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(b(a(b(b(a(End(x0)))))))))
Right2(b(a(End(x0))))	→	Left(a(b(b(a(b(b(a(End(x0)))))))))
Right3(b(a(b(End(x0)))))	→	Left(a(b(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(b(b(a(End(x0)))))))))
Right5(b(a(b(a(b(End(x0)))))))	→	Left(a(b(b(a(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(b(a(b(b(a(x0)))))))

Proof

1 Rule Removal

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

[Wait(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Right5(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	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

[Right1(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Right2(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Left(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[b(x₁)]

1	0	1
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

1	0	0
0	0	1
1	1	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Aa(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
1	0	0
1	0	0

[End(x₁)]

1	1	1
0	0	0
1	1	1

· x₁ +

1	0	0
0	0	0
1	0	0

[Right3(x₁)]

1	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
1	0	0

[Begin(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
1	0	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(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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(b(a(b(b(a(x0)))))))

remain.

1.1 Rule Removal

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

[Wait(x₁)]	=	0 · x₁ + -∞
[Right5(x₁)]	=	1 · 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))
Begin(b(x0))	→	Wait(Right5(x0))
Right2(b(a(End(x0))))	→	Left(a(b(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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(b(a(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(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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(b(a(b(b(a(x0)))))))

remain.

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

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

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(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(b(b(a(End(x0)))))))))
Right2(b(x0))	→	Ab(Right2(x0))
Right3(b(x0))	→	Ab(Right3(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(b(a(b(b(a(x0)))))))

remain.

1.1.1.1.1 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(a(b(b(a(Left(x0)))))))))
End(a(b(a(b(Right4(x0))))))	→	End(a(b(b(a(b(b(a(Left(x0)))))))))
b(Right2(x0))	→	Right2(Ab(x0))
b(Right3(x0))	→	Right3(Ab(x0))
b(Right4(x0))	→	Right4(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(a(b(b(a(x0)))))))

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 + 0
[a(x₁)]	=	1 · x₁ + 0
[Right1(x₁)]	=	12 · x₁ + 0
[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 + 4
[Right3(x₁)]	=	8 · x₁ + 2
[Begin(x₁)]	=	1 · x₁ + 0

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

remain.

1.1.1.1.1.1.1 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(b(a(b(b(a(End(x0)))))))))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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(b(a(b(b(a(x0)))))))

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^#(a(b(b(a(End(x0))))))
Right2^#(b(a(End(x0))))	→	b^#(b(a(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^#(a(b(b(a(End(x0))))))
Right4^#(b(a(b(a(End(x0))))))	→	b^#(b(a(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^#(a(b(b(a(x0)))))
b^#(a(b(a(b(b(x0))))))	→	b^#(b(a(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 (4 x 4)-matrices with strict dimension 1 over the naturals

[Wait^#(x₁)]

1	1	1
0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a(x₁)]

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Right2(x₁)]

1	0	0
1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Left(x₁)]

0	0	0	0
1	0	0	0
0	1	1	0
0	0	0	1

· x₁ +

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[b(x₁)]

1	0	1
1	0	1
0	0	0
0	1	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Right4(x₁)]

0	0	0
1	0	0
0	1	0
0	0	1

· x₁ +

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Aa(x₁)]

1	0	0	0
0	1	0	0
0	0	0	0
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Ab(x₁)]

1	0	0	0
0	1	0	1
0	1	0	1
0	0	1	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[End(x₁)]

1	1	1	1
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[Begin^#(x₁)]

0	1	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

together with the usable rules

Right2(b(a(End(x0))))	→	Left(a(b(b(a(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(b(a(b(b(a(x0)))))))
Aa(Left(x0))	→	Left(a(x0))
Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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 Knuth Bendix order with w0 = 1 and the following precedence and weight function

prec(Wait^#)	=	3	weight(Wait^#)	=	1
prec(Begin^#)	=	3	weight(Begin^#)	=	1
prec(Aa)	=	2	weight(Aa)	=	1
prec(Ab)	=	1	weight(Ab)	=	1
prec(Left)	=	0	weight(Left)	=	0
prec(End)	=	8	weight(End)	=	1
prec(Right4)	=	3	weight(Right4)	=	0
prec(a)	=	0	weight(a)	=	1
prec(b)	=	0	weight(b)	=	1

in combination with the following argument filter

π(Wait^#) = 1

π(Begin^#) = 1

π(Aa) = []

π(Ab) = [1]

π(Left) = 1

π(End) = 1

π(Right4) = [1]

π(a) = []

π(b) = [1]

together with the usable rules

Right4(b(a(b(a(End(x0))))))	→	Left(a(b(b(a(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(b(a(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^#(a(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)]	=	1 · x₁ + -∞
[b(x₁)]	=	3 · x₁ + -∞

together with the usable rule

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

→

a(b(b(a(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.

π(Wait^#)	=	1
π(Begin^#)	=	1
π(Aa)	=	[]
π(Ab)	=	[1]
π(Left)	=	1
π(End)	=	1
π(Right4)	=	[1]
π(a)	=	[]
π(b)	=	[1]