NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(0(x0)) → Wait(Right1(x0))
Begin(1(x0)) → Wait(Right2(x0))
Begin(m(x0)) → Wait(Right3(x0))
Begin(0(x0)) → Wait(Right4(x0))
Begin(1(x0)) → Wait(Right5(x0))
Begin(m(x0)) → Wait(Right6(x0))
Begin(b(x0)) → Wait(Right7(x0))
Begin(b(x0)) → Wait(Right8(x0))
Begin(qr(x0)) → Wait(Right9(x0))
Begin(qr(x0)) → Wait(Right10(x0))
Begin(qr(x0)) → Wait(Right11(x0))
Begin(ql(x0)) → Wait(Right12(x0))
Begin(ql(x0)) → Wait(Right13(x0))
Begin(ql(0(x0))) → Wait(Right14(x0))
Begin(0(x0)) → Wait(Right15(x0))
Begin(ql(1(x0))) → Wait(Right16(x0))
Begin(1(x0)) → Wait(Right17(x0))
Right1(r0(End(x0))) → Left(0(r0(End(x0))))
Right2(r0(End(x0))) → Left(1(r0(End(x0))))
Right3(r0(End(x0))) → Left(m(r0(End(x0))))
Right4(r1(End(x0))) → Left(0(r1(End(x0))))
Right5(r1(End(x0))) → Left(1(r1(End(x0))))
Right6(r1(End(x0))) → Left(m(r1(End(x0))))
Right7(r0(End(x0))) → Left(qr(0(b(End(x0)))))
Right8(r1(End(x0))) → Left(qr(1(b(End(x0)))))
Right9(0(End(x0))) → Left(qr(0(End(x0))))
Right10(1(End(x0))) → Left(qr(1(End(x0))))
Right11(m(End(x0))) → Left(ql(m(End(x0))))
Right12(0(End(x0))) → Left(ql(0(End(x0))))
Right13(1(End(x0))) → Left(ql(1(End(x0))))
Right14(b(End(x0))) → Left(0(b(r0(End(x0)))))
Right15(b(ql(End(x0)))) → Left(0(b(r0(End(x0)))))
Right16(b(End(x0))) → Left(1(b(r1(End(x0)))))
Right17(b(ql(End(x0)))) → Left(1(b(r1(End(x0)))))
Right1(r0(x0)) → Ar0(Right1(x0))
Right2(r0(x0)) → Ar0(Right2(x0))
Right3(r0(x0)) → Ar0(Right3(x0))
Right4(r0(x0)) → Ar0(Right4(x0))
Right5(r0(x0)) → Ar0(Right5(x0))
Right6(r0(x0)) → Ar0(Right6(x0))
Right7(r0(x0)) → Ar0(Right7(x0))
Right8(r0(x0)) → Ar0(Right8(x0))
Right9(r0(x0)) → Ar0(Right9(x0))
Right10(r0(x0)) → Ar0(Right10(x0))
Right11(r0(x0)) → Ar0(Right11(x0))
Right12(r0(x0)) → Ar0(Right12(x0))
Right13(r0(x0)) → Ar0(Right13(x0))
Right14(r0(x0)) → Ar0(Right14(x0))
Right15(r0(x0)) → Ar0(Right15(x0))
Right16(r0(x0)) → Ar0(Right16(x0))
Right17(r0(x0)) → Ar0(Right17(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))
Right12(0(x0)) → A0(Right12(x0))
Right13(0(x0)) → A0(Right13(x0))
Right14(0(x0)) → A0(Right14(x0))
Right15(0(x0)) → A0(Right15(x0))
Right16(0(x0)) → A0(Right16(x0))
Right17(0(x0)) → A0(Right17(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))
Right12(1(x0)) → A1(Right12(x0))
Right13(1(x0)) → A1(Right13(x0))
Right14(1(x0)) → A1(Right14(x0))
Right15(1(x0)) → A1(Right15(x0))
Right16(1(x0)) → A1(Right16(x0))
Right17(1(x0)) → A1(Right17(x0))
Right1(m(x0)) → Am(Right1(x0))
Right2(m(x0)) → Am(Right2(x0))
Right3(m(x0)) → Am(Right3(x0))
Right4(m(x0)) → Am(Right4(x0))
Right5(m(x0)) → Am(Right5(x0))
Right6(m(x0)) → Am(Right6(x0))
Right7(m(x0)) → Am(Right7(x0))
Right8(m(x0)) → Am(Right8(x0))
Right9(m(x0)) → Am(Right9(x0))
Right10(m(x0)) → Am(Right10(x0))
Right11(m(x0)) → Am(Right11(x0))
Right12(m(x0)) → Am(Right12(x0))
Right13(m(x0)) → Am(Right13(x0))
Right14(m(x0)) → Am(Right14(x0))
