NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(q0(1(x0))) → Wait(Right1(x0))
Begin(1(x0)) → Wait(Right2(x0))
Begin(q0(0(x0))) → Wait(Right3(x0))
Begin(0(x0)) → Wait(Right4(x0))
Begin(q1(1(x0))) → Wait(Right5(x0))
Begin(1(x0)) → Wait(Right6(x0))
Begin(q1(0(x0))) → Wait(Right7(x0))
Begin(0(x0)) → Wait(Right8(x0))
Begin(q1(x0)) → Wait(Right9(x0))
Begin(q2(x0)) → Wait(Right10(x0))
Begin(q2(x0)) → Wait(Right11(x0))
Right1(1(End(x0))) → Left(0(1(q1(End(x0)))))
Right2(1(q0(End(x0)))) → Left(0(1(q1(End(x0)))))
Right3(1(End(x0))) → Left(0(0(q1(End(x0)))))
Right4(1(q0(End(x0)))) → Left(0(0(q1(End(x0)))))
Right5(1(End(x0))) → Left(1(1(q1(End(x0)))))
Right6(1(q1(End(x0)))) → Left(1(1(q1(End(x0)))))
Right7(1(End(x0))) → Left(1(0(q1(End(x0)))))
Right8(1(q1(End(x0)))) → Left(1(0(q1(End(x0)))))
Right9(0(End(x0))) → Left(q2(1(End(x0))))
Right10(1(End(x0))) → Left(q2(1(End(x0))))
Right11(0(End(x0))) → Left(0(q0(End(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))
Right1(q0(x0)) → Aq0(Right1(x0))
Right2(q0(x0)) → Aq0(Right2(x0))
Right3(q0(x0)) → Aq0(Right3(x0))
Right4(q0(x0)) → Aq0(Right4(x0))
Right5(q0(x0)) → Aq0(Right5(x0))
Right6(q0(x0)) → Aq0(Right6(x0))
Right7(q0(x0)) → Aq0(Right7(x0))
Right8(q0(x0)) → Aq0(Right8(x0))
Right9(q0(x0)) → Aq0(Right9(x0))
Right10(q0(x0)) → Aq0(Right10(x0))
Right11(q0(x0)) → Aq0(Right11(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))
Right1(q1(x0)) → Aq1(Right1(x0))
Right2(q1(x0)) → Aq1(Right2(x0))
Right3(q1(x0)) → Aq1(Right3(x0))
Right4(q1(x0)) → Aq1(Right4(x0))
Right5(q1(x0)) → Aq1(Right5(x0))
Right6(q1(x0)) → Aq1(Right6(x0))
Right7(q1(x0)) → Aq1(Right7(x0))
Right8(q1(x0)) → Aq1(Right8(x0))
Right9(q1(x0)) → Aq1(Right9(x0))
Right10(q1(x0)) → Aq1(Right10(x0))
Right11(q1(x0)) → Aq1(Right11(x0))
Right1(q2(x0)) → Aq2(Right1(x0))
Right2(q2(x0)) → Aq2(Right2(x0))
Right3(q2(x0)) → Aq2(Right3(x0))
Right4(q2(x0)) → Aq2(Right4(x0))
Right5(q2(x0)) → Aq2(Right5(x0))
Right6(q2(x0)) → Aq2(Right6(x0))
Right7(q2(x0)) → Aq2(Right7(x0))
Right8(q2(x0)) → Aq2(Right8(x0))
Right9(q2(x0)) → Aq2(Right9(x0))
Right10(q2(x0)) → Aq2(Right10(x0))
Right11(q2(x0)) → Aq2(Right11(x0))
A1(Left(x0)) → Left(1(x0))
Aq0(Left(x0)) → Left(q0(x0))
A0(Left(x0)) → Left(0(x0))
Aq1(Left(x0)) → Left(q1(x0))
Aq2(Left(x0)) → Left(q2(x0))
Wait(Left(x0)) → Begin(x0)
1(q0(1(x0))) → 0(1(q1(x0)))
1(q0(0(x0))) → 0(0(q1(x0)))
1(q1(1(x0))) → 1(1(q1(x0)))
1(q1(0(x0))) → 1(0(q1(x0)))
0(q1(x0)) → q2(1(x0))
1(q2(x0)) → q2(1(x0))
0(q2(x0)) → 0(q0(x0))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(1(1(q1(End(x29627)))))
→ε Wait(Right6(1(q1(End(x29627)))))
→1 Wait(Left(1(1(q1(End(x29627))))))
→ε Begin(1(1(q1(End(x29627)))))
= t3
where t3 = t0σ and σ = {x29627/x29627}