YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right2(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(b(L(x0)))	→	Wait(Right8(x0))
Begin(L(x0))	→	Wait(Right9(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right2(a(End(x0)))	→	Left(L(Aa(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right4(c(End(x0)))	→	Left(L(Ac(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right8(a(End(x0)))	→	Left(b(a(a(R(End(x0))))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right10(c(End(x0)))	→	Left(b(b(c(R(End(x0))))))
Right11(c(b(End(x0))))	→	Left(b(b(c(R(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))
Right1(E(x0))	→	AE(Right1(x0))
Right2(E(x0))	→	AE(Right2(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right8(E(x0))	→	AE(Right8(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(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))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right2(c(x0))	→	AAc(Right2(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right8(c(x0))	→	AAc(Right8(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right2(Ac(x0))	→	AAAc(Right2(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right8(Ac(x0))	→	AAAc(Right8(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Begin(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Wait(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AL(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[End(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	1
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	1
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
1	0	0

[AE(x₁)]

1	0	1
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AR(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right2(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(b(L(x0)))	→	Wait(Right8(x0))
Begin(L(x0))	→	Wait(Right9(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right4(c(End(x0)))	→	Left(L(Ac(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right10(c(End(x0)))	→	Left(b(b(c(R(End(x0))))))
Right11(c(b(End(x0))))	→	Left(b(b(c(R(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))
Right1(E(x0))	→	AE(Right1(x0))
Right2(E(x0))	→	AE(Right2(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right8(E(x0))	→	AE(Right8(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(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))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right2(c(x0))	→	AAc(Right2(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right8(c(x0))	→	AAc(Right8(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right2(Ac(x0))	→	AAAc(Right2(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right8(Ac(x0))	→	AAAc(Right8(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	2 · x₁ + -∞
[L(x₁)]	=	2 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	2 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	2 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	2 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	2 · x₁ + -∞
[R(x₁)]	=	2 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	2 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	2 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(L(x0))	→	Wait(Right9(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right4(c(End(x0)))	→	Left(L(Ac(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right10(c(End(x0)))	→	Left(b(b(c(R(End(x0))))))
Right11(c(b(End(x0))))	→	Left(b(b(c(R(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))
Right1(E(x0))	→	AE(Right1(x0))
Right2(E(x0))	→	AE(Right2(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right8(E(x0))	→	AE(Right8(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(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))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right2(c(x0))	→	AAc(Right2(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right8(c(x0))	→	AAc(Right8(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right2(Ac(x0))	→	AAAc(Right2(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right8(Ac(x0))	→	AAAc(Right8(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
L(Begin(x0))	→	Right4(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
L(Begin(x0))	→	Right9(Wait(x0))
L(b(Begin(x0)))	→	Right10(Wait(x0))
L(Begin(x0))	→	Right11(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(c(Right4(x0)))	→	End(Ac(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
End(b(a(Right9(x0))))	→	End(R(a(a(b(Left(x0))))))
End(c(Right10(x0)))	→	End(R(c(b(b(Left(x0))))))
End(b(c(Right11(x0))))	→	End(R(c(b(b(Left(x0))))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right2(x0))	→	Right2(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right4(x0))	→	Right4(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right8(x0))	→	Right8(AR(x0))
R(Right9(x0))	→	Right9(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
R(Right11(x0))	→	Right11(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right2(x0))	→	Right2(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right8(x0))	→	Right8(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right2(x0))	→	Right2(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right4(x0))	→	Right4(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right8(x0))	→	Right8(AL(x0))
L(Right9(x0))	→	Right9(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
L(Right11(x0))	→	Right11(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right2(x0))	→	Right2(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right4(x0))	→	Right4(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right8(x0))	→	Right8(AAc(x0))
c(Right9(x0))	→	Right9(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
c(Right11(x0))	→	Right11(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right2(x0))	→	Right2(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right4(x0))	→	Right4(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right8(x0))	→	Right8(AAAc(x0))
Ac(Right9(x0))	→	Right9(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Ac(Right11(x0))	→	Right11(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 0
[L(x₁)]	=	4 · x₁ + 0
[Right8(x₁)]	=	4 · x₁ + 2
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 0
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 0
[AAAc(x₁)]	=	4 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 4
[Right10(x₁)]	=	4 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	4 · x₁ + 0
[Right4(x₁)]	=	4 · x₁ + 0
[R(x₁)]	=	4 · x₁ + 0
[AE(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	4 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	4 · x₁ + 0

the rules

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
L(Begin(x0))	→	Right4(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
L(Begin(x0))	→	Right9(Wait(x0))
L(b(Begin(x0)))	→	Right10(Wait(x0))
L(Begin(x0))	→	Right11(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(c(Right4(x0)))	→	End(Ac(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
End(b(a(Right9(x0))))	→	End(R(a(a(b(Left(x0))))))
End(c(Right10(x0)))	→	End(R(c(b(b(Left(x0))))))
End(b(c(Right11(x0))))	→	End(R(c(b(b(Left(x0))))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right2(x0))	→	Right2(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right4(x0))	→	Right4(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right9(x0))	→	Right9(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
R(Right11(x0))	→	Right11(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right2(x0))	→	Right2(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right2(x0))	→	Right2(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right4(x0))	→	Right4(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right9(x0))	→	Right9(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
L(Right11(x0))	→	Right11(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right2(x0))	→	Right2(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right4(x0))	→	Right4(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right9(x0))	→	Right9(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
c(Right11(x0))	→	Right11(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right2(x0))	→	Right2(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right4(x0))	→	Right4(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right9(x0))	→	Right9(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Ac(Right11(x0))	→	Right11(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 0
[L(x₁)]	=	1 · x₁ + 0
[Right8(x₁)]	=	4 · x₁ + 1
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	8 · x₁ + 0
[Wait(x₁)]	=	2 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 0
[Left(x₁)]	=	4 · x₁ + 0
[AL(x₁)]	=	1 · x₁ + 0
[AAAc(x₁)]	=	2 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	4 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 8
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	4 · x₁ + 0
[R(x₁)]	=	1 · x₁ + 0
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 0
[Ac(x₁)]	=	2 · x₁ + 0
[c(x₁)]	=	2 · x₁ + 0
[AR(x₁)]	=	1 · x₁ + 0
[Right6(x₁)]	=	4 · x₁ + 0
[Right5(x₁)]	=	4 · x₁ + 0
[Right7(x₁)]	=	8 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	4 · x₁ + 0

