Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z095)

The rewrite relation of the following TRS is considered.

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

→

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

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(b(b(x1))))	→	a^#(x1)	(2)
a^#(a(b(b(x1))))	→	a^#(a(x1))	(3)
a^#(a(b(b(x1))))	→	a^#(a(a(x1)))	(4)
a^#(a(b(b(x1))))	→	a^#(a(a(a(x1))))	(5)
a^#(a(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(6)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(b(b(x1))))	→	a^#(x1)	(2)
a^#(a(b(b(x1))))	→	a^#(a(x1))	(3)
a^#(a(b(b(x1))))	→	a^#(a(a(x1)))	(4)
a^#(a(b(b(x1))))	→	a^#(a(a(a(x1))))	(5)
a^#(a(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(6)

1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

=

0	1	2
1	0	0
0	0	0

· x₁ +

1

0

0

[a(x₁)]

=

0	0	2
0	0	1
0	1	0

· x₁ +

0

0

0

[a^#(x₁)]

=

0	1	1
0	0	0
0	0	0

· x₁ +

0

0

0

together with the usable rule

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

→

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

(1)

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

a^#(a(b(b(x1))))	→	a^#(x1)	(2)
a^#(a(b(b(x1))))	→	a^#(a(x1))	(3)
a^#(a(b(b(x1))))	→	a^#(a(a(x1)))	(4)
a^#(a(b(b(x1))))	→	a^#(a(a(a(x1))))	(5)
a^#(a(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(6)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.