NO Nontermination Proof

Nontermination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(2(x0))	→	Wait(Right1(x0))
Begin(3(x0))	→	Wait(Right2(x0))
Begin(1(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(L(x0))	→	Wait(Right5(x0))
Begin(L(x0))	→	Wait(Right6(x0))
Begin(b(x0))	→	Wait(Right7(x0))
Begin(c(x0))	→	Wait(Right8(x0))
Begin(c(1(x0)))	→	Wait(Right9(x0))
Begin(1(x0))	→	Wait(Right10(x0))
Begin(c(0(x0)))	→	Wait(Right11(x0))
Begin(0(x0))	→	Wait(Right12(x0))
Right1(R(End(x0)))	→	Left(2(R(End(x0))))
Right2(R(End(x0)))	→	Left(3(R(End(x0))))
Right3(R(End(x0)))	→	Left(L(3(End(x0))))
Right4(3(End(x0)))	→	Left(L(3(End(x0))))
Right5(2(End(x0)))	→	Left(L(2(End(x0))))
Right6(0(End(x0)))	→	Left(2(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(1(b(End(x0)))))
Right8(3(End(x0)))	→	Left(c(1(End(x0))))
Right9(2(End(x0)))	→	Left(c(0(R(1(End(x0))))))
Right10(2(c(End(x0))))	→	Left(c(0(R(1(End(x0))))))
Right11(2(End(x0)))	→	Left(c(0(0(End(x0)))))
Right12(2(c(End(x0))))	→	Left(c(0(0(End(x0)))))
Right1(R(x0))	→	AR(Right1(x0))
Right2(R(x0))	→	AR(Right2(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right4(R(x0))	→	AR(Right4(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right8(R(x0))	→	AR(Right8(x0))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right12(R(x0))	→	AR(Right12(x0))
Right1(2(x0))	→	A2(Right1(x0))
Right2(2(x0))	→	A2(Right2(x0))
Right3(2(x0))	→	A2(Right3(x0))
Right4(2(x0))	→	A2(Right4(x0))
Right5(2(x0))	→	A2(Right5(x0))
Right6(2(x0))	→	A2(Right6(x0))
Right7(2(x0))	→	A2(Right7(x0))
Right8(2(x0))	→	A2(Right8(x0))
Right9(2(x0))	→	A2(Right9(x0))
Right10(2(x0))	→	A2(Right10(x0))
Right11(2(x0))	→	A2(Right11(x0))
Right12(2(x0))	→	A2(Right12(x0))
Right1(3(x0))	→	A3(Right1(x0))
Right2(3(x0))	→	A3(Right2(x0))
Right3(3(x0))	→	A3(Right3(x0))
Right4(3(x0))	→	A3(Right4(x0))
Right5(3(x0))	→	A3(Right5(x0))
Right6(3(x0))	→	A3(Right6(x0))
Right7(3(x0))	→	A3(Right7(x0))
Right8(3(x0))	→	A3(Right8(x0))
Right9(3(x0))	→	A3(Right9(x0))
Right10(3(x0))	→	A3(Right10(x0))
Right11(3(x0))	→	A3(Right11(x0))
Right12(3(x0))	→	A3(Right12(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))
Right1(L(x0))	→	AL(Right1(x0))
Right2(L(x0))	→	AL(Right2(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right4(L(x0))	→	AL(Right4(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right8(L(x0))	→	AL(Right8(x0))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right12(L(x0))	→	AL(Right12(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))
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))
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))
Right6(c(x0))	→	Ac(Right6(x0))
Right7(c(x0))	→	Ac(Right7(x0))
Right8(c(x0))	→	Ac(Right8(x0))
Right9(c(x0))	→	Ac(Right9(x0))
Right10(c(x0))	→	Ac(Right10(x0))
Right11(c(x0))	→	Ac(Right11(x0))
Right12(c(x0))	→	Ac(Right12(x0))
AR(Left(x0))	→	Left(R(x0))
A2(Left(x0))	→	Left(2(x0))
A3(Left(x0))	→	Left(3(x0))
A1(Left(x0))	→	Left(1(x0))
AL(Left(x0))	→	Left(L(x0))
A0(Left(x0))	→	Left(0(x0))
Ab(Left(x0))	→	Left(b(x0))
Ac(Left(x0))	→	Left(c(x0))
Wait(Left(x0))	→	Begin(x0)
R(2(x0))	→	2(R(x0))
R(3(x0))	→	3(R(x0))
R(1(x0))	→	L(3(x0))
3(L(x0))	→	L(3(x0))
2(L(x0))	→	L(2(x0))
0(L(x0))	→	2(R(x0))
R(b(x0))	→	c(1(b(x0)))
3(c(x0))	→	c(1(x0))
2(c(1(x0)))	→	c(0(R(1(x0))))
2(c(0(x0)))	→	c(0(0(x0)))

Proof

1 Loop

The following loop proves nontermination.

t₀ = Begin(2(R(End(x43520))))

→_ε Wait(Right1(R(End(x43520))))

→₁ Wait(Left(2(R(End(x43520)))))

→_ε Begin(2(R(End(x43520))))

= t₃

where t₃ = t₀σ and σ = {x43520/x43520}

t₀	=	Begin(2(R(End(x43520))))
	→_ε	Wait(Right1(R(End(x43520))))
	→₁	Wait(Left(2(R(End(x43520)))))
	→_ε	Begin(2(R(End(x43520))))
	=	t₃