the rules

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
L(Begin(x0))	→	Right4(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
L(Begin(x0))	→	Right9(Wait(x0))
L(b(Begin(x0)))	→	Right10(Wait(x0))
L(Begin(x0))	→	Right11(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(c(Right4(x0)))	→	End(Ac(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
End(b(a(Right9(x0))))	→	End(R(a(a(b(Left(x0))))))
End(c(Right10(x0)))	→	End(R(c(b(b(Left(x0))))))
End(b(c(Right11(x0))))	→	End(R(c(b(b(Left(x0))))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right2(x0))	→	Right2(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right4(x0))	→	Right4(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right9(x0))	→	Right9(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
R(Right11(x0))	→	Right11(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right2(x0))	→	Right2(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right2(x0))	→	Right2(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right4(x0))	→	Right4(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right9(x0))	→	Right9(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
L(Right11(x0))	→	Right11(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right4(x0))	→	Right4(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right9(x0))	→	Right9(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
c(Right11(x0))	→	Right11(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right4(x0))	→	Right4(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right9(x0))	→	Right9(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Ac(Right11(x0))	→	Right11(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(L(x0))	→	Wait(Right9(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right4(c(End(x0)))	→	Left(L(Ac(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right10(c(End(x0)))	→	Left(b(b(c(R(End(x0))))))
Right11(c(b(End(x0))))	→	Left(b(b(c(R(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))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right2(E(x0))	→	AE(Right2(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(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))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 3
[L(x₁)]	=	4 · x₁ + 3
[Right8(x₁)]	=	1 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 3
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 3
[AAAc(x₁)]	=	4 · x₁ + 3
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	4 · x₁ + 3
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	6 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	2 · x₁ + 1
[Right4(x₁)]	=	4 · x₁ + 3
[R(x₁)]	=	4 · x₁ + 3
[AE(x₁)]	=	2 · x₁ + 1
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 3
[Ac(x₁)]	=	4 · x₁ + 3
[c(x₁)]	=	4 · x₁ + 3
[AR(x₁)]	=	4 · x₁ + 3
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 3
[Right3(x₁)]	=	4 · x₁ + 3
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	2 · x₁ + 1

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(L(x0))	→	Wait(Right9(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right4(c(End(x0)))	→	Left(L(Ac(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right10(c(End(x0)))	→	Left(b(b(c(R(End(x0))))))
Right11(c(b(End(x0))))	→	Left(b(b(c(R(End(x0))))))
Right1(R(x0))	→	AR(Right1(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))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(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))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[L(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Begin(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
0	0	0

[Wait(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	1	0
0	0	1
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[AL(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	1	0
0	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[End(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right10(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[AAa(x₁)]

1	0	1
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AE(x₁)]

1	0	1
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Ac(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[c(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[AR(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right5(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right7(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
0	0	0

[Right3(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right9(a(b(End(x0))))	→	Left(b(a(a(R(End(x0))))))
Right1(R(x0))	→	AR(Right1(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))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(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))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	1 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	0 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(L(x0))	→	Wait(Right4(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Begin(b(L(x0)))	→	Wait(Right10(x0))
Begin(L(x0))	→	Wait(Right11(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(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))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(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))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	2 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	2 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	0 · x₁ + -∞
[R(x₁)]	=	2 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	2 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	2 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(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))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(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))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right4(c(x0))	→	AAc(Right4(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right4(Ac(x0))	→	AAAc(Right4(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right4(x0))	→	Right4(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right9(x0))	→	Right9(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
R(Right11(x0))	→	Right11(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right4(x0))	→	Right4(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right9(x0))	→	Right9(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
L(Right11(x0))	→	Right11(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right4(x0))	→	Right4(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right9(x0))	→	Right9(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
c(Right11(x0))	→	Right11(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right4(x0))	→	Right4(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right9(x0))	→	Right9(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Ac(Right11(x0))	→	Right11(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	1 · x₁ + 0
[L(x₁)]	=	4 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	1 · x₁ + 0
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 0
[AAAc(x₁)]	=	4 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 9
[Right10(x₁)]	=	4 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	8 · x₁ + 3
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	2 · x₁ + 1
[R(x₁)]	=	4 · x₁ + 0
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	4 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 0

