Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z103)

The rewrite relation of the following TRS is considered.

a(d(x1)) d(b(x1)) (1)
a(x1) b(b(b(x1))) (2)
b(d(b(x1))) a(c(x1)) (3)
c(x1) d(x1) (4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(d(b(x1))) c#(x1) (5)
b#(d(b(x1))) a#(c(x1)) (6)
a#(x1) b#(x1) (7)
a#(x1) b#(b(x1)) (8)
a#(x1) b#(b(b(x1))) (9)
a#(d(x1)) b#(x1) (10)

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
[d(x1)] = x1 +
1
[c(x1)] = x1 +
1
[b(x1)] = x1 +
0
[a(x1)] = x1 +
0
[c#(x1)] = x1 +
0
[b#(x1)] = x1 +
1
[a#(x1)] = x1 +
1
together with the usable rules
a(d(x1)) d(b(x1)) (1)
a(x1) b(b(b(x1))) (2)
b(d(b(x1))) a(c(x1)) (3)
c(x1) d(x1) (4)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(d(b(x1))) c#(x1) (5)
a#(d(x1)) b#(x1) (10)
and no rules could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.