:S) A14(a14(x1:S)) -> A23(a12(a12(x1:S))) A14(a14(x1:S)) -> A23(a23(a12(a12(x1:S)))) A14(a14(x1:S)) -> A23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))) A14(a14(x1:S)) -> A23(a34(a34(a23(a23(a12(a12(x1:S))))))) A14(a14(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A14(a14(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A15(a15(x1:S)) -> A12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) A15(a15(x1:S)) -> A12(a12(x1:S)) A15(a15(x1:S)) -> A12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))) A15(a15(x1:S)) -> A12(x1:S) A15(a15(x1:S)) -> A23(a12(a12(x1:S))) A15(a15(x1:S)) -> A23(a23(a12(a12(x1:S)))) A15(a15(x1:S)) -> A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))) A15(a15(x1:S)) -> A23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))) A15(a15(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A15(a15(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A15(a15(x1:S)) -> A34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A15(a15(x1:S)) -> A34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))) A15(a15(x1:S)) -> A45(a34(a34(a23(a23(a12(a12(x1:S))))))) A15(a15(x1:S)) -> A45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))) A16(a16(x1:S)) -> A12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) A16(a16(x1:S)) -> A12(a12(x1:S)) A16(a16(x1:S)) -> A12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))))))) A16(a16(x1:S)) -> A12(x1:S) A16(a16(x1:S)) -> A23(a12(a12(x1:S))) A16(a16(x1:S)) -> A23(a23(a12(a12(x1:S)))) A16(a16(x1:S)) -> A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))) A16(a16(x1:S)) -> A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))))) A16(a16(x1:S)) -> A34(a23(a23(a12(a12(x1:S))))) A16(a16(x1:S)) -> A34(a34(a23(a23(a12(a12(x1:S)))))) A16(a16(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) A16(a16(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))))) A16(a16(x1:S)) -> A45(a34(a34(a23(a23(a12(a12(x1:S))))))) A16(a16(x1:S)) -> A45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))) A16(a16(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))) A16(a16(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))))) A16(a16(x1:S)) -> A56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S))))))))) A16(a16(x1:S)) -> A56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))) A23(a23(a45(a45(x1:S)))) -> A23(a23(x1:S)) A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a45(a45(x1:S)))) -> A45(a23(a23(x1:S))) A23(a23(a45(a45(x1:S)))) -> A45(a45(a23(a23(x1:S)))) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A56(a23(a23(x1:S))) A23(a23(a56(a56(x1:S)))) -> A56(a56(a23(a23(x1:S)))) A24(a24(x1:S)) -> A23(a23(a34(a34(a23(a23(x1:S)))))) A24(a24(x1:S)) -> A23(a23(x1:S)) A24(a24(x1:S)) -> A23(a34(a34(a23(a23(x1:S))))) A24(a24(x1:S)) -> A23(x1:S) A24(a24(x1:S)) -> A34(a23(a23(x1:S))) A24(a24(x1:S)) -> A34(a34(a23(a23(x1:S)))) A25(a25(x1:S)) -> A23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) A25(a25(x1:S)) -> A23(a23(x1:S)) A25(a25(x1:S)) -> A23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S))))))))) A25(a25(x1:S)) -> A23(x1:S) A25(a25(x1:S)) -> A34(a23(a23(x1:S))) A25(a25(x1:S)) -> A34(a34(a23(a23(x1:S)))) A25(a25(x1:S)) -> A34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))) A25(a25(x1:S)) -> A34(a45(a45(a34(a34(a23(a23(x1:S))))))) A25(a25(x1:S)) -> A45(a34(a34(a23(a23(x1:S))))) A25(a25(x1:S)) -> A45(a45(a34(a34(a23(a23(x1:S)))))) A26(a26(x1:S)) -> A23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) A26(a26(x1:S)) -> A23(a23(x1:S)) A26(a26(x1:S)) -> A23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))))))) A26(a26(x1:S)) -> A23(x1:S) A26(a26(x1:S)) -> A34(a23(a23(x1:S))) A26(a26(x1:S)) -> A34(a34(a23(a23(x1:S)))) A26(a26(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))) A26(a26