MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(1(2(3(4(5(1(x0))))))) Wait(Right1(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right2(x0))
Begin(3(4(5(1(x0))))) Wait(Right3(x0))
Begin(4(5(1(x0)))) Wait(Right4(x0))
Begin(5(1(x0))) Wait(Right5(x0))
Begin(1(x0)) Wait(Right6(x0))
Begin(1(2(3(4(5(1(x0))))))) Wait(Right7(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right8(x0))
Begin(3(4(5(1(x0))))) Wait(Right9(x0))
Begin(4(5(1(x0)))) Wait(Right10(x0))
Begin(5(1(x0))) Wait(Right11(x0))
Begin(1(x0)) Wait(Right12(x0))
Begin(1(2(3(4(5(1(x0))))))) Wait(Right13(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right14(x0))
Begin(3(4(5(1(x0))))) Wait(Right15(x0))
Begin(4(5(1(x0)))) Wait(Right16(x0))
Begin(5(1(x0))) Wait(Right17(x0))
Begin(1(x0)) Wait(Right18(x0))
Right1(0(End(x0))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right2(0(1(End(x0)))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right3(0(1(2(End(x0))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right4(0(1(2(3(End(x0)))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right5(0(1(2(3(4(End(x0))))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right6(0(1(2(3(4(5(End(x0)))))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right7(0(End(x0))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right8(0(1(End(x0)))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right9(0(1(2(End(x0))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right10(0(1(2(3(End(x0)))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right11(0(1(2(3(4(End(x0))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right12(0(1(2(3(4(5(End(x0)))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right13(0(End(x0))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right14(0(1(End(x0)))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right15(0(1(2(End(x0))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right16(0(1(2(3(End(x0)))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right17(0(1(2(3(4(End(x0))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right18(0(1(2(3(4(5(End(x0)))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right1(0(x0)) A0(Right1(x0))
Right2(0(x0)) A0(Right2(x0))
Right3(0(x0)) A0(Right3(x0))
Right4(0(x0)) A0(Right4(x0))
Right5(0(x0)) A0(Right5(x0))
Right6(0(x0)) A0(Right6(x0))
Right7(0(x0)) A0(Right7(x0))
Right8(0(x0)) A0(Right8(x0))
Right9(0(x0)) A0(Right9(x0))
Right10(0(x0)) A0(Right10(x0))
Right11(0(x0)) A0(Right11(x0))
Right12(0(x0)) A0(Right12(x0))
Right13(0(x0)) A0(Right13(x0))
Right14(0(x0)) A0(Right14(x0))
Right15(0(x0)) A0(Right15(x0))
Right16(0(x0)) A0(Right16(x0))
Right17(0(x0)) A0(Right17(x0))
Right18(0(x0)) A0(Right18(x0))
Right1(1(x0)) A1(Right1(x0))
Right2(1(x0)) A1(Right2(x0))
Right3(1(x0)) A1(Right3(x0))
Right4(1(x0)) A1(Right4(x0))
Right5(1(x0)) A1(Right5(x0))
Right6(1(x0)) A1(Right6(x0))
Right7(1(x0)) A1(Right7(x0))
Right8(1(x0)) A1(Right8(x0))
Right9(1(x0)) A1(Right9(x0))
Right10(1(x0)) A1(Right10(x0))
Right11(1(x0)) A1(Right11(x0))
Right12(1(x0)) A1(Right12(x0))
Right13(1(x0)) A1(Right13(x0))
Right14(1(x0)) A1(Right14(x0))
Right15(1(x0)) A1(Right15(x0))
Right16(1(x0)) A1(Right16(x0))
Right17(1(x0)) A1(Right17(x0))
Right18(1(x0)) A1(Right18(x0))
Right1(2(x0)) A2(Right1(x0))
Right2(2(x0)) A2(Right2(x0))
Right3(2(x0)) A2(Right3(x0))
Right4(2(x0)) A2(Right4(x0))
Right5(2(x0)) A2(Right5(x0))
Right6(2(x0)) A2(Right6(x0))
Right7(2(x0)) A2(Right7(x0))
Right8(2(x0)) A2(Right8(x0))
Right9(2(x0)) A2(Right9(x0))
Right10(2(x0)) A2(Right10(x0))
Right11(2(x0)) A2(Right11(x0))
Right12(2(x0)) A2(Right12(x0))
Right13(2(x0)) A2(Right13(x0))
Right14(2(x0)) A2(Right14(x0))
Right15(2(x0)) A2(Right15(x0))
Right16(2(x0)) A2(Right16(x0))
Right17(2(x0)) A2(Right17(x0))
Right18(2(x0)) A2(Right18(x0))
Right1(3(x0)) A3(Right1(x0))
Right2(3(x0)) A3(Right2(x0))
Right3(3(x0)) A3(Right3(x0))
Right4(3(x0)) A3(Right4(x0))
Right5(3(x0)) A3(Right5(x0))
Right6(3(x0)) A3(Right6(x0))
Right7(3(x0)) A3(Right7(x0))
Right8(3(x0)) A3(Right8(x0))
Right9(3(x0)) A3(Right9(x0))
Right10(3(x0)) A3(Right10(x0))
Right11(3(x0)) A3(Right11(x0))
Right12(3(x0)) A3(Right12(x0))
Right13(3(x0)) A3(Right13(x0))
Right14(3(x0)) A3(Right14(x0))
Right15(3(x0)) A3(Right15(x0))
Right16(3(x0)) A3(Right16(x0))
Right17(3(x0)) A3(Right17(x0))
Right18(3(x0)) A3(Right18(x0))
Right1(4(x0)) A4(Right1(x0))
Right2(4(x0)) A4(Right2(x0))
Right3(4(x0)) A4(Right3(x0))
Right4(4(x0)) A4(Right4(x0))
Right5(4(x0)) A4(Right5(x0))
Right6(4(x0)) A4(Right6(x0))
Right7(4(x0)) A4(Right7(x0))
Right8(4(x0)) A4(Right8(x0))
Right9(4(x0)) A4(Right9(x0))
Right10(4(x0)) A4(Right10(x0))
Right11(4(x0)) A4(Right11(x0))
Right12(4(x0)) A4(Right12(x0))
Right13(4(x0)) A4(Right13(x0))
Right14(4(x0)) A4(Right14(x0))
Right15(4(x0)) A4(Right15(x0))
Right16(4(x0)) A4(Right16(x0))
Right17(4(x0)) A4(Right17(x0))
Right18(4(x0)) A4(Right18(x0))
Right1(5(x0)) A5(Right1(x0))
Right2(5(x0)) A5(Right2(x0))
Right3(5(x0)) A5(Right3(x0))
Right4(5(x0)) A5(Right4(x0))
Right5(5(x0)) A5(Right5(x0))
Right6(5(x0)) A5(Right6(x0))
Right7(5(x0)) A5(Right7(x0))
Right8(5(x0)) A5(Right8(x0))
Right9(5(x0)) A5(Right9(x0))
Right10(5(x0)) A5(Right10(x0))
Right11(5(x0)) A5(Right11(x0))
Right12(5(x0)) A5(Right12(x0))
Right13(5(x0)) A5(Right13(x0))
Right14(5(x0)) A5(Right14(x0))
Right15(5(x0)) A5(Right15(x0))
Right16(5(x0)) A5(Right16(x0))
Right17(5(x0)) A5(Right17(x0))
Right18(5(x0)) A5(Right18(x0))
A0(Left(x0)) Left(0(x0))
A1(Left(x0)) Left(1(x0))
A2(Left(x0)) Left(2(x0))
A3(Left(x0)) Left(3(x0))
A4(Left(x0)) Left(4(x0))
A5(Left(x0)) Left(5(x0))
Wait(Left(x0)) Begin(x0)
0(1(2(3(4(5(1(x0))))))) 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))
0(1(2(3(4(5(1(x0))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))))))))
0(1(2(3(4(5(1(x0))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))))))))))))))

Proof

1 Termination Assumption

