NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(b(x0)) → Wait(Right1(x0))
Begin(b(s(x0))) → Wait(Right2(x0))
Begin(s(x0)) → Wait(Right3(x0))
Begin(b(x0)) → Wait(Right4(x0))
Begin(b(s(x0))) → Wait(Right5(x0))
Begin(s(x0)) → Wait(Right6(x0))
Begin(u(x0)) → Wait(Right7(x0))
Begin(s(x0)) → Wait(Right8(x0))
Begin(u(x0)) → Wait(Right9(x0))
Begin(u(x0)) → Wait(Right10(x0))
Right1(s(End(x0))) → Left(b(s(s(s(End(x0))))))
Right2(s(End(x0))) → Left(b(t(End(x0))))
Right3(s(b(End(x0)))) → Left(b(t(End(x0))))
Right4(t(End(x0))) → Left(b(s(End(x0))))
Right5(t(End(x0))) → Left(u(t(b(End(x0)))))
Right6(t(b(End(x0)))) → Left(u(t(b(End(x0)))))
Right7(b(End(x0))) → Left(b(s(End(x0))))
Right8(t(End(x0))) → Left(t(t(End(x0))))
Right9(t(End(x0))) → Left(u(t(End(x0))))
Right10(s(End(x0))) → Left(s(s(End(x0))))
Right1(s(x0)) → As(Right1(x0))
Right2(s(x0)) → As(Right2(x0))
Right3(s(x0)) → As(Right3(x0))
Right4(s(x0)) → As(Right4(x0))
Right5(s(x0)) → As(Right5(x0))
Right6(s(x0)) → As(Right6(x0))
Right7(s(x0)) → As(Right7(x0))
Right8(s(x0)) → As(Right8(x0))
Right9(s(x0)) → As(Right9(x0))
Right10(s(x0)) → As(Right10(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))
Right1(t(x0)) → At(Right1(x0))
Right2(t(x0)) → At(Right2(x0))
Right3(t(x0)) → At(Right3(x0))
Right4(t(x0)) → At(Right4(x0))
Right5(t(x0)) → At(Right5(x0))
Right6(t(x0)) → At(Right6(x0))
Right7(t(x0)) → At(Right7(x0))
Right8(t(x0)) → At(Right8(x0))
Right9(t(x0)) → At(Right9(x0))
Right10(t(x0)) → At(Right10(x0))
Right1(u(x0)) → Au(Right1(x0))
Right2(u(x0)) → Au(Right2(x0))
Right3(u(x0)) → Au(Right3(x0))
Right4(u(x0)) → Au(Right4(x0))
Right5(u(x0)) → Au(Right5(x0))
Right6(u(x0)) → Au(Right6(x0))
Right7(u(x0)) → Au(Right7(x0))
Right8(u(x0)) → Au(Right8(x0))
Right9(u(x0)) → Au(Right9(x0))
Right10(u(x0)) → Au(Right10(x0))
As(Left(x0)) → Left(s(x0))
Ab(Left(x0)) → Left(b(x0))
At(Left(x0)) → Left(t(x0))
Au(Left(x0)) → Left(u(x0))
Wait(Left(x0)) → Begin(x0)
s(b(x0)) → b(s(s(s(x0))))
s(b(s(x0))) → b(t(x0))
t(b(x0)) → b(s(x0))
t(b(s(x0))) → u(t(b(x0)))
b(u(x0)) → b(s(x0))
t(s(x0)) → t(t(x0))
t(u(x0)) → u(t(x0))
s(u(x0)) → s(s(x0))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(u(t(End(x24036))))
→ε Wait(Right9(t(End(x24036))))
→1 Wait(Left(u(t(End(x24036)))))
→ε Begin(u(t(End(x24036))))
= t3
where t3 = t0σ and σ = {x24036/x24036}