Certification Problem

Input (TPDB SRS_Standard/Wenzel_16/bababaaab-aabaaabababab.srs)

The rewrite relation of the following TRS is considered.

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

1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b0(x1)] = x1 +
1
[b1(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
all of the following rules can be deleted.
b0(b1(a0(b1(a0(b1(a1(a1(a0(b0(x1)))))))))) → b1(a1(a0(b1(a1(a1(a0(b1(a0(b1(a0(b1(a0(b0(x1)))))))))))))) (4)
b0(b1(a0(b1(a0(b1(a1(a1(a0(b1(x1)))))))))) → b1(a1(a0(b1(a1(a1(a0(b1(a0(b1(a0(b1(a0(b1(x1)))))))))))))) (5)

1.1.1.1 String Reversal

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

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b0#(a0(b1(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))))) (10)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))) (11)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))) (12)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(x1)))) (13)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(b1(a0(a1(a1(x1)))))))) (14)
b1#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))))) (15)
b1#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))) (16)
b1#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))) (17)
b1#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(x1)))) (18)
b1#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(b1(a0(a1(a1(x1)))))))) (19)

1.1.1.1.1.1 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
b0(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b0(a0(b1(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))))) (8)
b1(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1(a0(b1(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))))) (9)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))))) (11)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(b1(a0(a1(a1(b1(a0(a1(a1(x1)))))))))) (12)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(x1)))) (13)
b0#(a0(a1(a1(b1(a0(b1(a0(b1(a0(x1)))))))))) → b1#(a0(a1(a1(b1(a0(a1(a1(x1)))))))) (14)
and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.