Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x08)

The rewrite relation of the following TRS is considered.

a(x1)	→	x1	(1)
a(b(a(x1)))	→	a(x1)	(2)
b(a(a(b(x1))))	→	a(a(a(a(b(b(b(x1)))))))	(3)
a(a(a(a(x1))))	→	b(x1)	(4)

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(b(x1))))	→	b^#(b(x1))	(5)
b^#(a(a(b(x1))))	→	b^#(b(b(x1)))	(6)
b^#(a(a(b(x1))))	→	a^#(b(b(b(x1))))	(7)
b^#(a(a(b(x1))))	→	a^#(a(b(b(b(x1)))))	(8)
b^#(a(a(b(x1))))	→	a^#(a(a(b(b(b(x1))))))	(9)
b^#(a(a(b(x1))))	→	a^#(a(a(a(b(b(b(x1)))))))	(10)
a^#(a(a(a(x1))))	→	b^#(x1)	(11)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(a(a(b(x1))))	→	b^#(b(x1))	(5)
b^#(a(a(b(x1))))	→	b^#(b(b(x1)))	(6)
b^#(a(a(b(x1))))	→	a^#(b(b(b(x1))))	(7)
a^#(a(a(a(x1))))	→	b^#(x1)	(11)
b^#(a(a(b(x1))))	→	a^#(a(b(b(b(x1)))))	(8)
b^#(a(a(b(x1))))	→	a^#(a(a(b(b(b(x1))))))	(9)
b^#(a(a(b(x1))))	→	a^#(a(a(a(b(b(b(x1)))))))	(10)

1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

-∞

[a(x₁)]

0	0	4
0	0	0
-4	0	0
-4	-4	0

· x₁ +

-∞

[b^#(x₁)]

20	20	20	24
20	20	20	24
20	20	20	24
20	20	20	24

· x₁ +

-∞

[a^#(x₁)]

20	20	20	21
20	20	20	21
20	20	20	21
20	20	20	21

· x₁ +

-∞

together with the usable rules

a(x1)	→	x1	(1)
a(b(a(x1)))	→	a(x1)	(2)
b(a(a(b(x1))))	→	a(a(a(a(b(b(b(x1)))))))	(3)
a(a(a(a(x1))))	→	b(x1)	(4)

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

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

→

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

(7)

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(b(x1))))	→	b^#(b(x1))	(5)
b^#(a(a(b(x1))))	→	b^#(b(b(x1)))	(6)
b^#(a(a(b(x1))))	→	a^#(a(b(b(b(x1)))))	(8)
a^#(a(a(a(x1))))	→	b^#(x1)	(11)
b^#(a(a(b(x1))))	→	a^#(a(a(b(b(b(x1))))))	(9)
b^#(a(a(b(x1))))	→	a^#(a(a(a(b(b(b(x1)))))))	(10)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

[b(x₁)]

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

-∞

[a(x₁)]

0	0	4
0	0	0
-4	0	0
-4	-4	0

· x₁ +

-∞

[b^#(x₁)]

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

· x₁ +

-∞

[a^#(x₁)]

1	1	1	3
1	1	1	3
1	1	1	3
1	1	1	3

· x₁ +

-∞

together with the usable rules

a(x1)	→	x1	(1)
a(b(a(x1)))	→	a(x1)	(2)
b(a(a(b(x1))))	→	a(a(a(a(b(b(b(x1)))))))	(3)
a(a(a(a(x1))))	→	b(x1)	(4)

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

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

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(x1)	→	x1	(1)
a(b(a(x1)))	→	a(x1)	(2)
b(a(a(b(x1))))	→	a(a(a(a(b(b(b(x1)))))))	(3)
a(a(a(a(x1))))	→	b(x1)	(4)

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

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

→

b^#(x1)

(11)

and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.