Term Rewriting System R:
[X, Y, Z, X1, X2]
active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(2nd(X)) -> 2nd(active(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
top(ok(X)) -> top(active(X))
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(top(x1)) | = 4 + x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = 1 + 2·x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 2·x1 |
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:
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
where the Polynomial interpretation:
| POL(from(x1)) | = 1 + x1 |
| POL(top(x1)) | = 1 + x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
cons(mark(X1), X2) -> mark(cons(X1, X2))
top(mark(X)) -> top(proper(X))
from(mark(X)) -> mark(from(X))
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(top(x1)) | = 1 + x1 |
| POL(proper(x1)) | = x1 |
| POL(active(x1)) | = 1 + 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = 2·x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(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
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(proper(x1)) | = 2·x1 |
| POL(active(x1)) | = 1 + 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(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
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = 1 + x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(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(ok(X)) -> ok(from(X))
where the Polynomial interpretation:
| POL(from(x1)) | = 2·x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 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:
proper(from(X)) -> from(proper(X))
active(from(X)) -> mark(cons(X, from(s(X))))
where the Polynomial interpretation:
| POL(from(x1)) | = 1 + x1 |
| POL(proper(x1)) | = 2·x1 |
| POL(active(x1)) | = 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(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:
2nd(ok(X)) -> ok(2nd(X))
where the Polynomial interpretation:
| POL(active(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = 2·x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 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
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(mark(X)) -> mark(s(X))
where the Polynomial interpretation:
| POL(active(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(s(x1)) | = 2·x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(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
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(ok(X)) -> ok(s(X))
where the Polynomial interpretation:
| POL(active(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(s(x1)) | = 2·x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 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
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
where the Polynomial interpretation:
| POL(active(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = 2·x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 1 + 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
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(s(X)) -> s(active(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(active(x1)) | = 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(mark(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
...
→TRS13
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(s(X)) -> s(proper(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(mark(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
...
→TRS14
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
2nd(mark(X)) -> mark(2nd(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = 2·x1 |
| POL(mark(x1)) | = 1 + 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
...
→TRS15
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(cons1(x1, x2)) | = x1 + 2·x2 |
| POL(2nd(x1)) | = x1 |
| POL(mark(x1)) | = 1 + 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
...
→TRS16
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(2nd(X)) -> 2nd(proper(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = 2·x1 |
| POL(cons1(x1, x2)) | = x1 + x2 |
| POL(2nd(x1)) | = 1 + 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
...
→TRS17
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
where the Polynomial interpretation:
| POL(proper(x1)) | = 2·x1 |
| POL(cons1(x1, x2)) | = 1 + x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS18
↳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
...
→TRS19
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes