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