Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abaababaab-ababaabaababa.srs)

The rewrite relation of the following TRS is considered.

a(b(a(a(b(a(b(a(a(b(x1)))))))))) → a(b(a(b(a(a(b(a(a(b(a(b(a(x1))))))))))))) (1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{b(☐), a(☐)}

We obtain the transformed TRS
b(a(b(a(a(b(a(b(a(a(b(x1))))))))))) → b(a(b(a(b(a(a(b(a(a(b(a(b(a(x1)))))))))))))) (2)
a(a(b(a(a(b(a(b(a(a(b(x1))))))))))) → a(a(b(a(b(a(a(b(a(a(b(a(b(a(x1)))))))))))))) (3)

1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

[b(x1)] = 2x1 + 0
[a(x1)] = 2x1 + 1

We obtain the labeled TRS
a1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))))) (4)
a1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))) → a1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a0(x1)))))))))))))) (5)
b1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))))) (6)
b1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a0(x1)))))))))))))) (7)

1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
1
[b1(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
all of the following rules can be deleted.
a1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))) → a1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a0(x1)))))))))))))) (5)
b1(a0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a0(x1)))))))))))))) (7)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))))) (8)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a0(b1(a1(x1)))) (9)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))) (10)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(x1)) (11)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(a0(b1(a0(b1(a1(x1))))))) (12)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))) (13)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(x1) (14)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(a0(b1(a0(b1(a1(x1)))))) (15)
b1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(a0(b1(a1(a0(b1(a0(b1(a1(x1))))))))) (16)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a0(b1(a1(x1)))) (17)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))) (18)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(x1)) (19)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(a0(b1(a0(b1(a1(x1))))))) (20)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → b1#(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))) (21)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(x1) (22)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(a0(b1(a0(b1(a1(x1)))))) (23)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(a1(x1)))))))))))))) (24)
a1#(a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))) → a1#(a0(b1(a1(a0(b1(a0(b1(a1(x1))))))))) (25)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.