Certification Problem

Input (TPDB SRS_Standard/Bouchare_06/12)

The rewrite relation of the following TRS is considered.

b(b(x1))	→	a(a(a(x1)))	(1)
b(a(b(x1)))	→	a(x1)	(2)
b(a(a(x1)))	→	b(a(b(x1)))	(3)

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

2	2
0	0

· x₁ +

-∞

[a(x₁)]

0	2
0	0

· x₁ +

-∞

[b^#(x₁)]

3	3
3	3

· x₁ +

-∞

together with the usable rules

b(b(x1))	→	a(a(a(x1)))	(1)
b(a(b(x1)))	→	a(x1)	(2)
b(a(a(x1)))	→	b(a(b(x1)))	(3)

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

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

→

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^#(a(a(x1)))

→

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

(5)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

-∞	3
4	13

· x₁ +

[a(x₁)]

-∞	5
5	-∞

· x₁ +

-∞

[b^#(x₁)]

-∞	8
-∞	-∞

· x₁ +

-∞

together with the usable rules

b(b(x1))	→	a(a(a(x1)))	(1)
b(a(b(x1)))	→	a(x1)	(2)
b(a(a(x1)))	→	b(a(b(x1)))	(3)

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

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

→

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

(5)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.