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)

Innermost 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
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes