Certification Problem

Input (TPDB TRS_Standard/Rubio_04/aoto)

The rewrite relation of the following TRS is considered.

f(f(X)) → f(g(f(g(f(X))))) (1)
f(g(f(X))) → f(g(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 naturals
[g(x1)] =
1 0 0
0 0 0
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[f(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
f(f(X)) → f(g(f(g(f(X))))) (1)

1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(g) = 0 weight(g) = 2
prec(f) = 1 weight(f) = 2
all of the following rules can be deleted.
f(g(f(X))) → f(g(X)) (2)

1.1.1 R is empty

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