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 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

Begin^#(a(b(a(b(b(x0))))))	→	Right1^#(x0)
Begin^#(a(b(a(b(b(x0))))))	→	Wait^#(Right1(x0))
Begin^#(b(a(b(b(x0)))))	→	Right2^#(x0)
Begin^#(b(a(b(b(x0)))))	→	Wait^#(Right2(x0))
Begin^#(a(b(b(x0))))	→	Right3^#(x0)
Begin^#(a(b(b(x0))))	→	Wait^#(Right3(x0))
Begin^#(b(b(x0)))	→	Right4^#(x0)
Begin^#(b(b(x0)))	→	Wait^#(Right4(x0))
Begin^#(b(x0))	→	Right5^#(x0)
Begin^#(b(x0))	→	Wait^#(Right5(x0))
Right1^#(b(End(x0)))	→	b^#(a(End(x0)))
Right1^#(b(End(x0)))	→	b^#(b(a(End(x0))))
Right1^#(b(End(x0)))	→	b^#(b(b(a(End(x0)))))
Right1^#(b(End(x0)))	→	b^#(a(b(b(b(a(End(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)))))))
Right3^#(b(a(b(End(x0)))))	→	b^#(a(End(x0)))
Right3^#(b(a(b(End(x0)))))	→	b^#(b(a(End(x0))))
Right3^#(b(a(b(End(x0)))))	→	b^#(b(b(a(End(x0)))))
Right3^#(b(a(b(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)))))))
Right5^#(b(a(b(a(b(End(x0)))))))	→	b^#(a(End(x0)))
Right5^#(b(a(b(a(b(End(x0)))))))	→	b^#(b(a(End(x0))))
Right5^#(b(a(b(a(b(End(x0)))))))	→	b^#(b(b(a(End(x0)))))
Right5^#(b(a(b(a(b(End(x0)))))))	→	b^#(a(b(b(b(a(End(x0)))))))
Right1^#(b(x0))	→	Right1^#(x0)
Right1^#(b(x0))	→	Ab^#(Right1(x0))
Right2^#(b(x0))	→	Right2^#(x0)
Right2^#(b(x0))	→	Ab^#(Right2(x0))
Right3^#(b(x0))	→	Right3^#(x0)
Right3^#(b(x0))	→	Ab^#(Right3(x0))
Right4^#(b(x0))	→	Right4^#(x0)
Right4^#(b(x0))	→	Ab^#(Right4(x0))
Right5^#(b(x0))	→	Right5^#(x0)
Right5^#(b(x0))	→	Ab^#(Right5(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 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pairs

Wait^#(Left(x0))	→	Begin^#(x0)
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))

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)]

0	0	1	1
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

[Right5(x₁)]

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

· x₁ +

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

[a(x₁)]

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

· x₁ +

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

[Right1(x₁)]

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

· x₁ +

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

[Right2(x₁)]

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

· x₁ +

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

[Left(x₁)]

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

· x₁ +

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

[b(x₁)]

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

· x₁ +

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

[Aa(x₁)]

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

· x₁ +

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

[Ab(x₁)]

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

· x₁ +

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

[End(x₁)]

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

· x₁ +

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

[Begin^#(x₁)]

0	0	0	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

[Right3(x₁)]

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

· x₁ +

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

together with the usable rules

Right1(b(End(x0)))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right1(b(x0))	→	Ab(Right1(x0))
Right1(a(x0))	→	Aa(Right1(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))
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))
Right3(b(a(b(End(x0)))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right3(b(x0))	→	Ab(Right3(x0))
Right3(a(x0))	→	Aa(Right3(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))
Right5(b(a(b(a(b(End(x0)))))))	→	Left(a(b(a(b(b(b(a(End(x0)))))))))
Right5(b(x0))	→	Ab(Right5(x0))
Right5(a(x0))	→	Aa(Right5(x0))

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

Wait^#(Left(x0))	→	Begin^#(x0)
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))

remain.

1.1.1.1 Reduction Pair Processor with Usable Rules

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

[Wait^#(x₁)]

0	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

0	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[Right1(x₁)]

0	0	0
1	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[Right2(x₁)]

0	0	0
1	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[Left(x₁)]

0	0	1
0	0	0
0	1	1

· x₁ +

1	0	0
0	0	0
1	0	0

[b(x₁)]

0	0	0
0	0	0
0	1	1

· x₁ +

1	0	0
0	0	0
0	0	0

[Right4(x₁)]

0	0	1
1	0	1
0	0	1

· x₁ +

1	0	0
1	0	0
1	0	0

[Aa(x₁)]

0	0	1
0	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Ab(x₁)]

0	0	1
1	0	0
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

[End(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Begin^#(x₁)]

0	0	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Right3(x₁)]

0	0	1
0	0	1
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

together with the usable rules

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

remain.

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	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

[a(x₁)]

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

· x₁ +

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

[Right1(x₁)]

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

· x₁ +

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

[Right2(x₁)]

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

· x₁ +

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

[Left(x₁)]

0	0	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

[b(x₁)]

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

· x₁ +

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

[Right4(x₁)]

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

· 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	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

[End(x₁)]

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

· x₁ +

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

[Begin^#(x₁)]

0	0	1	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

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

remain.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[Wait^#(x₁)]

0	0	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[a(x₁)]

0	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
0	0	0

[Right2(x₁)]

0	0	1
1	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Left(x₁)]

0	0	1
1	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

0	0	1
0	1	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Right4(x₁)]

0	0	1
1	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[Aa(x₁)]

0	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[Ab(x₁)]

0	0	1
1	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[End(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[Begin^#(x₁)]

0	1	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	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 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 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))))

1.1.2 Reduction Pair Processor with Usable Rules

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

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

1.1.2.1 P is empty

There are no pairs anymore.