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 ttt2 @ 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^#(a(a(x1)))	(3)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(4)
a^#(a(a(c(a(b(a(a(x1))))))))	→	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^#(c(a(b(a(a(a(a(a(x1)))))))))	(7)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(c(a(b(a(a(a(a(a(x1))))))))))	(8)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(b(a(a(c(a(b(a(a(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(a(x1)))))	(5)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(4)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(3)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[c(x₁)]

0	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
2	0	0

[a^#(x₁)]

0	2	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a(x₁)]

0	0	1
1	0	0
0	2	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
1	0	0

together with the usable rule

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

→

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

(2)

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

a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(a(x1)))))	(5)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(4)
a^#(a(a(c(a(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(3)

could be deleted.

1.1.1.1.1 P is empty

There are no pairs anymore.