Term Rewriting System R:
[x, y]
natsactive -> addactive(zerosactive)
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(add(x)) -> addactive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
addactive(x) -> add(x)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
natsactive -> addactive(zerosactive)
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = x1 |
| POL(incr_active(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(nats) | = 1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(nats_active) | = 1 |
| POL(tl_active(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(incr_active(x1)) | = x1 |
| POL(hd_active(x1)) | = 2·x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = 1 + 2·x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 2·x1 |
| POL(nats) | = 0 |
| POL(s(x1)) | = x1 |
| POL(nats_active) | = 0 |
| POL(zeros) | = 0 |
| POL(tl_active(x1)) | = 2 + 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
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
zerosactive -> zeros
zerosactive -> cons(0, zeros)
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
mark(nats) -> natsactive
mark(0) -> 0
where the Polynomial interpretation:
| POL(zeros_active) | = 1 |
| POL(incr_active(x1)) | = x1 |
| POL(hd_active(x1)) | = 1 + x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(tl(x1)) | = x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 1 + x1 |
| POL(nats) | = 0 |
| POL(s(x1)) | = x1 |
| POL(nats_active) | = 0 |
| POL(zeros) | = 0 |
| POL(tl_active(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:
natsactive -> nats
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(incr_active(x1)) | = x1 |
| POL(hd_active(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(nats) | = 0 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(nats_active) | = 1 |
| POL(tl_active(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:
mark(incr(x)) -> incractive(mark(x))
addactive(x) -> add(x)
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = 2·x1 |
| POL(incr_active(x1)) | = 1 + x1 |
| POL(incr(x1)) | = 1 + x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = x1 |
| POL(add(x1)) | = 1 + x1 |
| POL(add_active(x1)) | = 2 + x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 2·x1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(tl_active(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:
hdactive(cons(x, y)) -> mark(x)
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = 1 + x1 |
| POL(incr_active(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 1 + x1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(tl_active(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:
mark(tl(x)) -> tlactive(mark(x))
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = x1 |
| POL(incr_active(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = 1 + x1 |
| POL(add(x1)) | = x1 |
| POL(add_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(tl_active(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:
addactive(cons(x, y)) -> incractive(cons(x, add(y)))
where the Polynomial interpretation:
| POL(add_active(x1)) | = 2 + x1 |
| POL(zeros_active) | = 0 |
| POL(incr_active(x1)) | = x1 |
| POL(hd_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(zeros) | = 0 |
| POL(add(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:
mark(hd(x)) -> hdactive(mark(x))
where the Polynomial interpretation:
| POL(add_active(x1)) | = x1 |
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = 1 + x1 |
| POL(incr_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 1 + x1 |
| POL(incr(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(zeros) | = 0 |
| POL(add(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:
mark(add(x)) -> addactive(mark(x))
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(add_active(x1)) | = x1 |
| POL(incr_active(x1)) | = x1 |
| POL(hd_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(add(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:
incractive(x) -> incr(x)
incractive(cons(x, y)) -> cons(s(x), incr(y))
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(incr_active(x1)) | = 1 + x1 |
| POL(hd_active(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(incr(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 0 |
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:
hdactive(x) -> hd(x)
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(hd_active(x1)) | = 1 + x1 |
| POL(hd(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 0 |
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:
mark(zeros) -> zerosactive
where the Polynomial interpretation:
| POL(zeros_active) | = 0 |
| POL(mark(x1)) | = 1 + x1 |
| POL(zeros) | = 0 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS14
↳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
...
→TRS15
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes