Certification Problem

Input (TPDB SRS_Relative/Waldmann_23/size-10-alpha-2-num-85)

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

a(b(a(x))) → x (1)

and S is the following TRS.

a(b(x)) → b(b(a(a(b(x))))) (2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 -∞ -∞
-∞ -∞ -∞
0 1 -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 -∞ 0
0 0 1
0 0 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
all of the following rules can be deleted.
a(b(a(x))) → x (1)

1.1 R is empty

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