Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abaaaabaa-aaabaabaaaab.srs)

The rewrite relation of the following TRS is considered.

a(b(a(a(a(a(b(a(a(x1))))))))) → a(a(a(b(a(a(b(a(a(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(a(a(b(a(a(x1)))))))))) → b(a(a(a(b(a(a(b(a(a(a(a(b(x1))))))))))))) (2)
a(a(b(a(a(a(a(b(a(a(x1)))))))))) → a(a(a(a(b(a(a(b(a(a(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(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))))) (4)
a1(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))))))) (5)
b1(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))))) (6)
b1(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1(a1(a1(a0(b1(a1(a0(b1(a1(a1(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(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1#(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))) (8)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1#(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))))))) (9)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1#(a1(a1(a1(a0(b0(x1)))))) (10)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b0(x1))) (11)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))))) (12)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b1(a1(a1(a1(a0(b0(x1)))))))) (13)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a0(b0(x1)))) (14)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1)))))))))))) (15)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a1(a0(b0(x1))))) (16)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(x1) (17)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))) (18)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))))) (19)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(a1(a1(a1(a0(b1(x1)))))) (20)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(x1))) (21)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))) (22)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(a1(a1(a1(a0(b1(x1)))))))) (23)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a0(b1(x1)))) (24)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1)))))))))))) (25)
b1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a1(a0(b1(x1))))) (26)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1#(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))) (27)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → b1#(a1(a1(a1(a0(b0(x1)))))) (28)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b0(x1))) (29)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))))) (30)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a0(b1(a1(a1(a1(a0(b0(x1)))))))) (31)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a0(b0(x1)))) (32)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1)))))))))))) (33)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a1(a0(b0(x1))))) (34)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a0(x1)))))))))) → a1#(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b0(x1))))))))))))) (35)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(x1) (36)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))) (37)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → b1#(a1(a1(a1(a0(b1(x1)))))) (38)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(x1))) (39)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))) (40)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a0(b1(a1(a1(a1(a0(b1(x1)))))))) (41)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a0(b1(x1)))) (42)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1)))))))))))) (43)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a1(a0(b1(x1))))) (44)
a1#(a0(b1(a1(a1(a1(a0(b1(a1(a1(x1)))))))))) → a1#(a1(a1(a0(b1(a1(a0(b1(a1(a1(a1(a0(b1(x1))))))))))))) (45)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.