Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z098)

The rewrite relation of the following TRS is considered.

a(a(a(b(b(b(x1))))))

→

b(b(b(b(a(a(a(a(x1))))))))

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

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

b(b(b(a(a(a(x1))))))

→

a(a(a(a(b(b(b(b(x1))))))))

(2)

1.1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS

b(b(b(b(a(a(a(x1)))))))	→	b(a(a(a(a(b(b(b(b(x1)))))))))	(3)
a(b(b(b(a(a(a(x1)))))))	→	a(a(a(a(a(b(b(b(b(x1)))))))))	(4)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

b_b(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_a(a_a(a_a(a_a(a_b(b_b(b_b(b_b(b_b(x1)))))))))	(5)
b_b(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_a(a_a(a_a(a_a(a_b(b_b(b_b(b_b(b_a(x1)))))))))	(6)
a_b(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_b(b_b(b_b(x1)))))))))	(7)
a_b(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_b(b_b(b_a(x1)))))))))	(8)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_b(x1)))))	(9)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(b_b(x1))))	(10)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(x1)))	(11)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(x1))	(12)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(x1)	(13)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(14)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_b(b_a(x1))))	(15)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_a(x1)))	(16)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_a(x1))	(17)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_b(x1)))))	(18)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(b_b(x1))))	(19)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(x1)))	(20)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(x1))	(21)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(x1)	(22)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(23)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_b(b_a(x1))))	(24)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_a(x1)))	(25)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_a(x1))	(26)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(23)
a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_b(b_a(x1))))	(24)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_b(x1)))))	(9)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(b_b(x1))))	(10)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(14)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_b(b_a(x1))))	(15)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(x1)))	(11)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(x1))	(12)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(x1)	(13)

1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[a_b^#(x₁)]

-∞

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b_b(x₁)]

-∞	0	-∞
0	-∞	0
0	0	-∞

· x₁

[b_a(x₁)]

-∞

-∞	-∞	-∞
-∞	-∞	-∞
0	0	0

· x₁

[a_a(x₁)]

-∞

0	0	0
0	0	0
0	0	0

· x₁

[b_b^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a_b(x₁)]

0	0	0
0	0	0
0	0	-∞

· x₁

the pairs

a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(23)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_b(b_b(b_a(x1)))))	(14)

could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[a_b^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b_b(x₁)]

-∞

-∞	0	-∞
0	-∞	0
0	0	-∞

· x₁

[b_a(x₁)]

-∞

0	-∞	-∞
0	0	-∞
1	0	0

· x₁

[a_a(x₁)]

-∞	0	-∞
0	-∞	0
0	0	-∞

· x₁

[b_b^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a_b(x₁)]

0	-∞	-∞
0	-∞	-∞
0	0	0

· x₁

the pair

a_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))

→

b_b^#(b_b(b_b(b_a(x1))))

(24)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(b_b(x1))))	(10)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))	→	b_b^#(b_b(b_b(b_a(x1))))	(15)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(x1)))	(11)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(x1))	(12)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(x1)	(13)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[b_b^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b_b(x₁)]

-∞

-∞	0	-∞
-∞	-∞	0
0	0	-∞

· x₁

[b_a(x₁)]

-∞

0	1	0
-∞	0	-∞
0	0	-∞

· x₁

[a_a(x₁)]

-∞	0	0
0	-∞	-∞
0	0	-∞

· x₁

[a_b(x₁)]

-∞	0	-∞
-∞	0	-∞
0	0	0

· x₁

the pairs

b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(b_b(x1))))	(10)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(b_b(x1)))	(11)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(b_b(x1))	(12)
b_b^#(b_b(b_b(b_a(a_a(a_a(a_b(x1)))))))	→	b_b^#(x1)	(13)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Switch to Innermost Termination

The TRS does not have overlaps with the pairs and is locally confluent:

Hence, it suffices to show innermost termination in the following.

1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

b_b(b_b(b_b(b_a(a_a(a_a(a_b(x0)))))))

b_b(b_b(b_b(b_a(a_a(a_a(a_a(x0)))))))

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[b_b^#(x₁)]	=	x₁
[b_b(x₁)]	=	-2 + 2 · x₁
[b_a(x₁)]	=	x₁
[a_a(x₁)]	=	2 + 2 · x₁

the pair

b_b^#(b_b(b_b(b_a(a_a(a_a(a_a(x1)))))))

→

b_b^#(b_b(b_b(b_a(x1))))

(15)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.