Term Rewriting System R:
[Z, X, Y, X1, X2]
fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
activate(nfst(X1, X2)) -> fst(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nadd(X1, X2)) -> add(X1, X2)
activate(nlen(X)) -> len(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:
fst(0, Z) -> nil
add(0, X) -> X
len(nil) -> 0
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(len(x1)) | = 2·x1 |
| POL(n__fst(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = 2·x1 + x2 |
| POL(n__from(x1)) | = x1 |
| POL(0) | = 1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(nil) | = 1 |
| POL(fst(x1, x2)) | = 2·x1 + 2·x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
| POL(n__add(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:
add(X1, X2) -> nadd(X1, X2)
add(s(X), Y) -> s(nadd(activate(X), Y))
where the Polynomial interpretation:
| POL(n__from(x1)) | = x1 |
| POL(from(x1)) | = 2·x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(len(x1)) | = 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(n__fst(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(fst(x1, x2)) | = 2·x1 + 2·x2 |
| POL(n__len(x1)) | = x1 |
| POL(n__add(x1, x2)) | = 1 + x1 + x2 |
| POL(add(x1, x2)) | = 2 + 2·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:
activate(nlen(X)) -> len(X)
where the Polynomial interpretation:
| POL(n__from(x1)) | = x1 |
| POL(from(x1)) | = 2·x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(len(x1)) | = 1 + 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(n__fst(x1, x2)) | = x1 + x2 |
| POL(fst(x1, x2)) | = 2·x1 + 2·x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = 1 + x1 |
| POL(n__add(x1, x2)) | = x1 + x2 |
| POL(add(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(nadd(X1, X2)) -> add(X1, X2)
where the Polynomial interpretation:
| POL(n__from(x1)) | = x1 |
| POL(from(x1)) | = 2·x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(len(x1)) | = 2·x1 |
| POL(n__fst(x1, x2)) | = x1 + x2 |
| POL(fst(x1, x2)) | = 2·x1 + 2·x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
| POL(n__add(x1, x2)) | = 1 + x1 + x2 |
| POL(add(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
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
fst(X1, X2) -> nfst(X1, X2)
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(n__from(x1)) | = x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(len(x1)) | = 2·x1 |
| POL(n__fst(x1, x2)) | = 1 + x1 + x2 |
| POL(fst(x1, x2)) | = 2 + 2·x1 + 2·x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
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
...
→TRS6
↳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(X) -> X
activate(nfst(X1, X2)) -> fst(X1, X2)
len(X) -> nlen(X)
where the Polynomial interpretation:
| POL(from(x1)) | = 1 + 2·x1 |
| POL(n__from(x1)) | = x1 |
| POL(activate(x1)) | = 1 + 2·x1 |
| POL(len(x1)) | = 1 + 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(n__fst(x1, x2)) | = x1 + x2 |
| POL(fst(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
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
...
→TRS7
↳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)) | = 1 + x1 |
| POL(from(x1)) | = x1 |
| POL(activate(x1)) | = x1 |
| POL(len(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
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
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
len(cons(X, Z)) -> s(nlen(activate(Z)))
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(len(x1)) | = 1 + x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(n__len(x1)) | = x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳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
...
→TRS10
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes