Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/baabbaab-aabbaabaabaabb.srs)

The rewrite relation of the following TRS is considered.

b(a(a(b(b(a(a(b(x1)))))))) → a(a(b(b(a(a(b(a(a(b(a(a(b(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(b(a(a(b(x1))))))))) → b(a(a(b(b(a(a(b(a(a(b(a(a(b(b(x1))))))))))))))) (2)
a(b(a(a(b(b(a(a(b(x1))))))))) → a(a(a(b(b(a(a(b(a(a(b(a(a(b(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(b0(b1(a1(a0(b0(x1))))))))) → b1(a1(a0(b0(b1(a1(a0(b1(a1(a0(b1(a1(a0(b0(b0(x1))))))))))))))) (4)
b0(b1(a1(a0(b0(b1(a1(a0(b1(x1))))))))) → b1(a1(a0(b0(b1(a1(a0(b1(a1(a0(b1(a1(a0(b0(b1(x1))))))))))))))) (5)
a0(b1(a1(a0(b0(b1(a1(a0(b0(x1))))))))) → a1(a1(a0(b0(b1(a1(a0(b1(a1(a0(b1(a1(a0(b0(b0(x1))))))))))))))) (6)
a0(b1(a1(a0(b0(b1(a1(a0(b1(x1))))))))) → a1(a1(a0(b0(b1(a1(a0(b1(a1(a0(b1(a1(a0(b0(b1(x1))))))))))))))) (7)

1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b0(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))))))) (8)
b1(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))))))) (9)
b0(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))))))) (10)
b1(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))))))) (11)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0#(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))))))) (12)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0#(a0(a1(b1(x1)))) (13)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0#(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1)))))))))))))) (14)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(x1) (15)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(b0(a0(a1(b1(x1))))) (16)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(a0(a1(b1(b0(a0(a1(b1(x1)))))))) (17)
b0#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))) (18)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0#(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))))))) (19)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0#(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1)))))))))))))) (20)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0#(a0(a1(a1(x1)))) (21)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(b0(a0(a1(a1(x1))))) (22)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(a0(a1(b1(b0(a0(a1(a1(x1)))))))) (23)
b0#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))) (24)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0#(a0(a1(b1(x1)))) (25)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b0#(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1)))))))))))))) (26)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(x1) (27)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(b0(a0(a1(b1(x1))))) (28)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))))))) (29)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(a0(a1(b1(b0(a0(a1(b1(x1)))))))) (30)
b1#(a0(a1(b1(b0(a0(a1(b1(b0(x1))))))))) → b1#(a0(a1(b1(a0(a1(b1(b0(a0(a1(b1(x1))))))))))) (31)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0#(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1)))))))))))))) (32)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b0#(a0(a1(a1(x1)))) (33)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(b0(a0(a1(b1(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))))))) (34)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(b0(a0(a1(a1(x1))))) (35)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(a0(a1(b1(b0(a0(a1(a1(x1)))))))) (36)
b1#(a0(a1(b1(b0(a0(a1(b1(a0(x1))))))))) → b1#(a0(a1(b1(a0(a1(b1(b0(a0(a1(a1(x1))))))))))) (37)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.