MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(a(b(x0))) → Wait(Right1(x0))
Begin(b(x0)) → Wait(Right2(x0))
Begin(e(b(x0))) → Wait(Right3(x0))
Begin(b(x0)) → Wait(Right4(x0))
Begin(d(x0)) → Wait(Right5(x0))
Begin(b(b(x0))) → Wait(Right6(x0))
Begin(b(x0)) → Wait(Right7(x0))
Begin(e(e(x0))) → Wait(Right8(x0))
Begin(e(x0)) → Wait(Right9(x0))
Begin(d(a(x0))) → Wait(Right10(x0))
Begin(a(x0)) → Wait(Right11(x0))
Begin(c(x0)) → Wait(Right12(x0))
Right1(a(End(x0))) → Left(c(d(End(x0))))
Right2(a(a(End(x0)))) → Left(c(d(End(x0))))
Right3(b(End(x0))) → Left(e(d(End(x0))))
Right4(b(e(End(x0)))) → Left(e(d(End(x0))))
Right5(b(End(x0))) → Left(e(b(End(x0))))
Right6(b(End(x0))) → Left(e(e(End(x0))))
Right7(b(b(End(x0)))) → Left(e(e(End(x0))))
Right8(e(End(x0))) → Left(d(e(End(x0))))
Right9(e(e(End(x0)))) → Left(d(e(End(x0))))
Right10(c(End(x0))) → Left(c(End(x0)))
Right11(c(d(End(x0)))) → Left(c(End(x0)))
Right12(d(End(x0))) → Left(c(d(a(End(x0)))))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Right6(a(x0)) → Aa(Right6(x0))
Right7(a(x0)) → Aa(Right7(x0))
Right8(a(x0)) → Aa(Right8(x0))
Right9(a(x0)) → Aa(Right9(x0))
Right10(a(x0)) → Aa(Right10(x0))
Right11(a(x0)) → Aa(Right11(x0))
Right12(a(x0)) → Aa(Right12(x0))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right6(b(x0)) → Ab(Right6(x0))
Right7(b(x0)) → Ab(Right7(x0))
Right8(b(x0)) → Ab(Right8(x0))
Right9(b(x0)) → Ab(Right9(x0))
Right10(b(x0)) → Ab(Right10(x0))
Right11(b(x0)) → Ab(Right11(x0))
Right12(b(x0)) → Ab(Right12(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right6(c(x0)) → Ac(Right6(x0))
Right7(c(x0)) → Ac(Right7(x0))
Right8(c(x0)) → Ac(Right8(x0))
Right9(c(x0)) → Ac(Right9(x0))
Right10(c(x0)) → Ac(Right10(x0))
Right11(c(x0)) → Ac(Right11(x0))
Right12(c(x0)) → Ac(Right12(x0))
Right1(d(x0)) → Ad(Right1(x0))
Right2(d(x0)) → Ad(Right2(x0))
Right3(d(x0)) → Ad(Right3(x0))
Right4(d(x0)) → Ad(Right4(x0))
Right5(d(x0)) → Ad(Right5(x0))
Right6(d(x0)) → Ad(Right6(x0))
Right7(d(x0)) → Ad(Right7(x0))
Right8(d(x0)) → Ad(Right8(x0))
Right9(d(x0)) → Ad(Right9(x0))
Right10(d(x0)) → Ad(Right10(x0))
Right11(d(x0)) → Ad(Right11(x0))
Right12(d(x0)) → Ad(Right12(x0))
Right1(e(x0)) → Ae(Right1(x0))
Right2(e(x0)) → Ae(Right2(x0))
Right3(e(x0)) → Ae(Right3(x0))
Right4(e(x0)) → Ae(Right4(x0))
Right5(e(x0)) → Ae(Right5(x0))
Right6(e(x0)) → Ae(Right6(x0))
Right7(e(x0)) → Ae(Right7(x0))
Right8(e(x0)) → Ae(Right8(x0))
Right9(e(x0)) → Ae(Right9(x0))
Right10(e(x0)) → Ae(Right10(x0))
Right11(e(x0)) → Ae(Right11(x0))
Right12(e(x0)) → Ae(Right12(x0))
Aa(Left(x0)) → Left(a(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Ad(Left(x0)) → Left(d(x0))
Ae(Left(x0)) → Left(e(x0))
Wait(Left(x0)) → Begin(x0)
a(a(b(x0))) → c(d(x0))
b(e(b(x0))) → e(d(x0))
b(d(x0)) → e(b(x0))
b(b(b(x0))) → e(e(x0))
e(e(e(x0))) → d(e(x0))
d(x0) → b(e(x0))
c(d(a(x0))) → c(x0)
d(c(x0)) → c(d(a(x0)))
a(x0) → e(b(x0))

Proof

1 Termination Assumption

We assume termination of the following TRS
Begin(a(b(x0))) → Wait(Right1(x0))
Begin(b(x0)) → Wait(Right2(x0))
Begin(e(b(x0))) → Wait(Right3(x0))
Begin(b(x0)) → Wait(Right4(x0))
Begin(d(x0)) → Wait(Right5(x0))
Begin(b(b(x0))) → Wait(Right6(x0))
Begin(b(x0)) → Wait(Right7(x0))
Begin(e(e(x0))) → Wait(Right8(x0))
Begin(e(x0)) → Wait(Right9(x0))
Begin(d(a(x0))) → Wait(Right10(x0))
Begin(a(x0)) → Wait(Right11(x0))
Begin(c(x0)) → Wait(Right12(x0))
Right1(a(End(x0))) → Left(c(d(End(x0))))
Right2(a(a(End(x0)))) → Left(c(d(End(x0))))
Right3(b(End(x0))) → Left(e(d(End(x0))))
Right4(b(e(End(x0)))) → Left(e(d(End(x0))))
Right5(b(End(x0))) → Left(e(b(End(x0))))
Right6(b(End(x0))) → Left(e(e(End(x0))))
Right7(b(b(End(x0)))) → Left(e(e(End(x0))))
Right8(e(End(x0))) → Left(d(e(End(x0))))
Right9(e(e(End(x0)))) → Left(d(e(End(x0))))
Right10(c(End(x0))) → Left(c(End(x0)))
Right11(c(d(End(x0)))) → Left(c(End(x0)))
Right12(d(End(x0))) → Left(c(d(a(End(x0)))))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Right6(a(x0)) → Aa(Right6(x0))
Right7(a(x0)) → Aa(Right7(x0))
Right8(a(x0)) → Aa(Right8(x0))
Right9(a(x0)) → Aa(Right9(x0))
Right10(a(x0)) → Aa(Right10(x0))
Right11(a(x0)) → Aa(Right11(x0))
Right12(a(x0)) → Aa(Right12(x0))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right6(b(x0)) → Ab(Right6(x0))
Right7(b(x0)) → Ab(Right7(x0))
Right8(b(x0)) → Ab(Right8(x0))
Right9(b(x0)) → Ab(Right9(x0))
Right10(b(x0)) → Ab(Right10(x0))
Right11(b(x0)) → Ab(Right11(x0))
Right12(b(x0)) → Ab(Right12(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right6(c(x0)) → Ac(Right6(x0))
Right7(c(x0)) → Ac(Right7(x0))
Right8(c(x0)) → Ac(Right8(x0))
Right9(c(x0)) → Ac(Right9(x0))
Right10(c(x0)) → Ac(Right10(x0))
Right11(c(x0)) → Ac(Right11(x0))
Right12(c(x0)) → Ac(Right12(x0))
Right1(d(x0)) → Ad(Right1(x0))
Right2(d(x0)) → Ad(Right2(x0))
Right3(d(x0)) → Ad(Right3(x0))
Right4(d(x0)) → Ad(Right4(x0))
Right5(d(x0)) → Ad(Right5(x0))
Right6(d(x0)) → Ad(Right6(x0))
Right7(d(x0)) → Ad(Right7(x0))
Right8(d(x0)) → Ad(Right8(x0))
Right9(d(x0)) → Ad(Right9(x0))
Right10(d(x0)) → Ad(Right10(x0))
Right11(d(x0)) → Ad(Right11(x0))
Right12(d(x0)) → Ad(Right12(x0))
Right1(e(x0)) → Ae(Right1(x0))
Right2(e(x0)) → Ae(Right2(x0))
Right3(e(x0)) → Ae(Right3(x0))
Right4(e(x0)) → Ae(Right4(x0))
Right5(e(x0)) → Ae(Right5(x0))
Right6(e(x0)) → Ae(Right6(x0))
Right7(e(x0)) → Ae(Right7(x0))
Right8(e(x0)) → Ae(Right8(x0))
Right9(e(x0)) → Ae(Right9(x0))
Right10(e(x0)) → Ae(Right10(x0))
Right11(e(x0)) → Ae(Right11(x0))
Right12(e(x0)) → Ae(Right12(x0))
Aa(Left(x0)) → Left(a(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Ad(Left(x0)) → Left(d(x0))
Ae(Left(x0)) → Left(e(x0))
Wait(Left(x0)) → Begin(x0)
a(a(b(x0))) → c(d(x0))
b(e(b(x0))) → e(d(x0))
b(d(x0)) → e(b(x0))
b(b(b(x0))) → e(e(x0))
e(e(e(x0))) → d(e(x0))
d(x0) → b(e(x0))
c(d(a(x0))) → c(x0)
d(c(x0)) → c(d(a(x0)))
a(x0) → e(b(x0))