the rules

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right9(x0))	→	Right9(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
R(Right11(x0))	→	Right11(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right9(x0))	→	Right9(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
L(Right11(x0))	→	Right11(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right9(x0))	→	Right9(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
c(Right11(x0))	→	Right11(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right9(x0))	→	Right9(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Ac(Right11(x0))	→	Right11(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right9(R(x0))	→	AR(Right9(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right11(R(x0))	→	AR(Right11(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right9(L(x0))	→	AL(Right9(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right11(L(x0))	→	AL(Right11(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right9(c(x0))	→	AAc(Right9(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right11(c(x0))	→	AAc(Right11(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right9(Ac(x0))	→	AAAc(Right9(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
Right11(Ac(x0))	→	AAAc(Right11(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 0
[L(x₁)]	=	4 · x₁ + 2
[Right8(x₁)]	=	2 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	2 · x₁ + 0
[Wait(x₁)]	=	2 · x₁ + 0
[Right9(x₁)]	=	5 · x₁ + 2
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 2
[AAAc(x₁)]	=	4 · x₁ + 2
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	4 · x₁ + 2
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	7 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	1 · x₁ + 0
[R(x₁)]	=	4 · x₁ + 2
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 2
[Ac(x₁)]	=	4 · x₁ + 2
[c(x₁)]	=	4 · x₁ + 2
[AR(x₁)]	=	4 · x₁ + 2
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 2
[Right3(x₁)]	=	4 · x₁ + 2
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 0

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right4(E(x0))	→	AE(Right4(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right9(E(x0))	→	AE(Right9(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right11(E(x0))	→	AE(Right11(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right4(x0))	→	Right4(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right9(x0))	→	Right9(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
E(Right11(x0))	→	Right11(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 6
[L(x₁)]	=	4 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 8
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	1 · x₁ + 5
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 0
[AAAc(x₁)]	=	4 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	1 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	4 · x₁ + 0
[Right4(x₁)]	=	2 · x₁ + 2
[R(x₁)]	=	4 · x₁ + 0
[AE(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	4 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	4 · x₁ + 0

the rules

E(Begin(x0))	→	Right1(Wait(x0))
L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right1(x0)))	→	End(E(L(Left(x0))))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
R(Right10(x0))	→	Right10(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
E(Right10(x0))	→	Right10(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
L(Right10(x0))	→	Right10(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
c(Right10(x0))	→	Right10(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Ac(Right10(x0))	→	Right10(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right10(R(x0))	→	AR(Right10(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right10(E(x0))	→	AE(Right10(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right10(L(x0))	→	AL(Right10(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right10(c(x0))	→	AAc(Right10(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
Right10(Ac(x0))	→	AAAc(Right10(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 8
[L(x₁)]	=	4 · x₁ + 3
[Right8(x₁)]	=	8 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	3 · x₁ + 8
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 3
[AAAc(x₁)]	=	2 · x₁ + 1
[End(x₁)]	=	2 · x₁ + 0
[Right10(x₁)]	=	4 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	8 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	2 · x₁ + 1
[Right4(x₁)]	=	4 · x₁ + 8
[R(x₁)]	=	4 · x₁ + 3
[AE(x₁)]	=	2 · x₁ + 1
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 1
[Ac(x₁)]	=	2 · x₁ + 1
[c(x₁)]	=	2 · x₁ + 1
[AR(x₁)]	=	4 · x₁ + 3
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	2 · x₁ + 1
[Right3(x₁)]	=	4 · x₁ + 3
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	2 · x₁ + 1

the rules

Begin(E(x0))	→	Wait(Right1(x0))
Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Begin(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Wait(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AL(x₁)]

1	0	0
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[End(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
1	1	1
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	1
0	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[AE(x₁)]

1	0	0
1	1	1
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[c(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[AR(x₁)]

1	0	1
0	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right6(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right1(R(End(x0)))	→	Left(L(E(End(x0))))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	5 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	1 · x₁ + -∞
[Right4(x₁)]	=	0 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	1 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	4 · x₁ + -∞

the rules

Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right1(R(x0))	→	AR(Right1(x0))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right1(L(x0))	→	AL(Right1(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right1(c(x0))	→	AAc(Right1(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right1(Ac(x0))	→	AAAc(Right1(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right1(x0))	→	Right1(AR(x0))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
L(Right1(x0))	→	Right1(AL(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right1(x0))	→	Right1(AAc(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right1(x0))	→	Right1(AAAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	6 · x₁ + 8
[L(x₁)]	=	2 · x₁ + 0
[Right8(x₁)]	=	4 · x₁ + 4
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 5
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	2 · x₁ + 0
[AAAc(x₁)]	=	4 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	8 · x₁ + 5
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	1 · x₁ + 8
[R(x₁)]	=	2 · x₁ + 0
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	2 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 0
[Right3(x₁)]	=	2 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	4 · x₁ + 6

the rules

L(Begin(x0))	→	Right3(Wait(x0))
Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(b(Right3(x0)))	→	End(Ab(L(Left(x0))))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right3(b(End(x0)))	→	Left(L(Ab(End(x0))))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	1
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Begin(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Wait(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AL(x₁)]

1	0	1
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[End(x₁)]

1	0	0
1	0	0
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right10(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	1	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	1	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	1
0	0	0
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AE(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	1	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AR(x₁)]

