Certification Problem

Input (TPDB SRS_Standard/Mixed_SRS/06)

The rewrite relation of the following TRS is considered.

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

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

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)]

0	0
-2	-2

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[b^#(x₁)]

2	2
2	2

· x₁ +

-∞

[a^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

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

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

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

→

a^#(b(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

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

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)]

0	2
0	0

· x₁ +

-∞

[a(x₁)]

0	2
-2	0

· x₁ +

-∞

[b^#(x₁)]

12	13
12	13

· x₁ +

-∞

[a^#(x₁)]

11	13
11	13

· x₁ +

-∞

together with the usable rules

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

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

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

could be deleted.

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

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

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

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

→

b^#(b(x1))

(8)

and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(x1))

→

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

(3)

1.1.1.1.1.1.1.1 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)]

0	0
0	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

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

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

b^#(b(x1))

→

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

(3)

could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.