Certification Problem

Input (TPDB SRS_Standard/Trafo_06/un10)

The rewrite relation of the following TRS is considered.

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

→

b(b(b(b(b(a(b(b(a(b(b(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.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

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
-3	-3	0
-3	-3	-3

· x₁ +

-∞

[a(x₁)]

0	0	3
0	0	3
0	0	3

· x₁ +

-∞

[b^#(x₁)]

3	4	6
3	4	6
3	4	6

· x₁ +

-∞

together with the usable rule

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

→

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

(1)

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

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

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

b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(x1))))	(5)
b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(b(a(b(x1)))))))	(6)
b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(b(a(b(b(a(b(x1))))))))))	(7)

1.1.1.1.1 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	3
0	0	0
-3	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

17	17	20
17	17	20
17	17	20

· x₁ +

-∞

together with the usable rule

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

→

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

(1)

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

b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(x1))))	(5)
b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(b(a(b(x1)))))))	(6)
b^#(b(a(b(b(a(b(b(b(b(x1))))))))))	→	b^#(b(a(b(b(a(b(b(a(b(x1))))))))))	(7)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.