Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z033)

The rewrite relation of the following TRS is considered.

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

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → b0#(b1(a0(x1))) (8)
b0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → b0#(b1(a0(b0(b1(a0(x1)))))) (9)
b0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b0(b1(a0(x1)))) (10)
b0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b0(b1(a0(b0(b1(a0(x1))))))) (11)
b0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b1(a1(a0(b0(b1(a0(b0(b1(a0(x1)))))))))) (12)
b0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → b0#(b1(a0(b0(b1(a1(x1)))))) (13)
b0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → b0#(b1(a1(x1))) (14)
b0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b0(b1(a0(b0(b1(a1(x1))))))) (15)
b0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b0(b1(a1(x1)))) (16)
b0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b1(a1(a0(b0(b1(a0(b0(b1(a1(x1)))))))))) (17)
a0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → b0#(b1(a0(x1))) (18)
a0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → b0#(b1(a0(b0(b1(a0(x1)))))) (19)
a0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b0(b1(a0(x1)))) (20)
a0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b0(b1(a0(b0(b1(a0(x1))))))) (21)
a0#(b1(a0(b0(b1(a0(b1(a0(x1)))))))) → a0#(b1(a1(a0(b0(b1(a0(b0(b1(a0(x1)))))))))) (22)
a0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → b0#(b1(a0(b0(b1(a1(x1)))))) (23)
a0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → b0#(b1(a1(x1))) (24)
a0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b0(b1(a0(b0(b1(a1(x1))))))) (25)
a0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b0(b1(a1(x1)))) (26)
a0#(b1(a0(b0(b1(a0(b1(a1(x1)))))))) → a0#(b1(a1(a0(b0(b1(a0(b0(b1(a1(x1)))))))))) (27)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.