1	0	1
0	0	0
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	1	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Begin(L(x0))	→	Wait(Right3(x0))
Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	1 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	1 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	8 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	4 · x₁ + -∞
[R(x₁)]	=	1 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	1 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right3(R(x0))	→	AR(Right3(x0))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right3(L(x0))	→	AL(Right3(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right3(c(x0))	→	AAc(Right3(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right3(Ac(x0))	→	AAAc(Right3(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right3(x0))	→	Right3(AR(x0))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
L(Right3(x0))	→	Right3(AL(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right3(x0))	→	Right3(AAc(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right3(x0))	→	Right3(AAAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 0
[L(x₁)]	=	2 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	1 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	8 · x₁ + 0
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	2 · x₁ + 0
[AAAc(x₁)]	=	4 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	2 · x₁ + 2
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	2 · x₁ + 8
[R(x₁)]	=	2 · x₁ + 0
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 0
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	2 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 1
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 0

the rules

Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right1(x0))	→	Right1(AE(x0))
E(Right3(x0))	→	Right3(AE(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right1(E(x0))	→	AE(Right1(x0))
Right3(E(x0))	→	AE(Right3(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	5 · x₁ + 14
[L(x₁)]	=	8 · x₁ + 0
[Right8(x₁)]	=	4 · x₁ + 10
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	4 · x₁ + 0
[Wait(x₁)]	=	4 · x₁ + 0
[Right9(x₁)]	=	1 · x₁ + 0
[Left(x₁)]	=	1 · x₁ + 0
[AL(x₁)]	=	8 · x₁ + 0
[AAAc(x₁)]	=	1 · x₁ + 0
[End(x₁)]	=	3 · x₁ + 0
[Right10(x₁)]	=	9 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	9 · x₁ + 14
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	2 · x₁ + 1
[Right4(x₁)]	=	1 · x₁ + 0
[R(x₁)]	=	8 · x₁ + 0
[AE(x₁)]	=	2 · x₁ + 1
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	1 · x₁ + 0
[Ac(x₁)]	=	1 · x₁ + 0
[c(x₁)]	=	1 · x₁ + 0
[AR(x₁)]	=	8 · x₁ + 0
[Right6(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	1 · x₁ + 0
[Right7(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	12 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	14 · x₁ + 0

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right7(E(x0))	→	AE(Right7(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
E(Right7(x0))	→	Right7(AE(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	11 · x₁ + 0
[L(x₁)]	=	1 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	2 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	2 · x₁ + 0
[Left(x₁)]	=	2 · x₁ + 0
[AL(x₁)]	=	1 · x₁ + 0
[AAAc(x₁)]	=	2 · x₁ + 1
[End(x₁)]	=	2 · x₁ + 0
[Right10(x₁)]	=	8 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	4 · x₁ + 0
[Right4(x₁)]	=	12 · x₁ + 0
[R(x₁)]	=	1 · x₁ + 0
[AE(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 1
[Ac(x₁)]	=	2 · x₁ + 2
[c(x₁)]	=	2 · x₁ + 2
[AR(x₁)]	=	1 · x₁ + 0
[Right6(x₁)]	=	2 · x₁ + 0
[Right5(x₁)]	=	2 · x₁ + 0
[Right7(x₁)]	=	4 · x₁ + 2
[Right3(x₁)]	=	2 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	2 · x₁ + 0

the rules

Aa(Begin(x0))	→	Right5(Wait(x0))
Ab(Begin(x0))	→	Right6(Wait(x0))
Ac(Begin(x0))	→	Right7(Wait(x0))
End(R(Right5(x0)))	→	End(R(a(Left(x0))))
End(R(Right6(x0)))	→	End(R(b(Left(x0))))
End(R(Right7(x0)))	→	End(R(c(Left(x0))))
R(Right5(x0))	→	Right5(AR(x0))
R(Right6(x0))	→	Right6(AR(x0))
R(Right7(x0))	→	Right7(AR(x0))
E(Right5(x0))	→	Right5(AE(x0))
E(Right6(x0))	→	Right6(AE(x0))
L(Right5(x0))	→	Right5(AL(x0))
L(Right6(x0))	→	Right6(AL(x0))
L(Right7(x0))	→	Right7(AL(x0))
a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
c(Right5(x0))	→	Right5(AAc(x0))
c(Right6(x0))	→	Right6(AAc(x0))
c(Right7(x0))	→	Right7(AAc(x0))
Ac(Right5(x0))	→	Right5(AAAc(x0))
Ac(Right6(x0))	→	Right6(AAAc(x0))
Ac(Right7(x0))	→	Right7(AAAc(x0))
Left(AR(x0))	→	R(Left(x0))
Left(AE(x0))	→	E(Left(x0))
Left(AL(x0))	→	L(Left(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
Left(AAc(x0))	→	c(Left(x0))
Left(AAAc(x0))	→	Ac(Left(x0))
Left(Wait(x0))	→	Begin(x0)
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Begin(Ac(x0))	→	Wait(Right7(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right7(R(x0))	→	AR(Right7(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right7(L(x0))	→	AL(Right7(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
Right7(Ac(x0))	→	AAAc(Right7(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 0
[L(x₁)]	=	1 · x₁ + 2
[Right8(x₁)]	=	2 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	8 · x₁ + 0
[Wait(x₁)]	=	2 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 0
[Left(x₁)]	=	4 · x₁ + 0
[AL(x₁)]	=	1 · x₁ + 8
[AAAc(x₁)]	=	2 · x₁ + 8
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	1 · x₁ + 4
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	4 · x₁ + 0
[Right4(x₁)]	=	1 · x₁ + 0
[R(x₁)]	=	1 · x₁ + 2
[AE(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 0
[Ac(x₁)]	=	2 · x₁ + 2
[c(x₁)]	=	2 · x₁ + 0
[AR(x₁)]	=	1 · x₁ + 8
[Right6(x₁)]	=	4 · x₁ + 0
[Right5(x₁)]	=	4 · x₁ + 0
[Right7(x₁)]	=	8 · x₁ + 0
[Right3(x₁)]	=	8 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	8 · x₁ + 0

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right7(R(End(x0)))	→	Left(c(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	2 · x₁ + -∞
[Wait(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	8 · x₁ + -∞
[Left(x₁)]	=	2 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	8 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	1 · x₁ + -∞
[Right4(x₁)]	=	2 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	1 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	2 · x₁ + -∞
[Right5(x₁)]	=	2 · x₁ + -∞
[Right7(x₁)]	=	8 · x₁ + -∞
[Right3(x₁)]	=	10 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	13 · x₁ + -∞

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right7(c(x0))	→	AAc(Right7(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	4 · x₁ + 0
[L(x₁)]	=	2 · x₁ + 1
[Right8(x₁)]	=	2 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	8 · x₁ + 1
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	4 · x₁ + 0
[Left(x₁)]	=	8 · x₁ + 1
[AL(x₁)]	=	2 · x₁ + 7
[AAAc(x₁)]	=	2 · x₁ + 7
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	2 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	2 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	8 · x₁ + 0
[R(x₁)]	=	2 · x₁ + 1
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 7
[Ac(x₁)]	=	2 · x₁ + 1
[c(x₁)]	=	2 · x₁ + 1
[AR(x₁)]	=	2 · x₁ + 7
[Right6(x₁)]	=	8 · x₁ + 1
[Right5(x₁)]	=	8 · x₁ + 1
[Right7(x₁)]	=	10 · x₁ + 1
[Right3(x₁)]	=	1 · x₁ + 8
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	2 · x₁ + 0

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Begin(Ab(x0))	→	Wait(Right6(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
1	1	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	1	0
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Begin(x₁)]

1	1	0
1	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Wait(x₁)]

1	0	1
1	0	0
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AL(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[End(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

[AAa(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
1	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
1	1	0
0	0	1

· x₁ +

0	0	0
1	0	0
1	0	0

[E(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	1
0	1	0

· x₁ +

0	0	0
1	0	0
1	0	0

[R(x₁)]

1	0	0
1	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AE(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[AAAb(x₁)]

1	0	0
0	0	0
1	0	1

· x₁ +

0	0	0
0	0	0
1	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[AR(x₁)]

1	0	0
0	1	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	1
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right5(x₁)]

1	0	0
1	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

[Right3(x₁)]

1	0	0
1	1	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
1	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right6(R(End(x0)))	→	Left(b(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	14 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	4 · x₁ + -∞
[Wait(x₁)]	=	4 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	2 · x₁ + -∞
[End(x₁)]	=	1 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	0 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	2 · x₁ + -∞
[Ac(x₁)]	=	2 · x₁ + -∞
[c(x₁)]	=	2 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	2 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	12 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	4 · x₁ + -∞

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right6(E(x0))	→	AE(Right6(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 0
[L(x₁)]	=	2 · x₁ + 0
[Right8(x₁)]	=	4 · x₁ + 4
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	8 · x₁ + 2
[Wait(x₁)]	=	1 · x₁ + 2
[Right9(x₁)]	=	8 · x₁ + 2
[Left(x₁)]	=	8 · x₁ + 0
[AL(x₁)]	=	2 · x₁ + 0
[AAAc(x₁)]	=	2 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	4 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	1 · x₁ + 8
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 1
[Right4(x₁)]	=	4 · x₁ + 0
[R(x₁)]	=	2 · x₁ + 0
[AE(x₁)]	=	1 · x₁ + 8
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	2 · x₁ + 0
[Ac(x₁)]	=	2 · x₁ + 0
[c(x₁)]	=	2 · x₁ + 0
[AR(x₁)]	=	2 · x₁ + 0
[Right6(x₁)]	=	12 · x₁ + 0
[Right5(x₁)]	=	8 · x₁ + 0
[Right7(x₁)]	=	1 · x₁ + 4
[Right3(x₁)]	=	8 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 7

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right6(R(x0))	→	AR(Right6(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right6(L(x0))	→	AL(Right6(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right6(c(x0))	→	AAc(Right6(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
Right6(Ac(x0))	→	AAAc(Right6(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 2
[L(x₁)]	=	4 · x₁ + 2
[Right8(x₁)]	=	1 · x₁ + 8
[Aa(x₁)]	=	1 · x₁ + 0
[Begin(x₁)]	=	2 · x₁ + 0
[Wait(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	8 · x₁ + 4
[Left(x₁)]	=	2 · x₁ + 0
[AL(x₁)]	=	4 · x₁ + 4
[AAAc(x₁)]	=	4 · x₁ + 4
[End(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	1 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	4 · x₁ + 3
[R(x₁)]	=	4 · x₁ + 2
[AE(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 4
[Ac(x₁)]	=	4 · x₁ + 2
[c(x₁)]	=	4 · x₁ + 2
[AR(x₁)]	=	4 · x₁ + 4
[Right6(x₁)]	=	6 · x₁ + 2
[Right5(x₁)]	=	2 · x₁ + 0
[Right7(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	1 · x₁ + 1
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	8 · x₁ + 2

the rules

Begin(Aa(x0))	→	Wait(Right5(x0))
Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	1	0
1	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Aa(x₁)]

1	0	0
1	1	0
1	0	1

· x₁ +

0	0	0
1	0	0
1	0	0

[Begin(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Wait(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right9(x₁)]

1	0	0
0	0	1
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AL(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAc(x₁)]

1	0	0
1	1	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[End(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Right10(x₁)]

1	0	0
0	0	1
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
1	1	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	1
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
1	1	1
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AE(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
1	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAc(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

1	0	0
1	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	1	0
1	0	1

· x₁ +

1	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
1	1	0
0	0	1

· x₁ +

1	0	0
1	0	0
1	0	0

[AR(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
1	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Right5(R(End(x0)))	→	Left(a(R(End(x0))))
Right5(R(x0))	→	AR(Right5(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	8 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	4 · x₁ + -∞
[Wait(x₁)]	=	4 · x₁ + -∞
[Right9(x₁)]	=	1 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	8 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	4 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	7 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	0 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	8 · x₁ + -∞
[Ac(x₁)]	=	8 · x₁ + -∞
[c(x₁)]	=	8 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	2 · x₁ + -∞
[Right7(x₁)]	=	2 · x₁ + -∞
[Right3(x₁)]	=	2 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Right5(R(x0))	→	AR(Right5(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
Wait(Left(x0))	→	Begin(x0)
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	15 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	2 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	0 · x₁ + -∞
[Wait(x₁)]	=	15 · x₁ + -∞
[Right9(x₁)]	=	1 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	1 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	12 · x₁ + -∞
[Right4(x₁)]	=	8 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	12 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	2 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Right5(R(x0))	→	AR(Right5(x0))
Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
Right5(Ac(x0))	→	AAAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AE(Left(x0))	→	Left(E(x0))
AL(Left(x0))	→	Left(L(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAc(Left(x0))	→	Left(c(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	1 · x₁ + 2
[L(x₁)]	=	4 · x₁ + 4
[Right8(x₁)]	=	8 · x₁ + 1
[Aa(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	2 · x₁ + 14
[Left(x₁)]	=	4 · x₁ + 4
[AL(x₁)]	=	4 · x₁ + 8
[AAAc(x₁)]	=	4 · x₁ + 0
[Right10(x₁)]	=	2 · x₁ + 8
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 0
[E(x₁)]	=	4 · x₁ + 0
[Right4(x₁)]	=	1 · x₁ + 8
[R(x₁)]	=	4 · x₁ + 4
[AE(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 0
[AAc(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	4 · x₁ + 3
[c(x₁)]	=	4 · x₁ + 0
[AR(x₁)]	=	4 · x₁ + 4
[Right6(x₁)]	=	1 · x₁ + 5
[Right5(x₁)]	=	2 · x₁ + 0
[Right7(x₁)]	=	7 · x₁ + 2
[Right3(x₁)]	=	4 · x₁ + 9
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 0

the rules

Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
AAAc(Left(x0))	→	Left(Ac(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[AL(x₁)]	=	0 · x₁ + -∞
[AAAc(x₁)]	=	10 · x₁ + -∞
[Right10(x₁)]	=	4 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	1 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	1 · x₁ + -∞
[Ac(x₁)]	=	1 · x₁ + -∞
[c(x₁)]	=	1 · x₁ + -∞
[AR(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	4 · x₁ + -∞
[Right3(x₁)]	=	2 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Right5(E(x0))	→	AE(Right5(x0))
Right5(L(x0))	→	AL(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
AR(Left(x0))	→	Left(R(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	2 · x₁ + -∞
[Right8(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	3 · x₁ + -∞
[AL(x₁)]	=	1 · x₁ + -∞
[Right10(x₁)]	=	1 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	2 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	9 · x₁ + -∞
[Right4(x₁)]	=	5 · x₁ + -∞
[R(x₁)]	=	2 · x₁ + -∞
[AE(x₁)]	=	9 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[AR(x₁)]	=	12 · x₁ + -∞
[Right6(x₁)]	=	3 · x₁ + -∞
[Right5(x₁)]	=	1 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	11 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	3 · x₁ + -∞

the rules

Right5(E(x0))	→	AE(Right5(x0))
Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	12 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	2 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	6 · x₁ + -∞
[Left(x₁)]	=	1 · x₁ + -∞
[Right10(x₁)]	=	8 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	13 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	2 · x₁ + -∞
[Right4(x₁)]	=	7 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AE(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	2 · x₁ + -∞
[Ac(x₁)]	=	2 · x₁ + -∞
[c(x₁)]	=	2 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	6 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
Right5(c(x0))	→	AAc(Right5(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	1 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Right8(x₁)]	=	2 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Right9(x₁)]	=	5 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[Right10(x₁)]	=	4 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	4 · x₁ + -∞
[Right4(x₁)]	=	12 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[AAAb(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[AAc(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	8 · x₁ + -∞
[c(x₁)]	=	8 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right7(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[AAb(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	8 · x₁ + -∞

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAb(Left(x0))	→	Left(b(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAb(x0))	→	b(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	1	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	1
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	1	0
0	1	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right3(x0))	→	Right3(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
Left(AAa(x0))	→	a(Left(x0))
Left(AAAa(x0))	→	Aa(Left(x0))
Left(AAAb(x0))	→	Ab(Left(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right3(Ab(x0))	→	AAAb(Right3(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
1	1	1
1	1	1

· x₁ +

0	0	0
1	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[AAAa(x₁)]

1	0	1
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
1	0	1

· x₁ +

0	0	0
1	0	0
1	0	0

[E(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[AAAb(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	1
0	0	0
1	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	1
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right3(x₁)]

1	0	1
0	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAa(Left(x0))	→	Left(Aa(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Left(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	1	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	1
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
AAa(Left(x0))	→	Left(a(x0))
AAAb(Left(x0))	→	Left(Ab(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Left(x₁)]

1	1	0
0	0	1
1	0	1

· x₁ +

0	0	0
0	0	0
1	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	1
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[AAb(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
AAa(Left(x0))	→	Left(a(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
Left(AAa(x0))	→	a(Left(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

