Certification Problem

Input (TPDB SRS_Relative/Mixed_relative_SRS/zr07)

The relative rewrite relation R/S is considered where R is the following TRS

a(c(b(x1))) → b(a(b(a(x1)))) (1)
a(a(x1)) → a(b(a(x1))) (2)

and S is the following TRS.

b(x1) → b(c(x1)) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 3 with strict dimension 1 over the integers
[a(x1)] =
0
0
0
+
1 0 1
0 0 1
0 0 0
· x1
[c(x1)] =
0
0
0
+
1 0 0
0 0 0
0 1 0
· x1
[b(x1)] =
0
2
0
+
1 0 0
0 2 1
0 0 0
· x1
all of the following rules can be deleted.
a(c(b(x1))) → b(a(b(a(x1)))) (1)

1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[a(x1)] =
0
2
+
1 2
0 2
· x1
[b(x1)] =
0
0
+
1 0
0 0
· x1
[c(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
a(a(x1)) → a(b(a(x1))) (2)

1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.