(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))))) A26(a26(x1:S)) -> A45(a34(a34(a23(a23(x1:S))))) A26(a26(x1:S)) -> A45(a45(a34(a34(a23(a23(x1:S)))))) A26(a26(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))) A26(a26(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S))))))))) A26(a26(x1:S)) -> A56(a45(a45(a34(a34(a23(a23(x1:S))))))) A26(a26(x1:S)) -> A56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))) A34(a34(a56(a56(x1:S)))) -> A34(a34(x1:S)) A34(a34(a56(a56(x1:S)))) -> A34(x1:S) A34(a34(a56(a56(x1:S)))) -> A56(a34(a34(x1:S))) A34(a34(a56(a56(x1:S)))) -> A56(a56(a34(a34(x1:S)))) A35(a35(x1:S)) -> A34(a34(a45(a45(a34(a34(x1:S)))))) A35(a35(x1:S)) -> A34(a34(x1:S)) A35(a35(x1:S)) -> A34(a45(a45(a34(a34(x1:S))))) A35(a35(x1:S)) -> A34(x1:S) A35(a35(x1:S)) -> A45(a34(a34(x1:S))) A35(a35(x1:S)) -> A45(a45(a34(a34(x1:S)))) A36(a36(x1:S)) -> A34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) A36(a36(x1:S)) -> A34(a34(x1:S)) A36(a36(x1:S)) -> A34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S))))))))) A36(a36(x1:S)) -> A34(x1:S) A36(a36(x1:S)) -> A45(a34(a34(x1:S))) A36(a36(x1:S)) -> A45(a45(a34(a34(x1:S)))) A36(a36(x1:S)) -> A45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))) A36(a36(x1:S)) -> A45(a56(a56(a45(a45(a34(a34(x1:S))))))) A36(a36(x1:S)) -> A56(a45(a45(a34(a34(x1:S))))) A36(a36(x1:S)) -> A56(a56(a45(a45(a34(a34(x1:S)))))) A46(a46(x1:S)) -> A45(a45(a56(a56(a45(a45(x1:S)))))) A46(a46(x1:S)) -> A45(a45(x1:S)) A46(a46(x1:S)) -> A45(a56(a56(a45(a45(x1:S))))) A46(a46(x1:S)) -> A45(x1:S) A46(a46(x1:S)) -> A56(a45(a45(x1:S))) A46(a46(x1:S)) -> A56(a56(a45(a45(x1:S)))) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A34(a34(a56(a56(x1:S)))) -> A34(a34(x1:S)) A34(a34(a56(a56(x1:S)))) -> A34(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->->Cycle: ->->-> Pairs: A23(a23(a45(a45(x1:S)))) -> A23(a23(x1:S)) A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->->Cycle: ->->-> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(a12(x1:S)) A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S The problem is decomposed in 3 subproblems. Problem 1.1: Reduction Pair Processor: -> Pairs: A34(a34(a56(a56(x1:S)))) -> A34(a34(x1:S)) A34(a34(a56(a56(x1:S)))) -> A34(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a34](X) = 2.X [a45](X) = 2.X [a56](X) = 2.X + 2 [A34](X) = 2.X Problem 1.1: SCC Processor: -> Pairs: A34(a34(a56(a56(x1:S)))) -> A34(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A34(a34(a56(a56(x1:S)))) -> A34(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.1: Subterm Processor: -> Pairs: A34(a34(a56(a56(x1:S)))) -> A34(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Projection: pi(A34) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: A23(a23(a45(a45(x1:S)))) -> A23(a23(x1:S)) A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a23](X) = 2.X + 2 [a34](X) = 2.X [a45](X) = 2.X + 2 [a56](X) = X [A23](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.2: Reduction Pair Processor: -> Pairs: A23(a23(a45(a45(x1:S)))) -> A23(x1:S) A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a23](X) = 2.X + 1 [a34](X) = 2.X [a45](X) = 2.X + 2 [a56](X) = X + 1 [A23](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.2: Reduction Pair Processor: -> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(a23(x1:S)) A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a23](X) = 2.X [a34](X) = 2.X [a45](X) = 2.X + 2 [a56](X) = 2.X + 2 [A23](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.2: Subterm Processor: -> Pairs: A23(a23(a56(a56(x1:S)))) -> A23(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Projection: pi(A23) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Reduction Pair Processor: -> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(a12(x1:S)) A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a12](X) = 2.X + 1 [a23](X) = X [a34](X) = 2.X + 1 [a45](X) = 2.X + 2 [a56](X) = 2.X + 1 [A12](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.3: Reduction Pair Processor: -> Pairs: A12(a12(a34(a34(x1:S)))) -> A12(x1:S) A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a12](X) = 2.X + 1 [a23](X) = X [a34](X) = X [a45](X) = X + 1 [a56](X) = X [A12](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.3: Reduction Pair Processor: -> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(a12(x1:S)) A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a12](X) = 2.X + 1 [a23](X) = 2.X [a34](X) = 2.X + 2 [a45](X) = 2.X + 2 [a56](X) = X + 1 [A12](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.3: Reduction Pair Processor: -> Pairs: A12(a12(a45(a45(x1:S)))) -> A12(x1:S) A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a12](X) = 2.X + 2 [a23](X) = 2.X [a34](X) = 2.X + 2 [a45](X) = X + 1 [a56](X) = 2.X + 2 [A12](X) = X Problem 1.3: SCC Processor: -> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.3: Reduction Pair Processor: -> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(a12(x1:S)) A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S -> Usable rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a56(a56(a56(a56(x1:S)))) -> x1:S ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a12](X) = X [a23](X) = X [a34](X) = 2.X [a45](X) = 2.X [a56](X) = 2.X + 1 [A12](X) = 2.X Problem 1.3: SCC Processor: -> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(x1:S) ->->-> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S Problem 1.3: Subterm Processor: -> Pairs: A12(a12(a56(a56(x1:S)))) -> A12(x1:S) -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Projection: pi(A12) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: a12(a12(a12(a12(x1:S)))) -> x1:S a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1:S)))))))))))) -> x1:S a12(a12(a34(a34(x1:S)))) -> a34(a34(a12(a12(x1:S)))) a12(a12(a45(a45(x1:S)))) -> a45(a45(a12(a12(x1:S)))) a12(a12(a56(a56(x1:S)))) -> a56(a56(a12(a12(x1:S)))) a13(a13(a13(a13(x1:S)))) -> x1:S a13(a13(x1:S)) -> a12(a12(a23(a23(a12(a12(x1:S)))))) a14(a14(a14(a14(x1:S)))) -> x1:S a14(a14(x1:S)) -> a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1:S)))))))))) a15(a15(a15(a15(x1:S)))) -> x1:S a15(a15(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))) a16(a16(a16(a16(x1:S)))) -> x1:S a16(a16(x1:S)) -> a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1:S)))))))))))))))))) a23(a23(a23(a23(x1:S)))) -> x1:S a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1:S)))))))))))) -> x1:S a23(a23(a45(a45(x1:S)))) -> a45(a45(a23(a23(x1:S)))) a23(a23(a56(a56(x1:S)))) -> a56(a56(a23(a23(x1:S)))) a24(a24(a24(a24(x1:S)))) -> x1:S a24(a24(x1:S)) -> a23(a23(a34(a34(a23(a23(x1:S)))))) a25(a25(a25(a25(x1:S)))) -> x1:S a25(a25(x1:S)) -> a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1:S)))))))))) a26(a26(a26(a26(x1:S)))) -> x1:S a26(a26(x1:S)) -> a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1:S)))))))))))))) a34(a34(a34(a34(x1:S)))) -> x1:S a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1:S)))))))))))) -> x1:S a34(a34(a56(a56(x1:S)))) -> a56(a56(a34(a34(x1:S)))) a35(a35(a35(a35(x1:S)))) -> x1:S a35(a35(x1:S)) -> a34(a34(a45(a45(a34(a34(x1:S)))))) a36(a36(a36(a36(x1:S)))) -> x1:S a36(a36(x1:S)) -> a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1:S)))))))))) a45(a45(a45(a45(x1:S)))) -> x1:S a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1:S)))))))))))) -> x1:S a46(a46(a46(a46(x1:S)))) -> x1:S a46(a46(x1:S)) -> a45(a45(a56(a56(a45(a45(x1:S)))))) a56(a56(a56(a56(x1:S)))) -> x1:S ->Strongly Connected Components: There is no strongly connected component The problem is finite.