Certification Problem

Input (Example 2 in Hofbauer and Waldmann "Termination of String Rewriting with Matrix Interpretations")

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

a(a(x)) → a(b(a(x))) (1)

and S is the following TRS.

b(x) → b(b(x)) (2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof

1 Rule Removal

Using the matrix interpretations (as in "Termination of String Rewriting with Matrix Interpretations") of dimension 2 with I = {1} and domain EI
[a(x1)] =
1 1
1 0
· x1
[b(x1)] =
1 0
0 0
· x1
all of the following rules can be deleted.
a(a(x)) → a(b(a(x))) (1)

1.1 R is empty

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