We assume termination of the following TRS
Begin(1(2(3(4(5(1(x0))))))) Wait(Right1(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right2(x0))
Begin(3(4(5(1(x0))))) Wait(Right3(x0))
Begin(4(5(1(x0)))) Wait(Right4(x0))
Begin(5(1(x0))) Wait(Right5(x0))
Begin(1(x0)) Wait(Right6(x0))
Begin(1(2(3(4(5(1(x0))))))) Wait(Right7(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right8(x0))
Begin(3(4(5(1(x0))))) Wait(Right9(x0))
Begin(4(5(1(x0)))) Wait(Right10(x0))
Begin(5(1(x0))) Wait(Right11(x0))
Begin(1(x0)) Wait(Right12(x0))
Begin(1(2(3(4(5(1(x0))))))) Wait(Right13(x0))
Begin(2(3(4(5(1(x0)))))) Wait(Right14(x0))
Begin(3(4(5(1(x0))))) Wait(Right15(x0))
Begin(4(5(1(x0)))) Wait(Right16(x0))
Begin(5(1(x0))) Wait(Right17(x0))
Begin(1(x0)) Wait(Right18(x0))
Right1(0(End(x0))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right2(0(1(End(x0)))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right3(0(1(2(End(x0))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right4(0(1(2(3(End(x0)))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right5(0(1(2(3(4(End(x0))))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right6(0(1(2(3(4(5(End(x0)))))))) Left(0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))
Right7(0(End(x0))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right8(0(1(End(x0)))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right9(0(1(2(End(x0))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right10(0(1(2(3(End(x0)))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right11(0(1(2(3(4(End(x0))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right12(0(1(2(3(4(5(End(x0)))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))
Right13(0(End(x0))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right14(0(1(End(x0)))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right15(0(1(2(End(x0))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right16(0(1(2(3(End(x0)))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right17(0(1(2(3(4(End(x0))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right18(0(1(2(3(4(5(End(x0)))))))) Left(1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(End(x0)))))))))))))))))))))))))))))))))
Right1(0(x0)) A0(Right1(x0))
Right2(0(x0)) A0(Right2(x0))
Right3(0(x0)) A0(Right3(x0))
Right4(0(x0)) A0(Right4(x0))
Right5(0(x0)) A0(Right5(x0))
Right6(0(x0)) A0(Right6(x0))
Right7(0(x0)) A0(Right7(x0))
Right8(0(x0)) A0(Right8(x0))
Right9(0(x0)) A0(Right9(x0))
Right10(0(x0)) A0(Right10(x0))
Right11(0(x0)) A0(Right11(x0))
Right12(0(x0)) A0(Right12(x0))
Right13(0(x0)) A0(Right13(x0))
Right14(0(x0)) A0(Right14(x0))
Right15(0(x0)) A0(Right15(x0))
Right16(0(x0)) A0(Right16(x0))
Right17(0(x0)) A0(Right17(x0))
Right18(0(x0)) A0(Right18(x0))
Right1(1(x0)) A1(Right1(x0))
Right2(1(x0)) A1(Right2(x0))
Right3(1(x0)) A1(Right3(x0))
Right4(1(x0)) A1(Right4(x0))
Right5(1(x0)) A1(Right5(x0))
Right6(1(x0)) A1(Right6(x0))
Right7(1(x0)) A1(Right7(x0))
Right8(1(x0)) A1(Right8(x0))
Right9(1(x0)) A1(Right9(x0))
Right10(1(x0)) A1(Right10(x0))
Right11(1(x0)) A1(Right11(x0))
Right12(1(x0)) A1(Right12(x0))
Right13(1(x0)) A1(Right13(x0))
Right14(1(x0)) A1(Right14(x0))
Right15(1(x0)) A1(Right15(x0))
Right16(1(x0)) A1(Right16(x0))
Right17(1(x0)) A1(Right17(x0))
Right18(1(x0)) A1(Right18(x0))
Right1(2(x0)) A2(Right1(x0))
Right2(2(x0)) A2(Right2(x0))
Right3(2(x0)) A2(Right3(x0))
Right4(2(x0)) A2(Right4(x0))
Right5(2(x0)) A2(Right5(x0))
Right6(2(x0)) A2(Right6(x0))
Right7(2(x0)) A2(Right7(x0))
Right8(2(x0)) A2(Right8(x0))
Right9(2(x0)) A2(Right9(x0))
Right10(2(x0)) A2(Right10(x0))
Right11(2(x0)) A2(Right11(x0))
Right12(2(x0)) A2(Right12(x0))
Right13(2(x0)) A2(Right13(x0))
Right14(2(x0)) A2(Right14(x0))
Right15(2(x0)) A2(Right15(x0))
Right16(2(x0)) A2(Right16(x0))
Right17(2(x0)) A2(Right17(x0))
Right18(2(x0)) A2(Right18(x0))
Right1(3(x0)) A3(Right1(x0))
Right2(3(x0)) A3(Right2(x0))
Right3(3(x0)) A3(Right3(x0))
Right4(3(x0)) A3(Right4(x0))
Right5(3(x0)) A3(Right5(x0))
Right6(3(x0)) A3(Right6(x0))
Right7(3(x0)) A3(Right7(x0))
Right8(3(x0)) A3(Right8(x0))
Right9(3(x0)) A3(Right9(x0))
Right10(3(x0)) A3(Right10(x0))
Right11(3(x0)) A3(Right11(x0))
Right12(3(x0)) A3(Right12(x0))
Right13(3(x0)) A3(Right13(x0))
Right14(3(x0)) A3(Right14(x0))
Right15(3(x0)) A3(Right15(x0))
Right16(3(x0)) A3(Right16(x0))
Right17(3(x0)) A3(Right17(x0))
Right18(3(x0)) A3(Right18(x0))
Right1(4(x0)) A4(Right1(x0))
Right2(4(x0)) A4(Right2(x0))
Right3(4(x0)) A4(Right3(x0))
Right4(4(x0)) A4(Right4(x0))
Right5(4(x0)) A4(Right5(x0))
Right6(4(x0)) A4(Right6(x0))
Right7(4(x0)) A4(Right7(x0))
Right8(4(x0)) A4(Right8(x0))
Right9(4(x0)) A4(Right9(x0))
Right10(4(x0)) A4(Right10(x0))
Right11(4(x0)) A4(Right11(x0))
Right12(4(x0)) A4(Right12(x0))
Right13(4(x0)) A4(Right13(x0))
Right14(4(x0)) A4(Right14(x0))
Right15(4(x0)) A4(Right15(x0))
Right16(4(x0)) A4(Right16(x0))
Right17(4(x0)) A4(Right17(x0))
Right18(4(x0)) A4(Right18(x0))
Right1(5(x0)) A5(Right1(x0))
Right2(5(x0)) A5(Right2(x0))
Right3(5(x0)) A5(Right3(x0))
Right4(5(x0)) A5(Right4(x0))
Right5(5(x0)) A5(Right5(x0))
Right6(5(x0)) A5(Right6(x0))
Right7(5(x0)) A5(Right7(x0))
Right8(5(x0)) A5(Right8(x0))
Right9(5(x0)) A5(Right9(x0))
Right10(5(x0)) A5(Right10(x0))
Right11(5(x0)) A5(Right11(x0))
Right12(5(x0)) A5(Right12(x0))
Right13(5(x0)) A5(Right13(x0))
Right14(5(x0)) A5(Right14(x0))
Right15(5(x0)) A5(Right15(x0))
Right16(5(x0)) A5(Right16(x0))
Right17(5(x0)) A5(Right17(x0))
Right18(5(x0)) A5(Right18(x0))
A0(Left(x0)) Left(0(x0))
A1(Left(x0)) Left(1(x0))
A2(Left(x0)) Left(2(x0))
A3(Left(x0)) Left(3(x0))
A4(Left(x0)) Left(4(x0))
A5(Left(x0)) Left(5(x0))
Wait(Left(x0)) Begin(x0)
0(1(2(3(4(5(1(x0))))))) 0(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))
0(1(2(3(4(5(1(x0))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))))))))
0(1(2(3(4(5(1(x0))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x0)))))))))))))))))))))))))))))))