Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/bababaaaba-aabaaababababa.srs)

The rewrite relation of the following TRS is considered.

b(a(b(a(b(a(a(a(b(a(x1))))))))))

→

a(a(b(a(a(a(b(a(b(a(b(a(b(a(x1))))))))))))))

(1)

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

a(b(a(a(a(b(a(b(a(b(x1))))))))))

→

a(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))

(2)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(x1)	(3)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))	(4)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))	(5)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(a(a(b(a(a(x1))))))))))	(6)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(x1))))	(7)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(a(b(a(a(x1))))))))	(8)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(x1))	(9)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(b(a(a(x1)))))	(10)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(a(b(a(a(x1))))))	(11)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(x1)	(3)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))	(4)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))	(5)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(b(a(a(a(b(a(a(x1))))))))))	(6)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(x1))))	(7)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(a(b(a(a(x1))))))))	(8)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(x1))	(9)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(b(a(a(x1)))))	(10)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(a(b(a(a(x1))))))	(11)

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 +

-∞

[a^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rule

a(b(a(a(a(b(a(b(a(b(x1))))))))))

→

a(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))

(2)

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

a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(x1))	(9)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(b(a(a(x1)))))	(10)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(a(a(b(a(a(x1))))))	(11)

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

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	2
0	2

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a^#(x₁)]

9	11
9	11

· x₁ +

-∞

together with the usable rule

a(b(a(a(a(b(a(b(a(b(x1))))))))))

→

a(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))

(2)

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

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

→

a^#(x1)

(3)

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

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

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 naturals

[b(x₁)]

2	0
0	0

· x₁ +

[a(x₁)]

0	1
0	0

· x₁ +

[a^#(x₁)]

1	1
0	0

· x₁ +

together with the usable rule

a(b(a(a(a(b(a(b(a(b(x1))))))))))

→

a(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))

(2)

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

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

could be deleted.

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

a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(x1))))	(7)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(a(b(a(a(x1))))))))	(8)

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

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[a^#(x₁)]

9	9	13	13
9	9	13	13
9	9	13	13
9	9	13	13

· x₁ +

-∞

together with the usable rule

a(b(a(a(a(b(a(b(a(b(x1))))))))))

→

a(b(a(b(a(b(a(b(a(a(a(b(a(a(x1))))))))))))))

(2)

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

a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(x1))))	(7)
a^#(b(a(a(a(b(a(b(a(b(x1))))))))))	→	a^#(b(a(a(a(b(a(a(x1))))))))	(8)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.