MAYBE Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(a(a(b(b(x0))))) → Wait(Right1(x0))
Begin(a(b(b(x0)))) → Wait(Right2(x0))
Begin(b(b(x0))) → Wait(Right3(x0))
Begin(b(x0)) → Wait(Right4(x0))
Begin(a(a(a(b(x0))))) → Wait(Right5(x0))
Begin(a(a(b(x0)))) → Wait(Right6(x0))
Begin(a(b(x0))) → Wait(Right7(x0))
Begin(b(x0)) → Wait(Right8(x0))
Begin(a(a(x0))) → Wait(Right9(x0))
Begin(a(x0)) → Wait(Right10(x0))
Begin(b(x0)) → Wait(Right11(x0))
Right1(a(End(x0))) → Left(b(b(b(End(x0)))))
Right2(a(a(End(x0)))) → Left(b(b(b(End(x0)))))
Right3(a(a(a(End(x0))))) → Left(b(b(b(End(x0)))))
Right4(a(a(a(b(End(x0)))))) → Left(b(b(b(End(x0)))))
Right5(b(End(x0))) → Left(a(a(a(b(a(a(a(End(x0)))))))))
Right6(b(a(End(x0)))) → Left(a(a(a(b(a(a(a(End(x0)))))))))
Right7(b(a(a(End(x0))))) → Left(a(a(a(b(a(a(a(End(x0)))))))))
Right8(b(a(a(a(End(x0)))))) → Left(a(a(a(b(a(a(a(End(x0)))))))))
Right9(a(End(x0))) → Left(a(a(End(x0))))
Right10(a(a(End(x0)))) → Left(a(a(End(x0))))
Right11(b(End(x0))) → Left(a(b(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))
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))
Aa(Left(x0)) → Left(a(x0))
Ab(Left(x0)) → Left(b(x0))
Wait(Left(x0)) → Begin(x0)
a(a(a(b(b(x0))))) → b(b(b(x0)))
b(a(a(a(b(x0))))) → a(a(a(b(a(a(a(x0)))))))
a(a(a(x0))) → a(a(x0))
b(b(x0)) → a(b(a(x0)))

Proof

1 Termination Assumption

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