Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_9_Luc02c_Z)
The rewrite relation of the following TRS is considered.
2nd(cons1(X,cons(Y,Z))) |
→ |
Y |
(1) |
2nd(cons(X,X1)) |
→ |
2nd(cons1(X,activate(X1))) |
(2) |
from(X) |
→ |
cons(X,n__from(s(X))) |
(3) |
from(X) |
→ |
n__from(X) |
(4) |
activate(n__from(X)) |
→ |
from(X) |
(5) |
activate(X) |
→ |
X |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
[from(x1)] |
= |
· x1 +
|
[cons1(x1, x2)] |
= |
· x1 + · x2 +
|
[activate(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[n__from(x1)] |
= |
· x1 +
|
[2nd(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
2nd(cons1(X,cons(Y,Z))) |
→ |
Y |
(1) |
2nd(cons(X,X1)) |
→ |
2nd(cons1(X,activate(X1))) |
(2) |
from(X) |
→ |
n__from(X) |
(4) |
activate(X) |
→ |
X |
(6) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[from(x1)] |
= |
7 · x1 + 17 |
[activate(x1)] |
= |
13 · x1 + 4 |
[s(x1)] |
= |
1 · x1 + 3 |
[cons(x1, x2)] |
= |
4 · x1 + 1 · x2 + 0 |
[n__from(x1)] |
= |
2 · x1 + 10 |
all of the following rules can be deleted.
from(X) |
→ |
cons(X,n__from(s(X))) |
(3) |
activate(n__from(X)) |
→ |
from(X) |
(5) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.