Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/aabacaaa-aaaaabacaabac.srs)

The rewrite relation of the following TRS is considered.

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

→

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

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

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

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

→

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

(2)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(8)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(a(x1)))))	(7)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(9)

1.1.1.1 Reduction Pair Processor

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

[a^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞	0	0
0	-∞	-∞
0	1	-∞

· x₁

[c(x₁)]

-∞	0	-∞
-∞	-∞	-∞
0	0	-∞

· x₁

[b(x₁)]

0	-∞	-∞
0	0	0
1	0	0

· x₁

the pair

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

→

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

(8)

could be deleted.

1.1.1.1.1 Reduction Pair Processor

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

[a^#(x₁)]

-∞

0	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[a(x₁)]

-∞

-∞	0	-∞
-∞	-∞	0
0	-∞	-∞

· x₁

[c(x₁)]

-∞	-∞	-∞
0	-∞	-∞
-∞	0	0

· x₁

[b(x₁)]

-∞	1	0
-∞	0	-∞
0	0	1

· x₁

the pair

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

→

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

(7)

could be deleted.

1.1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(9)

1	>	1

As there is no critical graph in the transitive closure, there are no infinite chains.