Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x09)

The rewrite relation of the following TRS is considered.

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

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(b(a(a(a(x1)))))	→	a(a(a(b(b(a(b(b(x1))))))))	(5)
a(b(b(x1)))	→	x1	(6)
a(x1)	→	b(b(b(x1)))	(3)
a(x1)	→	b(x1)	(4)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(b(a(a(a(x1)))))	→	b^#(x1)	(7)
b^#(b(a(a(a(x1)))))	→	b^#(b(x1))	(8)
b^#(b(a(a(a(x1)))))	→	b^#(b(a(b(b(x1)))))	(9)
b^#(b(a(a(a(x1)))))	→	b^#(a(b(b(x1))))	(10)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(x1)))	(11)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(a(b(b(x1))))))	(12)
b^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(b(b(x1)))))))	(13)
b^#(b(a(a(a(x1)))))	→	a^#(a(a(b(b(a(b(b(x1))))))))	(14)
a^#(x1)	→	b^#(x1)	(15)
a^#(x1)	→	b^#(b(x1))	(16)
a^#(x1)	→	b^#(b(b(x1)))	(17)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(a(a(a(x1)))))	→	b^#(x1)	(7)
b^#(b(a(a(a(x1)))))	→	b^#(b(x1))	(8)
b^#(b(a(a(a(x1)))))	→	b^#(b(a(b(b(x1)))))	(9)
b^#(b(a(a(a(x1)))))	→	b^#(a(b(b(x1))))	(10)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(x1)))	(11)
a^#(x1)	→	b^#(x1)	(15)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(a(b(b(x1))))))	(12)
a^#(x1)	→	b^#(b(x1))	(16)
b^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(b(b(x1)))))))	(13)
a^#(x1)	→	b^#(b(b(x1)))	(17)
b^#(b(a(a(a(x1)))))	→	a^#(a(a(b(b(a(b(b(x1))))))))	(14)

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

· x₁ +

-∞

[a(x₁)]

0	3	3
0	0	3
0	0	0

· x₁ +

-∞

[b^#(x₁)]

2	5	5
2	5	5
2	5	5

· x₁ +

-∞

[a^#(x₁)]

5	5	5
5	5	5
5	5	5

· x₁ +

-∞

together with the usable rules

b(b(a(a(a(x1)))))	→	a(a(a(b(b(a(b(b(x1))))))))	(5)
a(b(b(x1)))	→	x1	(6)
a(x1)	→	b(b(b(x1)))	(3)
a(x1)	→	b(x1)	(4)

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

b^#(b(a(a(a(x1)))))	→	b^#(x1)	(7)
b^#(b(a(a(a(x1)))))	→	b^#(b(x1))	(8)
b^#(b(a(a(a(x1)))))	→	b^#(b(a(b(b(x1)))))	(9)
b^#(b(a(a(a(x1)))))	→	b^#(a(b(b(x1))))	(10)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(x1)))	(11)
b^#(b(a(a(a(x1)))))	→	a^#(b(b(a(b(b(x1))))))	(12)
b^#(b(a(a(a(x1)))))	→	a^#(a(b(b(a(b(b(x1)))))))	(13)
a^#(x1)	→	b^#(b(b(x1)))	(17)

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

a^#(x1)	→	b^#(x1)	(15)
b^#(b(a(a(a(x1)))))	→	a^#(a(a(b(b(a(b(b(x1))))))))	(14)
a^#(x1)	→	b^#(b(x1))	(16)

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 arctic semiring over the naturals

[b(x₁)]

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

· x₁ +

-∞

[a(x₁)]

0	3	3
0	0	3
0	0	0

· x₁ +

-∞

[b^#(x₁)]

2	2	2
2	2	2
2	2	2

· x₁ +

-∞

[a^#(x₁)]

3	3	3
3	3	3
3	3	3

· x₁ +

-∞

together with the usable rules

b(b(a(a(a(x1)))))	→	a(a(a(b(b(a(b(b(x1))))))))	(5)
a(b(b(x1)))	→	x1	(6)
a(x1)	→	b(b(b(x1)))	(3)
a(x1)	→	b(x1)	(4)

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

a^#(x1)	→	b^#(x1)	(15)
b^#(b(a(a(a(x1)))))	→	a^#(a(a(b(b(a(b(b(x1))))))))	(14)
a^#(x1)	→	b^#(b(x1))	(16)

could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.