Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/baababaab-ababaabaabab.srs)

The rewrite relation of the following TRS is considered.

b(a(a(b(a(b(a(a(b(x1))))))))) → a(b(a(b(a(a(b(a(a(b(a(b(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(b(a(a(b(a(b(a(a(b(x1)))))))))) → b(a(b(a(b(a(a(b(a(a(b(a(b(x1))))))))))))) (2)
a(b(a(a(b(a(b(a(a(b(x1)))))))))) → a(a(b(a(b(a(a(b(a(a(b(a(b(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
b0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b0(x1))))))))))))) (4)
b0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(x1))))))))))))) (5)
a0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → a1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b0(x1))))))))))))) (6)
a0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → a1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(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.
b0(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b0(x1))))))))))))) (4)
b0(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → b1(a0(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(x1))))))))))))) (5)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → a0#(b1(a0(b0(x1)))) (8)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → a0#(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b0(x1)))))))))))) (9)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → a0#(b1(a1(a0(b1(a0(b0(x1))))))) (10)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) → a0#(b1(a1(a0(b1(a1(a0(b1(a0(b0(x1)))))))))) (11)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → a0#(b1(a0(b1(x1)))) (12)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → a0#(b1(a0(b1(a1(a0(b1(a1(a0(b1(a0(b1(x1)))))))))))) (13)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → a0#(b1(a1(a0(b1(a0(b1(x1))))))) (14)
a0#(b1(a1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) → a0#(b1(a1(a0(b1(a1(a0(b1(a0(b1(x1)))))))))) (15)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.