Right15(m(x0)) → Am(Right15(x0))
Right16(m(x0)) → Am(Right16(x0))
Right17(m(x0)) → Am(Right17(x0))
Right1(r1(x0)) → Ar1(Right1(x0))
Right2(r1(x0)) → Ar1(Right2(x0))
Right3(r1(x0)) → Ar1(Right3(x0))
Right4(r1(x0)) → Ar1(Right4(x0))
Right5(r1(x0)) → Ar1(Right5(x0))
Right6(r1(x0)) → Ar1(Right6(x0))
Right7(r1(x0)) → Ar1(Right7(x0))
Right8(r1(x0)) → Ar1(Right8(x0))
Right9(r1(x0)) → Ar1(Right9(x0))
Right10(r1(x0)) → Ar1(Right10(x0))
Right11(r1(x0)) → Ar1(Right11(x0))
Right12(r1(x0)) → Ar1(Right12(x0))
Right13(r1(x0)) → Ar1(Right13(x0))
Right14(r1(x0)) → Ar1(Right14(x0))
Right15(r1(x0)) → Ar1(Right15(x0))
Right16(r1(x0)) → Ar1(Right16(x0))
Right17(r1(x0)) → Ar1(Right17(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))
Right13(b(x0)) → Ab(Right13(x0))
Right14(b(x0)) → Ab(Right14(x0))
Right15(b(x0)) → Ab(Right15(x0))
Right16(b(x0)) → Ab(Right16(x0))
Right17(b(x0)) → Ab(Right17(x0))
Right1(qr(x0)) → Aqr(Right1(x0))
Right2(qr(x0)) → Aqr(Right2(x0))
Right3(qr(x0)) → Aqr(Right3(x0))
Right4(qr(x0)) → Aqr(Right4(x0))
Right5(qr(x0)) → Aqr(Right5(x0))
Right6(qr(x0)) → Aqr(Right6(x0))
Right7(qr(x0)) → Aqr(Right7(x0))
Right8(qr(x0)) → Aqr(Right8(x0))
Right9(qr(x0)) → Aqr(Right9(x0))
Right10(qr(x0)) → Aqr(Right10(x0))
Right11(qr(x0)) → Aqr(Right11(x0))
Right12(qr(x0)) → Aqr(Right12(x0))
Right13(qr(x0)) → Aqr(Right13(x0))
Right14(qr(x0)) → Aqr(Right14(x0))
Right15(qr(x0)) → Aqr(Right15(x0))
Right16(qr(x0)) → Aqr(Right16(x0))
Right17(qr(x0)) → Aqr(Right17(x0))
Right1(ql(x0)) → Aql(Right1(x0))
Right2(ql(x0)) → Aql(Right2(x0))
Right3(ql(x0)) → Aql(Right3(x0))
Right4(ql(x0)) → Aql(Right4(x0))
Right5(ql(x0)) → Aql(Right5(x0))
Right6(ql(x0)) → Aql(Right6(x0))
Right7(ql(x0)) → Aql(Right7(x0))
Right8(ql(x0)) → Aql(Right8(x0))
Right9(ql(x0)) → Aql(Right9(x0))
Right10(ql(x0)) → Aql(Right10(x0))
Right11(ql(x0)) → Aql(Right11(x0))
Right12(ql(x0)) → Aql(Right12(x0))
Right13(ql(x0)) → Aql(Right13(x0))
Right14(ql(x0)) → Aql(Right14(x0))
Right15(ql(x0)) → Aql(Right15(x0))
Right16(ql(x0)) → Aql(Right16(x0))
Right17(ql(x0)) → Aql(Right17(x0))
Ar0(Left(x0)) → Left(r0(x0))
A0(Left(x0)) → Left(0(x0))
A1(Left(x0)) → Left(1(x0))
Am(Left(x0)) → Left(m(x0))
Ar1(Left(x0)) → Left(r1(x0))
Ab(Left(x0)) → Left(b(x0))
Aqr(Left(x0)) → Left(qr(x0))
Aql(Left(x0)) → Left(ql(x0))
Wait(Left(x0)) → Begin(x0)
r0(0(x0)) → 0(r0(x0))
r0(1(x0)) → 1(r0(x0))
r0(m(x0)) → m(r0(x0))
r1(0(x0)) → 0(r1(x0))
r1(1(x0)) → 1(r1(x0))
r1(m(x0)) → m(r1(x0))
r0(b(x0)) → qr(0(b(x0)))
r1(b(x0)) → qr(1(b(x0)))
0(qr(x0)) → qr(0(x0))
1(qr(x0)) → qr(1(x0))
m(qr(x0)) → ql(m(x0))
0(ql(x0)) → ql(0(x0))
1(ql(x0)) → ql(1(x0))
b(ql(0(x0))) → 0(b(r0(x0)))
b(ql(1(x0))) → 1(b(r1(x0)))

Proof

1 Loop

The following loop proves nontermination.

t0 = Begin(0(r0(End(x83826))))
→ε Wait(Right1(r0(End(x83826))))
→1 Wait(Left(0(r0(End(x83826)))))
→ε Begin(0(r0(End(x83826))))
= t3
where t3 = t0σ and σ = {x83826/x83826}