Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abbaaaa-aaaaabbaabbabb.srs)

The rewrite relation of the following TRS is considered.

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

→

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

(1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

a_b(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_a(x1))))))))))))))	(2)
a_b(b_b(b_a(a_a(a_a(a_a(a_b(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_b(x1))))))))))))))	(3)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(x1)))	(4)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_b(b_b(b_a(x1))))))	(5)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_a(x1))))))))))	(6)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_b(x1)))	(7)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_a(a_b(b_b(b_b(x1))))))	(8)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_b(x1)))))))	→	a_b^#(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_b(x1))))))))))	(9)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_a(x1))))))))))	(6)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_b(b_b(b_a(x1))))))	(5)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(x1)))	(4)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

[b_a(x₁)]

0	0	0	0
0	0	0	1
0	0	0	1
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
1	0	0	0

[b_b(x₁)]

0	1	0	0
0	0	0	0
0	0	0	1
0	0	0	1

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a_a(x₁)]

0	0	1	0
0	0	0	1
0	1	0	0
1	0	0	0

· x₁ +

1	0	0	0
0	0	0	0
1	0	0	0
0	0	0	0

[a_b^#(x₁)]

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a_b(x₁)]

1	0	0
1	1	0
1	0	1
1	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

together with the usable rules

a_b(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_a(x1))))))))))))))	(2)
a_b(b_b(b_a(a_a(a_a(a_a(a_b(x1)))))))	→	a_a(a_a(a_a(a_a(a_b(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_b(x1))))))))))))))	(3)

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

a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_a(a_b(b_b(b_a(a_b(b_b(b_a(x1))))))))))	(6)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(a_b(b_b(b_a(x1))))))	(5)
a_b^#(b_b(b_a(a_a(a_a(a_a(a_a(x1)))))))	→	a_b^#(b_b(b_a(x1)))	(4)

could be deleted.

1.1.1.1.1 P is empty

There are no pairs anymore.