YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(a(b(a(b(b(x0))))))	→	Wait(Right1(x0))
Begin(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(a(b(b(x0))))	→	Wait(Right3(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Begin(b(x0))	→	Wait(Right5(x0))
Right1(b(End(x0)))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right3(b(a(b(End(x0)))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right5(b(a(b(a(b(End(x0)))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right1(b(x0))	→	Ab(Right1(x0))
Right2(b(x0))	→	Ab(Right2(x0))
Right3(b(x0))	→	Ab(Right3(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right5(b(x0))	→	Ab(Right5(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals

[Wait(x₁)]

1	0
0	0

· x₁ +

0	0
2	0

[Right5(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[a(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[Right1(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[Right2(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[Left(x₁)]

1	0
0	3

· x₁ +

0	0
0	0

[b(x₁)]

1	2
0	2

· x₁ +

0	0
0	0

[Right4(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[Aa(x₁)]

1	2
0	0

· x₁ +

0	0
0	0

[Ab(x₁)]

1	1
0	2

· x₁ +

0	0
0	0

[End(x₁)]

2	0
0	0

· x₁ +

0	0
1	0

[Right3(x₁)]

1	0
0	0

· x₁ +

0	0
0	0

[Begin(x₁)]

1	0
0	0

· x₁ +

0	0
2	0

the rules

Begin(a(b(a(b(b(x0))))))	→	Wait(Right1(x0))
Begin(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(a(b(b(x0))))	→	Wait(Right3(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Begin(b(x0))	→	Wait(Right5(x0))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right1(b(x0))	→	Ab(Right1(x0))
Right2(b(x0))	→	Ab(Right2(x0))
Right3(b(x0))	→	Ab(Right3(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right5(b(x0))	→	Ab(Right5(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

remain.

1.1 Rule Removal

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

[Wait(x₁)]	=	7 · x₁ + -∞
[Right5(x₁)]	=	3 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	7 · x₁ + -∞
[Left(x₁)]	=	1 · x₁ + -∞
[b(x₁)]	=	2 · x₁ + -∞
[Right4(x₁)]	=	5 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	2 · x₁ + -∞
[End(x₁)]	=	13 · x₁ + -∞
[Right3(x₁)]	=	5 · x₁ + -∞
[Begin(x₁)]	=	8 · x₁ + -∞

the rules

Begin(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(a(b(b(x0))))	→	Wait(Right3(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Begin(b(x0))	→	Wait(Right5(x0))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right1(b(x0))	→	Ab(Right1(x0))
Right2(b(x0))	→	Ab(Right2(x0))
Right3(b(x0))	→	Ab(Right3(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right5(b(x0))	→	Ab(Right5(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

remain.

1.1.1 Rule Removal

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

[Wait(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[Right1(x₁)]	=	0 · x₁ + -∞
[Right2(x₁)]	=	1 · x₁ + -∞
[Left(x₁)]	=	1 · x₁ + -∞
[b(x₁)]	=	0 · x₁ + -∞
[Right4(x₁)]	=	1 · x₁ + -∞
[Aa(x₁)]	=	0 · x₁ + -∞
[Ab(x₁)]	=	0 · x₁ + -∞
[End(x₁)]	=	0 · x₁ + -∞
[Right3(x₁)]	=	0 · x₁ + -∞
[Begin(x₁)]	=	1 · x₁ + -∞

the rules

Begin(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right1(b(x0))	→	Ab(Right1(x0))
Right2(b(x0))	→	Ab(Right2(x0))
Right3(b(x0))	→	Ab(Right3(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right5(b(x0))	→	Ab(Right5(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

remain.

1.1.1.1 String Reversal

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

b(b(a(b(Begin(x0)))))	→	Right2(Wait(x0))
b(b(Begin(x0)))	→	Right4(Wait(x0))
End(a(b(Right2(x0))))	→	End(a(b(b(b(a(b(a(Left(x0)))))))))
End(a(b(a(b(Right4(x0))))))	→	End(a(b(b(b(a(b(a(Left(x0)))))))))
b(Right1(x0))	→	Right1(Ab(x0))
b(Right2(x0))	→	Right2(Ab(x0))
b(Right3(x0))	→	Right3(Ab(x0))
b(Right4(x0))	→	Right4(Ab(x0))
b(Right5(x0))	→	Right5(Ab(x0))
a(Right1(x0))	→	Right1(Aa(x0))
a(Right2(x0))	→	Right2(Aa(x0))
a(Right3(x0))	→	Right3(Aa(x0))
a(Right4(x0))	→	Right4(Aa(x0))
a(Right5(x0))	→	Right5(Aa(x0))
Left(Ab(x0))	→	b(Left(x0))
Left(Aa(x0))	→	a(Left(x0))
Left(Wait(x0))	→	Begin(x0)
b(b(a(b(a(b(x0))))))	→	a(b(b(b(a(b(a(x0)))))))

