Certification Problem

Input (TPDB TRS_Standard/AG01/#3.56)

The rewrite relation of the following TRS is considered.

g(c(x,s(y))) → g(c(s(x),y)) (1)
f(c(s(x),y)) → f(c(x,s(y))) (2)
f(f(x)) → f(d(f(x))) (3)
f(x) → x (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
[d(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[c(x1, x2)] =
1 0 1
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 1
· x2 +
0 0 0
0 0 0
0 0 0
[f(x1)] =
1 0 1
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
1 0 0
[s(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 0
1 0 1
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(f(x)) → f(d(f(x))) (3)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1, x2)] =
1 1 1
1 0 1
0 0 1
· x1 +
1 0 1
0 1 1
1 0 1
· x2 +
0 0 0
0 0 0
0 0 0
[f(x1)] =
1 1 1
0 1 0
1 1 1
· x1 +
1 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 1
0 1 1
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 1
0 1 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(x) → x (4)

1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1, x2)] =
1 0 0
1 0 0
0 0 0
· x1 +
1 1 1
0 0 0
0 1 0
· x2 +
1 0 0
0 0 0
0 0 0
[f(x1)] =
1 1 0
0 0 0
1 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
0 0 1
0 1 0
· x1 +
1 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(c(s(x),y)) → f(c(x,s(y))) (2)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c(x1, x2)] =
1 0 0
0 1 0
0 0 0
· x1 +
1 0 0
0 1 1
1 0 0
· x2 +
1 0 0
0 0 0
0 0 0
[s(x1)] =
1 0 0
1 1 0
0 1 0
· x1 +
1 0 0
1 0 0
0 0 0
[g(x1)] =
1 0 1
1 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
g(c(x,s(y))) → g(c(s(x),y)) (1)

1.1.1.1.1 R is empty

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