1	1	1
1	0	0
0	0	1

· x₁ +

1	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	1	1
1	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[a(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
1	0	1
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAb(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
b(Right1(x0))	→	Right1(AAb(x0))
b(Right2(x0))	→	Right2(AAb(x0))
b(Right3(x0))	→	Right3(AAb(x0))
b(Right4(x0))	→	Right4(AAb(x0))
b(Right5(x0))	→	Right5(AAb(x0))
b(Right6(x0))	→	Right6(AAb(x0))
b(Right7(x0))	→	Right7(AAb(x0))
b(Right8(x0))	→	Right8(AAb(x0))
b(Right9(x0))	→	Right9(AAb(x0))
b(Right10(x0))	→	Right10(AAb(x0))
b(Right11(x0))	→	Right11(AAb(x0))
Ab(Right1(x0))	→	Right1(AAAb(x0))
Ab(Right2(x0))	→	Right2(AAAb(x0))
Ab(Right4(x0))	→	Right4(AAAb(x0))
Ab(Right5(x0))	→	Right5(AAAb(x0))
Ab(Right6(x0))	→	Right6(AAAb(x0))
Ab(Right7(x0))	→	Right7(AAAb(x0))
Ab(Right8(x0))	→	Right8(AAAb(x0))
Ab(Right9(x0))	→	Right9(AAAb(x0))
Ab(Right10(x0))	→	Right10(AAAb(x0))
Ab(Right11(x0))	→	Right11(AAAb(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))
L(b(c(x0)))	→	R(c(b(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right2(b(x0))	→	AAb(Right2(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right4(b(x0))	→	AAb(Right4(x0))
Right5(b(x0))	→	AAb(Right5(x0))
Right6(b(x0))	→	AAb(Right6(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right8(b(x0))	→	AAb(Right8(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right11(b(x0))	→	AAb(Right11(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right2(Ab(x0))	→	AAAb(Right2(x0))
Right4(Ab(x0))	→	AAAb(Right4(x0))
Right5(Ab(x0))	→	AAAb(Right5(x0))
Right6(Ab(x0))	→	AAAb(Right6(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right8(Ab(x0))	→	AAAb(Right8(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
Right11(Ab(x0))	→	AAAb(Right11(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	8 · x₁ + 0
[L(x₁)]	=	1 · x₁ + 0
[Right8(x₁)]	=	15 · x₁ + 1
[Aa(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	1 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	4 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 1
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	4 · x₁ + 0
[R(x₁)]	=	1 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 1
[b(x₁)]	=	1 · x₁ + 1
[Ac(x₁)]	=	2 · x₁ + 0
[c(x₁)]	=	2 · x₁ + 0
[Right6(x₁)]	=	4 · x₁ + 1
[Right5(x₁)]	=	2 · x₁ + 4
[Right7(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	1 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 1
[Right1(x₁)]	=	1 · x₁ + 0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
Right1(b(x0))	→	AAb(Right1(x0))
Right3(b(x0))	→	AAb(Right3(x0))
Right7(b(x0))	→	AAb(Right7(x0))
Right9(b(x0))	→	AAb(Right9(x0))
Right10(b(x0))	→	AAb(Right10(x0))
Right1(Ab(x0))	→	AAAb(Right1(x0))
Right7(Ab(x0))	→	AAAb(Right7(x0))
Right9(Ab(x0))	→	AAAb(Right9(x0))
Right10(Ab(x0))	→	AAAb(Right10(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))
c(b(L(x0)))	→	b(b(c(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	7 · x₁ + 0
[L(x₁)]	=	4 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Right9(x₁)]	=	1 · x₁ + 0
[Right10(x₁)]	=	2 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[AAAa(x₁)]	=	1 · x₁ + 0
[Right2(x₁)]	=	2 · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 1
[E(x₁)]	=	2 · x₁ + 0
[Right4(x₁)]	=	10 · x₁ + 0
[R(x₁)]	=	4 · x₁ + 0
[AAAb(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 4
[Ac(x₁)]	=	4 · x₁ + 2
[c(x₁)]	=	4 · x₁ + 8
[Right6(x₁)]	=	2 · x₁ + 0
[Right5(x₁)]	=	2 · x₁ + 0
[Right7(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	2 · x₁ + 0
[AAb(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	1 · x₁ + 0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right1(Aa(x0))	→	AAAa(Right1(x0))
Right2(Aa(x0))	→	AAAa(Right2(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right4(Aa(x0))	→	AAAa(Right4(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
Right11(Aa(x0))	→	AAAa(Right11(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right1(x0))	→	Right1(AAAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
Aa(Right11(x0))	→	Right11(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	1
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
1	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	1	1
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
1	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	1	1
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	1	0
0	0	0
1	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right2(x0))	→	Right2(AAAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right4(x0))	→	Right4(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	1	1
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	1	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right10(x₁)]

1	0	1
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	1	1
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	1
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right7(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
0	1	1
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right2(x0))	→	Right2(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right7(x0))	→	Right7(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right10(x0))	→	Right10(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right7(x0))	→	Right7(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
Aa(Right10(x0))	→	Right10(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Right1(a(x0))	→	AAa(Right1(x0))
Right2(a(x0))	→	AAa(Right2(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right7(a(x0))	→	AAa(Right7(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right10(a(x0))	→	AAa(Right10(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right7(Aa(x0))	→	AAAa(Right7(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
Right10(Aa(x0))	→	AAAa(Right10(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[L(x₁)]

1	1	0
0	1	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Aa(x₁)]

1	1	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
1	0	0

[Right9(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right10(x₁)]

1	0	1
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0

[AAAa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right2(x₁)]

1	0	1
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

1	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
1	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Ac(x₁)]

1	0	0
0	0	0
1	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right5(x₁)]

1	0	0
1	0	0
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right7(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	0	0
1	0	0
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right8(Aa(x0))	→	AAAa(Right8(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(Right1(x0))	→	Right1(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right8(x0))	→	Right8(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Aa(x₁)]

