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 ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 2
0 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.