Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z040)

The rewrite relation of the following TRS is considered.

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

1.1.1 Dependency Pair Transformation

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

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.