Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z008)

The rewrite relation of the following TRS is considered.

a(s(x1)) → s(a(x1)) (1)
b(a(b(s(x1)))) → a(b(s(a(x1)))) (2)
b(a(b(b(x1)))) → a(b(a(b(x1)))) (3)
a(b(a(a(x1)))) → b(a(b(a(x1)))) (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
[a(x1)] =
1 1 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 1
0 1 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[b(x1)] =
1 0 1
0 1 0
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
a(s(x1)) → s(a(x1)) (1)
b(a(b(s(x1)))) → a(b(s(a(x1)))) (2)

1.1 Rule Removal

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

1.1.1 R is empty

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