Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/multum2)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(a(a(x1)))	→	b^#(x1)	(5)
b^#(a(a(x1)))	→	b^#(b(x1))	(6)
b^#(a(a(x1)))	→	a^#(b(b(x1)))	(7)
b^#(a(a(x1)))	→	a^#(a(b(b(x1))))	(8)
a^#(b(a(b(x1))))	→	b^#(a(a(a(x1))))	(9)
a^#(b(a(b(x1))))	→	a^#(x1)	(10)
a^#(b(a(b(x1))))	→	a^#(a(x1))	(11)
a^#(b(a(b(x1))))	→	a^#(a(a(x1)))	(12)

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^#(x1)	(5)
b^#(a(a(x1)))	→	b^#(b(x1))	(6)
b^#(a(a(x1)))	→	a^#(b(b(x1)))	(7)
a^#(b(a(b(x1))))	→	b^#(a(a(a(x1))))	(9)
b^#(a(a(x1)))	→	a^#(a(b(b(x1))))	(8)
a^#(b(a(b(x1))))	→	a^#(x1)	(10)
a^#(b(a(b(x1))))	→	a^#(a(x1))	(11)
a^#(b(a(b(x1))))	→	a^#(a(a(x1)))	(12)

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
-4	0	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

2	2	2	2
2	2	2	2
2	2	2	2
2	2	2	2

· x₁ +

-∞

[a^#(x₁)]

1	1	2	2
1	1	2	2
1	1	2	2
1	1	2	2

· x₁ +

-∞

together with the usable rules

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

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

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

→

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

(12)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1.1.1 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
-4	0	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

3	3	3	3
3	3	3	3
3	3	3	3
3	3	3	3

· 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

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

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

b^#(a(a(x1)))	→	a^#(a(b(b(x1))))	(8)
a^#(b(a(b(x1))))	→	a^#(a(x1))	(11)

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

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[b(x₁)]

0	0	0
0	0	1
1	0	0

· x₁ +

[a(x₁)]

0	1	1
0	0	1
0	0	0

· x₁ +

[b^#(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

[a^#(x₁)]

0	0	1
0	0	0
0	0	0

· x₁ +

together with the usable rules

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

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

b^#(a(a(x1)))	→	a^#(b(b(x1)))	(7)
a^#(b(a(b(x1))))	→	a^#(x1)	(10)

could be deleted.

1.1.1.1.1.1.1.1.1 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

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

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

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

→

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

(9)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1.1.1.1.1.1.1.1 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	-∞	-∞
2	-∞	0	2

· x₁ +

-∞

[a(x₁)]

1	-∞	0	-∞
-∞	-∞	-∞	4
-∞	-∞	1	-∞
-∞	-∞	0	-∞

· x₁ +

-∞

[b^#(x₁)]

2	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

-∞

together with the usable rules

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

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

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

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.