Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z107)

The rewrite relation of the following TRS is considered.

a(a(x1)) b(b(b(x1))) (1)
b(b(x1)) c(c(c(x1))) (2)
c(c(x1)) d(d(d(x1))) (3)
b(x1) d(d(x1)) (4)
c(d(d(x1))) a(x1) (5)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a(x1)] = 1 · x1 + 80
[b(x1)] = 1 · x1 + 53
[c(x1)] = 1 · x1 + 35
[d(x1)] = 1 · x1 + 23
all of the following rules can be deleted.
a(a(x1)) b(b(b(x1))) (1)
b(b(x1)) c(c(c(x1))) (2)
c(c(x1)) d(d(d(x1))) (3)
b(x1) d(d(x1)) (4)
c(d(d(x1))) a(x1) (5)

1.1 R is empty

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