Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z094)

The rewrite relation of the following TRS is considered.

f(x1) → n(c(c(x1))) (1)
c(f(x1)) → f(c(c(x1))) (2)
c(c(x1)) → c(x1) (3)
n(s(x1)) → f(s(s(x1))) (4)
n(f(x1)) → f(n(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 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 0
1 1 0
· x1 +
0 0 0
1 0 0
0 0 0
[f(x1)] =
1 0 0
0 1 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[n(x1)] =
1 1 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
n(s(x1)) → f(s(s(x1))) (4)

1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(n) = 1 weight(n) = 2
prec(c) = 3 weight(c) = 0
prec(f) = 0 weight(f) = 3
all of the following rules can be deleted.
f(x1) → n(c(c(x1))) (1)
c(f(x1)) → f(c(c(x1))) (2)
c(c(x1)) → c(x1) (3)
n(f(x1)) → f(n(x1)) (5)

1.1.1 R is empty

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