Certification Problem

Input (TPDB TRS_Standard/Rubio_04/p266)

The rewrite relation of the following TRS is considered.

f(f(X)) → f(a(b(f(X)))) (1)
f(a(g(X))) → b(X) (2)
b(X) → a(X) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
f(f(X)) → f(b(a(f(X)))) (4)
g(a(f(X))) → b(X) (5)
b(X) → a(X) (3)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[b(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
1 0 0
[g(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
1 0 0
[f(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[a(x1)] =
1 0 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.
g(a(f(X))) → b(X) (5)
b(X) → a(X) (3)

1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[b(x1)] =
1 0 0
0 0 0
1 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[f(x1)] =
1 1 0
1 1 0
1 0 0
· x1 +
1 0 0
1 0 0
0 0 0
[a(x1)] =
1 0 0
1 0 1
0 1 1
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(f(X)) → f(b(a(f(X)))) (4)

1.1.1.1 R is empty

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