Term Rewriting System R:
[X, Y, Z, X1, X2]
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(X) -> X
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
first(0, X) -> nil
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(n__from(x1)) | = x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(first(x1, x2)) | = x1 + 2·x2 |
| POL(0) | = 1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(s(x1)) | = x1 |
| POL(n__first(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
first(X1, X2) -> nfirst(X1, X2)
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(n__from(x1)) | = x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(first(x1, x2)) | = 2 + x1 + 2·x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(n__first(x1, x2)) | = 1 + x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(X) -> X
where the Polynomial interpretation:
| POL(from(x1)) | = 1 + 2·x1 |
| POL(n__from(x1)) | = x1 |
| POL(activate(x1)) | = 1 + 2·x1 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(n__first(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(nfrom(X)) -> from(X)
where the Polynomial interpretation:
| POL(n__from(x1)) | = x1 |
| POL(from(x1)) | = x1 |
| POL(activate(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes