Certification Problem

Input (TPDB SRS_Relative/Zantema_06_relative/rel05)

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

b(c(a(x1))) → d(d(x1)) (1)
b(x1) → c(c(x1)) (2)
a(a(x1)) → a(x1) (3)

and S is the following TRS.

a(b(x1)) → d(x1) (4)
d(x1) → a(b(x1)) (5)

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

1.1 Rule Removal

Using the linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1 over the naturals
[c(x1)] =
1 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
· x1 +
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
[d(x1)] =
1 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
· x1 +
1 0 0 0 0
0 0 0 0 0
1 0 0 0 0
1 0 0 0 0
1 0 0 0 0
[a(x1)] =
1 0 1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 0 1 0
· x1 +
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
[b(x1)] =
1 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
· x1 +
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
all of the following rules can be deleted.
b(x1) → c(c(x1)) (2)
a(a(x1)) → a(x1) (3)

1.1.1 R is empty

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