Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z032)

The rewrite relation of the following TRS is considered.

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

→

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

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(a(b(a(a(x1)))))	→	b^#(b(a(b(a(b(a(x1)))))))	(2)
b^#(a(b(a(a(x1)))))	→	b^#(a(b(a(b(a(x1))))))	(3)
b^#(a(b(a(a(x1)))))	→	b^#(a(b(a(x1))))	(4)
b^#(a(b(a(a(x1)))))	→	b^#(a(x1))	(5)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(a(b(a(a(x1)))))	→	b^#(a(b(a(x1))))	(4)
b^#(a(b(a(a(x1)))))	→	b^#(a(b(a(b(a(x1))))))	(3)
b^#(a(b(a(a(x1)))))	→	b^#(a(x1))	(5)

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^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

0	-∞	-∞
1	-∞	0
-∞	0	0

· x₁

[b(x₁)]

-∞	-∞	0
0	-∞	-∞
0	-∞	-∞

· x₁

the pairs

b^#(a(b(a(a(x1)))))	→	b^#(a(b(a(b(a(x1))))))	(3)
b^#(a(b(a(a(x1)))))	→	b^#(a(x1))	(5)

could be deleted.

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^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

1	-∞	-∞
0	-∞	-∞
-∞	0	-∞

· x₁

[b(x₁)]

-∞	-∞	0
1	0	-∞
-∞	-∞	-∞

· x₁

the pair

b^#(a(b(a(a(x1)))))

→

b^#(a(b(a(x1))))

(4)

could be deleted.

1.1.1.1.1.1 P is empty

There are no pairs anymore.