Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/abababaab-aabaabababab.srs)

The rewrite relation of the following TRS is considered.

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

1.1.1 String Reversal

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

1.1.1.1 Dependency Pair Transformation

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

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
0
[b1(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
[b0#(x1)] = x1 +
1
[b1#(x1)] = x1 +
0
together with the usable rules
b1(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b1(a0(b1(a0(b1(a0(b1(a0(a1(b1(a0(a1(a1(x1))))))))))))) (8)
b0(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b0(a0(b1(a0(b1(a0(b1(a0(a1(b1(a0(a1(a1(x1))))))))))))) (9)
b1(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b1(a0(b1(a0(b1(a0(b1(a0(a1(b1(a0(a1(b1(x1))))))))))))) (10)
b0(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b0(a0(b1(a0(b1(a0(b1(a0(a1(b1(a0(a1(b1(x1))))))))))))) (11)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b0#(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b1#(a0(b1(a0(b1(a0(a1(b1(a0(a1(b1(x1))))))))))) (13)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b1#(a0(b1(a0(a1(b1(a0(a1(b1(x1))))))))) (14)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b1#(a0(a1(b1(x1)))) (15)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(b1(x1)))))))))) → b1#(a0(a1(b1(a0(a1(b1(x1))))))) (16)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b1#(a0(b1(a0(b1(a0(a1(b1(a0(a1(a1(x1))))))))))) (18)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b1#(a0(b1(a0(a1(b1(a0(a1(a1(x1))))))))) (19)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b1#(a0(a1(b1(a0(a1(a1(x1))))))) (20)
b0#(a0(a1(b1(a0(b1(a0(b1(a0(a1(x1)))))))))) → b1#(a0(a1(a1(x1)))) (21)
and no rules could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.