Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/baabababba-abababbaababab.srs)

The rewrite relation of the following TRS is considered.

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

1.1.1 String Reversal

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

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(x1)) (12)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(x1)))) (13)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(a0(b1(x1)))))) (14)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))))))) (15)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))))) (16)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))) (17)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a1#(b1(b0(a0(b1(a0(b1(a0(b1(x1))))))))) (18)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(x1)))))) (19)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))))))) (20)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(a1(x1)))) (21)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))))) (22)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(a1(x1)) (23)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))) (24)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a1#(x1) (25)
a0#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a1#(b1(b0(a0(b1(a0(b1(a0(a1(x1))))))))) (26)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(x1)) (27)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(x1)))) (28)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(a0(b1(x1)))))) (29)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))))))) (30)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))))) (31)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a0#(a1(b1(b0(a0(b1(a0(b1(a0(b1(x1)))))))))) (32)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(b0(x1))))))))))) → a1#(b1(b0(a0(b1(a0(b1(a0(b1(x1))))))))) (33)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(x1)))))) (34)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))))))) (35)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(a1(x1)))) (36)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(b1(a0(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))))) (37)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(a1(x1)) (38)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a0#(a1(b1(b0(a0(b1(a0(b1(a0(a1(x1)))))))))) (39)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a1#(x1) (40)
a1#(b1(b0(a0(b1(a0(b1(a0(a1(b1(a0(x1))))))))))) → a1#(b1(b0(a0(b1(a0(b1(a0(a1(x1))))))))) (41)

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.