YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(b(c(a(b(c(x0)))))) → Wait(Right1(x0))
Begin(c(a(b(c(x0))))) → Wait(Right2(x0))
Begin(a(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Begin(c(x0)) → Wait(Right5(x0))
Right1(b(End(x0))) → Left(a(b(b(c(b(c(a(End(x0)))))))))
Right2(b(b(End(x0)))) → Left(a(b(b(c(b(c(a(End(x0)))))))))
Right3(b(b(c(End(x0))))) → Left(a(b(b(c(b(c(a(End(x0)))))))))
Right4(b(b(c(a(End(x0)))))) → Left(a(b(b(c(b(c(a(End(x0)))))))))
Right5(b(b(c(a(b(End(x0))))))) → Left(a(b(b(c(b(c(a(End(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))
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))
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))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(b(c(a(b(c(x0)))))) → a(b(b(c(b(c(a(x0)))))))

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
c(b(a(c(b(Begin(x0)))))) → Right1(Wait(x0))
c(b(a(c(Begin(x0))))) → Right2(Wait(x0))
c(b(a(Begin(x0)))) → Right3(Wait(x0))
c(b(Begin(x0))) → Right4(Wait(x0))
c(Begin(x0)) → Right5(Wait(x0))
End(b(Right1(x0))) → End(a(c(b(c(b(b(a(Left(x0)))))))))
End(b(b(Right2(x0)))) → End(a(c(b(c(b(b(a(Left(x0)))))))))
End(c(b(b(Right3(x0))))) → End(a(c(b(c(b(b(a(Left(x0)))))))))
End(a(c(b(b(Right4(x0)))))) → End(a(c(b(c(b(b(a(Left(x0)))))))))
End(b(a(c(b(b(Right5(x0))))))) → End(a(c(b(c(b(b(a(Left(x0)))))))))
b(Right1(x0)) → Right1(Ab(x0))
b(Right2(x0)) → Right2(Ab(x0))
b(Right3(x0)) → Right3(Ab(x0))
b(Right4(x0)) → Right4(Ab(x0))
b(Right5(x0)) → Right5(Ab(x0))
c(Right1(x0)) → Right1(Ac(x0))
c(Right2(x0)) → Right2(Ac(x0))
c(Right3(x0)) → Right3(Ac(x0))
c(Right4(x0)) → Right4(Ac(x0))
c(Right5(x0)) → Right5(Ac(x0))
a(Right1(x0)) → Right1(Aa(x0))
a(Right2(x0)) → Right2(Aa(x0))
a(Right3(x0)) → Right3(Aa(x0))
a(Right4(x0)) → Right4(Aa(x0))
a(Right5(x0)) → Right5(Aa(x0))
Left(Ab(x0)) → b(Left(x0))
Left(Ac(x0)) → c(Left(x0))
Left(Aa(x0)) → a(Left(x0))
Left(Wait(x0)) → Begin(x0)
c(b(a(c(b(b(x0)))))) → a(c(b(c(b(b(a(x0)))))))

1.1 Bounds

The given TRS is match-bounded by 2. This is shown by the following automaton.