Certification Problem

Input (TPDB TRS_Relative/Relative_05/rt3-7)

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

f(b(x),y) → f(x,b(y)) (1)
f(x,a(y)) → f(a(x),y) (2)

and S is the following TRS.

f(x,y) → f(x,b(y)) (3)
f(x,y) → f(a(x),y) (4)

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

1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[f(x1, x2)] =
1
1
+
1 0
1 0
· x1 +
1 1
1 1
· x2
[a(x1)] =
0
1
+
1 0
0 1
· x1
[b(x1)] =
0
0
+
1 1
0 0
· x1
all of the following rules can be deleted.
f(x,a(y)) → f(a(x),y) (2)

1.1.1 R is empty

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