Certification Problem

Input (TPDB TRS_Relative/Relative_05/rt2-4)

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

f(g(x)) → x (1)

and S is the following TRS.

a → h(g(f(a))) (2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

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

1.1 R is empty

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