Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/test1)

The rewrite relation of the following TRS is considered.

f(s(x),y) f(x,s(s(x))) (1)
f(x,s(s(y))) f(y,x) (2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals
[f(x1, x2)] =
1 0 1 1
1 1 0 1
0 0 0 0
1 0 1 1
· x1 +
1 0 0 0
1 0 0 0
0 0 0 0
1 0 0 0
· x2 +
1 0 0 0
0 0 0 0
1 0 0 0
0 0 0 0
[s(x1)] =
1 0 0 1
1 1 0 1
1 0 1 1
0 1 1 1
· x1 +
0 0 0 0
0 0 0 0
1 0 0 0
0 0 0 0
all of the following rules can be deleted.
f(s(x),y) f(x,s(s(x))) (1)

1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(f) = 0 weight(f) = 0
prec(s) = 1 weight(s) = 2
all of the following rules can be deleted.
f(x,s(s(y))) f(y,x) (2)

1.1.1 R is empty

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