Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abaaba-aabababababaab.srs)

The rewrite relation of the following TRS is considered.

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

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))))) (8)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) (9)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a1(a0(b0(x1)))))))) (10)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a1(a0(b0(x1)))))) (11)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a1(a0(b0(x1)))) (12)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a1(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))))))) (13)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → a1#(a0(b0(x1))) (14)
b1#(a0(b1(a1(a0(b1(a0(x1))))))) → a1#(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))))))) (15)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(x1) (16)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))))) (17)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) (18)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a1(a0(b1(x1)))))))) (19)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a1(a0(b1(x1)))))) (20)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a1(a0(b1(x1)))) (21)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a1(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))))))) (22)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → a1#(a0(b1(x1))) (23)
b1#(a0(b1(a1(a0(b1(a1(x1))))))) → a1#(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))))))) (24)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))))) (25)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))) (26)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a0(b1(a1(a0(b0(x1)))))))) (27)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a0(b1(a1(a0(b0(x1)))))) (28)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → b1#(a1(a0(b0(x1)))) (29)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → a1#(a0(b0(x1))) (30)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → a1#(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1)))))))))))))) (31)
a1#(a0(b1(a1(a0(b1(a0(x1))))))) → a1#(a1(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b0(x1))))))))))))))) (32)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(x1) (33)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))))) (34)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))) (35)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a0(b1(a1(a0(b1(x1)))))))) (36)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a0(b1(a1(a0(b1(x1)))))) (37)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → b1#(a1(a0(b1(x1)))) (38)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → a1#(a0(b1(x1))) (39)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → a1#(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1)))))))))))))) (40)
a1#(a0(b1(a1(a0(b1(a1(x1))))))) → a1#(a1(a0(b1(a0(b1(a0(b1(a0(b1(a0(b1(a1(a0(b1(x1))))))))))))))) (41)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.