Certification Problem

Input (TPDB TRS_Relative/Relative_05/rt2-1)

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

T(I(x),y) → T(x,y) (1)

and S is the following TRS.

T(x,y) → T(x,I(y)) (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
[T(x1, x2)] =
1
0
+
1 1
1 1
· x1 +
1 0
1 0
· x2
[I(x1)] =
0
1
+
1 0
0 1
· x1
all of the following rules can be deleted.
T(I(x),y) → T(x,y) (1)

1.1 R is empty

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