Certification Problem

Input (TPDB SRS_Relative/Waldmann_06_relative/r3)

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

a(a(x1)) → x1 (1)

and S is the following TRS.

a(a(x1)) → b(a(a(a(b(x1))))) (2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[b(x1)] =
0 -∞ -∞
-5 -2 -∞
0 -∞ 0
· x1
[a(x1)] =
0 -2 -5
4 -1 -1
-1 2 0
· x1
all of the following rules can be deleted.
a(a(x1)) → x1 (1)

1.1 R is empty

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