Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/aabaaaaa-aaaaaaabaab.srs)

The rewrite relation of the following TRS is considered.

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

→

a(a(a(a(a(a(a(b(a(a(b(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(a(a(a(a(a(x1))))))))	→	a^#(b(x1))	(2)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(b(a(a(b(x1)))))	(3)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(b(x1)))	(4)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(b(a(a(b(x1))))))	(5)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(b(a(a(b(x1)))))))	(6)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(a(a(b(x1))))))))	(7)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(a(a(b(x1)))))))))	(8)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(b(a(a(b(x1))))))))))	(9)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(a(b(a(a(b(x1)))))))))))	(10)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(a(b(a(a(b(x1)))))))))))	(10)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(b(x1)))	(4)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(b(a(a(b(x1))))))	(5)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(b(a(a(b(x1)))))))	(6)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(a(a(b(x1))))))))	(7)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(a(a(b(x1)))))))))	(8)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(b(a(a(b(x1))))))))))	(9)

1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

-∞

[a(x₁)]

0	0	0
-3	-3	0
-3	-3	-3

· x₁ +

-∞

[a^#(x₁)]

1	2	4
1	2	4
1	2	4

· x₁ +

-∞

together with the usable rule

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

→

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

(1)

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

a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(a(b(a(a(b(x1)))))))))))	(10)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(b(a(a(b(x1)))))))	(6)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(b(a(a(b(x1))))))))	(7)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(b(a(a(b(x1)))))))))	(8)
a^#(a(b(a(a(a(a(a(x1))))))))	→	a^#(a(a(a(a(a(b(a(a(b(x1))))))))))	(9)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 5 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

0	0	0	0	0
-5	-5	-5	-5	-5
-5	-5	-5	-5	-5
-5	-5	-5	-5	-5
-5	-5	-5	-5	-5

· x₁ +

-∞

[a(x₁)]

0	0	0	5
0	0	0	0
-5	0	0	0
-5	-5	0	0
-5	-5	-5	0

· x₁ +

-∞

[a^#(x₁)]

1	1	1	1	5
1	1	1	1	5
1	1	1	1	5
1	1	1	1	5
1	1	1	1	5

· x₁ +

-∞

together with the usable rule

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

→

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

(1)

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

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

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.