Certification Problem

Input (TPDB TRS_Relative/Relative_05/rt3-5)

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

f(g(f(x))) → f(g(g(g(f(x))))) (1)

and S is the following TRS.

g(x) → g(g(x)) (2)
g(x) → g(g(f(g(g(x))))) (3)
f(x) → g(g(f(g(g(x))))) (4)

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 1
0 0 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
0 0 0
1 0 0
all of the following rules can be deleted.
f(g(f(x))) → f(g(g(g(f(x))))) (1)

1.1 R is empty

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