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Wait(x₁)]	=	2 · x₁ + 1
[Right5(x₁)]	=	6 · x₁ + 0
[a(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	4 · x₁ + 12
[Right2(x₁)]	=	8 · x₁ + 0
[Left(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	2 · x₁ + 0
[Right4(x₁)]	=	4 · x₁ + 0
[Aa(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	2 · x₁ + 0
[End(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	4 · x₁ + 3
[Begin(x₁)]	=	2 · x₁ + 1

the rules

b(b(a(b(Begin(x0)))))	→	Right2(Wait(x0))
b(b(Begin(x0)))	→	Right4(Wait(x0))
End(a(b(Right2(x0))))	→	End(a(b(b(b(a(b(a(Left(x0)))))))))
End(a(b(a(b(Right4(x0))))))	→	End(a(b(b(b(a(b(a(Left(x0)))))))))
b(Right2(x0))	→	Right2(Ab(x0))
b(Right4(x0))	→	Right4(Ab(x0))
b(Right5(x0))	→	Right5(Ab(x0))
a(Right1(x0))	→	Right1(Aa(x0))
a(Right2(x0))	→	Right2(Aa(x0))
a(Right3(x0))	→	Right3(Aa(x0))
a(Right4(x0))	→	Right4(Aa(x0))
a(Right5(x0))	→	Right5(Aa(x0))
Left(Ab(x0))	→	b(Left(x0))
Left(Aa(x0))	→	a(Left(x0))
Left(Wait(x0))	→	Begin(x0)
b(b(a(b(a(b(x0))))))	→	a(b(b(b(a(b(a(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(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right2(b(x0))	→	Ab(Right2(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right5(b(x0))	→	Ab(Right5(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[Wait(x₁)]	=	1 · x₁ + 0
[Right5(x₁)]	=	8 · x₁ + 7
[a(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	2 · x₁ + 5
[Right2(x₁)]	=	8 · x₁ + 14
[Left(x₁)]	=	1 · x₁ + 0
[b(x₁)]	=	2 · x₁ + 2
[Right4(x₁)]	=	4 · x₁ + 6
[Aa(x₁)]	=	1 · x₁ + 0
[Ab(x₁)]	=	2 · x₁ + 2
[End(x₁)]	=	1 · x₁ + 0
[Right3(x₁)]	=	8 · x₁ + 2
[Begin(x₁)]	=	1 · x₁ + 0

the rules

Begin(b(a(b(b(x0)))))	→	Wait(Right2(x0))
Begin(b(b(x0)))	→	Wait(Right4(x0))
Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right2(b(x0))	→	Ab(Right2(x0))
Right4(b(x0))	→	Ab(Right4(x0))
Right1(a(x0))	→	Aa(Right1(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Right3(a(x0))	→	Aa(Right3(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Right5(a(x0))	→	Aa(Right5(x0))
Ab(Left(x0))	→	Left(b(x0))
Aa(Left(x0))	→	Left(a(x0))
Wait(Left(x0))	→	Begin(x0)
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))

remain.

1.1.1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

Begin^#(b(a(b(b(x0)))))	→	Right2^#(x0)
Begin^#(b(a(b(b(x0)))))	→	Wait^#(Right2(x0))
Begin^#(b(b(x0)))	→	Right4^#(x0)
Begin^#(b(b(x0)))	→	Wait^#(Right4(x0))
Right2^#(b(a(End(x0))))	→	b^#(b(a(End(x0))))
Right2^#(b(a(End(x0))))	→	b^#(b(b(a(End(x0)))))
Right2^#(b(a(End(x0))))	→	b^#(a(b(b(b(a(End(x0)))))))
Right4^#(b(a(b(a(End(x0))))))	→	b^#(b(a(End(x0))))
Right4^#(b(a(b(a(End(x0))))))	→	b^#(b(b(a(End(x0)))))
Right4^#(b(a(b(a(End(x0))))))	→	b^#(a(b(b(b(a(End(x0)))))))
Right2^#(b(x0))	→	Right2^#(x0)
Right2^#(b(x0))	→	Ab^#(Right2(x0))
Right4^#(b(x0))	→	Right4^#(x0)
Right4^#(b(x0))	→	Ab^#(Right4(x0))
Right1^#(a(x0))	→	Right1^#(x0)
Right1^#(a(x0))	→	Aa^#(Right1(x0))
Right2^#(a(x0))	→	Right2^#(x0)
Right2^#(a(x0))	→	Aa^#(Right2(x0))
Right3^#(a(x0))	→	Right3^#(x0)
Right3^#(a(x0))	→	Aa^#(Right3(x0))
Right4^#(a(x0))	→	Right4^#(x0)
Right4^#(a(x0))	→	Aa^#(Right4(x0))
Right5^#(a(x0))	→	Right5^#(x0)
Right5^#(a(x0))	→	Aa^#(Right5(x0))
Ab^#(Left(x0))	→	b^#(x0)
Wait^#(Left(x0))	→	Begin^#(x0)
b^#(a(b(a(b(b(x0))))))	→	b^#(a(x0))
b^#(a(b(a(b(b(x0))))))	→	b^#(b(a(x0)))
b^#(a(b(a(b(b(x0))))))	→	b^#(b(b(a(x0))))
b^#(a(b(a(b(b(x0))))))	→	b^#(a(b(b(b(a(x0))))))

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pairs

Wait^#(Left(x0))	→	Begin^#(x0)
Begin^#(b(a(b(b(x0)))))	→	Wait^#(Right2(x0))
Begin^#(b(b(x0)))	→	Wait^#(Right4(x0))

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[Wait^#(x₁)]	=	-7 · x₁ + 0
[a(x₁)]	=	-16 · x₁ + 0
[Right2(x₁)]	=	0 · x₁ + -∞
[Left(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	8 · x₁ + -∞
[Right4(x₁)]	=	1 · x₁ + -∞
[Aa(x₁)]	=	-16 · x₁ + 0
[Ab(x₁)]	=	8 · x₁ + -∞
[End(x₁)]	=	-∞ · x₁ + 1
[Begin^#(x₁)]	=	-7 · x₁ + 0

together with the usable rules

Right2(b(a(End(x0))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right2(b(x0))	→	Ab(Right2(x0))
Right2(a(x0))	→	Aa(Right2(x0))
Ab(Left(x0))	→	Left(b(x0))
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))
Aa(Left(x0))	→	Left(a(x0))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(x0))	→	Ab(Right4(x0))
Right4(a(x0))	→	Aa(Right4(x0))

(w.r.t. the implicit argument filter of the reduction pair), the pairs

Wait^#(Left(x0))	→	Begin^#(x0)
Begin^#(b(b(x0)))	→	Wait^#(Right4(x0))

remain.

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[Wait^#(x₁)]	=	12 · x₁ + 0
[a(x₁)]	=	-∞ · x₁ + 0
[Left(x₁)]	=	0 · x₁ + -∞
[b(x₁)]	=	1 · x₁ + 1
[Right4(x₁)]	=	0 · x₁ + 0
[Aa(x₁)]	=	-∞ · x₁ + 0
[Ab(x₁)]	=	1 · x₁ + 1
[End(x₁)]	=	8 · x₁ + 0
[Begin^#(x₁)]	=	11 · x₁ + 0

together with the usable rules

Right4(b(a(b(a(End(x0))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right4(b(x0))	→	Ab(Right4(x0))
Right4(a(x0))	→	Aa(Right4(x0))
Ab(Left(x0))	→	Left(b(x0))
b(a(b(a(b(b(x0))))))	→	a(b(a(b(b(b(a(x0)))))))
Aa(Left(x0))	→	Left(a(x0))

(w.r.t. the implicit argument filter of the reduction pair), the pair

Wait^#(Left(x0))

→

Begin^#(x0)

remains.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pairs

b^#(a(b(a(b(b(x0))))))	→	b^#(a(x0))
b^#(a(b(a(b(b(x0))))))	→	b^#(b(a(x0)))
b^#(a(b(a(b(b(x0))))))	→	b^#(b(b(a(x0))))

1.1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[a(x₁)]	=	0 · x₁ + -∞
[b^#(x₁)]	=	8 · x₁ + -∞
[b(x₁)]	=	4 · x₁ + -∞

together with the usable rule

b(a(b(a(b(b(x0))))))

→

a(b(a(b(b(b(a(x0)))))))

(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1.1.1.1.1.1.1.2.1 P is empty

There are no pairs anymore.