1	0	1
1	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
1	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	1	1
0	1	1
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right3(x₁)]

1	0	0
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

1	1	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right3(x0))	→	Right3(AAAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

Right1(a(x0))	→	AAa(Right1(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right3(Aa(x0))	→	AAAa(Right3(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[Right11(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[L(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right8(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right9(x₁)]

1	0	0
0	1	0
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[AAa(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[AAAa(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[E(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	1	0
0	1	0
1	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[R(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ac(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[c(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right6(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Right3(x₁)]

1	1	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right1(x₁)]

1	1	0
1	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

the rules

Right1(a(x0))	→	AAa(Right1(x0))
Right3(a(x0))	→	AAa(Right3(x0))
Right4(a(x0))	→	AAa(Right4(x0))
Right5(a(x0))	→	AAa(Right5(x0))
Right6(a(x0))	→	AAa(Right6(x0))
Right8(a(x0))	→	AAa(Right8(x0))
Right9(a(x0))	→	AAa(Right9(x0))
Right11(a(x0))	→	AAa(Right11(x0))
Right5(Aa(x0))	→	AAAa(Right5(x0))
Right6(Aa(x0))	→	AAAa(Right6(x0))
Right9(Aa(x0))	→	AAAa(Right9(x0))
R(E(x0))	→	L(E(x0))
a(L(x0))	→	L(Aa(x0))
b(L(x0))	→	L(Ab(x0))
c(L(x0))	→	L(Ac(x0))
R(Aa(x0))	→	a(R(x0))
R(Ab(x0))	→	b(R(x0))
R(Ac(x0))	→	c(R(x0))
a(b(L(x0)))	→	b(a(a(R(x0))))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(Right1(x0))	→	Right1(AAa(x0))
a(Right3(x0))	→	Right3(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right8(x0))	→	Right8(AAa(x0))
a(Right9(x0))	→	Right9(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
Aa(Right9(x0))	→	Right9(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))
L(b(a(x0)))	→	R(a(a(b(x0))))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Right11(x₁)]	=	2 · x₁ + 1
[L(x₁)]	=	2 · x₁ + 0
[Right8(x₁)]	=	1 · x₁ + 10
[Aa(x₁)]	=	1 · x₁ + 4
[Right9(x₁)]	=	1 · x₁ + 0
[AAa(x₁)]	=	1 · x₁ + 1
[a(x₁)]	=	1 · x₁ + 2
[AAAa(x₁)]	=	1 · x₁ + 2
[Ab(x₁)]	=	4 · x₁ + 0
[E(x₁)]	=	1 · x₁ + 0
[Right4(x₁)]	=	2 · x₁ + 1
[R(x₁)]	=	2 · x₁ + 0
[b(x₁)]	=	4 · x₁ + 0
[Ac(x₁)]	=	2 · x₁ + 4
[c(x₁)]	=	2 · x₁ + 2
[Right6(x₁)]	=	2 · x₁ + 0
[Right5(x₁)]	=	2 · x₁ + 0
[Right3(x₁)]	=	1 · x₁ + 2
[Right1(x₁)]	=	2 · x₁ + 4

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
E(R(x0))	→	E(L(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	2 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[E(x₁)]	=	3 · x₁ + -∞
[Right4(x₁)]	=	2 · x₁ + -∞
[R(x₁)]	=	4 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	1 · x₁ + -∞
[c(x₁)]	=	1 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
Aa(Right5(x0))	→	Right5(AAAa(x0))
Aa(Right6(x0))	→	Right6(AAAa(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	0 · x₁ + -∞
[L(x₁)]	=	0 · x₁ + -∞
[Aa(x₁)]	=	3 · x₁ + -∞
[AAa(x₁)]	=	3 · x₁ + -∞
[a(x₁)]	=	3 · x₁ + -∞
[AAAa(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	4 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[Ac(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	0 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	8 · x₁ + -∞

the rules

a(Right1(x0))	→	Right1(AAa(x0))
a(Right4(x0))	→	Right4(AAa(x0))
a(Right5(x0))	→	Right5(AAa(x0))
a(Right6(x0))	→	Right6(AAa(x0))
a(Right11(x0))	→	Right11(AAa(x0))
L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers

[Right11(x₁)]	=	1 · x₁ + -∞
[L(x₁)]	=	4 · x₁ + -∞
[Aa(x₁)]	=	1 · x₁ + -∞
[AAa(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	1 · x₁ + -∞
[Ab(x₁)]	=	3 · x₁ + -∞
[Right4(x₁)]	=	10 · x₁ + -∞
[R(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	3 · x₁ + -∞
[Ac(x₁)]	=	1 · x₁ + -∞
[c(x₁)]	=	1 · x₁ + -∞
[Right6(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	6 · x₁ + -∞
[Right1(x₁)]	=	4 · x₁ + -∞

the rules

L(a(x0))	→	Aa(L(x0))
L(b(x0))	→	Ab(L(x0))
L(c(x0))	→	Ac(L(x0))
Aa(R(x0))	→	R(a(x0))
Ab(R(x0))	→	R(b(x0))
Ac(R(x0))	→	R(c(x0))

remain.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight function

prec(c)	=	0	weight(c)	=	1
prec(a)	=	0	weight(a)	=	1
prec(R)	=	0	weight(R)	=	1
prec(b)	=	0	weight(b)	=	1
prec(Ac)	=	2	weight(Ac)	=	1
prec(Ab)	=	2	weight(Ab)	=	1
prec(Aa)	=	2	weight(Aa)	=	1
prec(L)	=	3	weight(L)	=	0

all rules could be removed.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.