Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-87)

The rewrite relation of the following TRS is considered.

a(a(b(a(x1)))) a(b(a(a(x1)))) (1)
b(a(b(b(x1)))) b(b(a(b(x1)))) (2)
a(b(a(b(x1)))) a(b(b(b(x1)))) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[b(x1)] = x1 +
0
[a(x1)] = x1 +
1
all of the following rules can be deleted.
a(b(a(b(x1)))) a(b(b(b(x1)))) (3)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(a(b(b(x1)))) b#(b(a(b(x1)))) (4)
b#(a(b(b(x1)))) b#(a(b(x1))) (5)
b#(a(b(b(x1)))) a#(b(x1)) (6)
a#(a(b(a(x1)))) b#(a(a(x1))) (7)
a#(a(b(a(x1)))) a#(b(a(a(x1)))) (8)
a#(a(b(a(x1)))) a#(a(x1)) (9)

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
[b(x1)] = x1 +
1
[a(x1)] = x1 +
1
[b#(x1)] = x1 +
0
[a#(x1)] = x1 +
0
together with the usable rules
a(a(b(a(x1)))) a(b(a(a(x1)))) (1)
b(a(b(b(x1)))) b(b(a(b(x1)))) (2)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(a(b(b(x1)))) b#(a(b(x1))) (5)
b#(a(b(b(x1)))) a#(b(x1)) (6)
a#(a(b(a(x1)))) b#(a(a(x1))) (7)
a#(a(b(a(x1)))) a#(a(x1)) (9)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.