Term Rewriting System R:
[X, Y, X1, X2]
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Termination of R to be shown.



   R
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

anats -> aadx(azeros)

where the Polynomial interpretation:
  POL(a__nats)=  1  
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(incr(x1))=  x1  
  POL(a__hd(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__adx(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(a__incr(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:

mark(nats) -> anats

where the Polynomial interpretation:
  POL(a__nats)=  1  
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(incr(x1))=  x1  
  POL(a__hd(x1))=  2·x1  
  POL(mark(x1))=  2·x1  
  POL(tl(x1))=  2·x1  
  POL(a__tl(x1))=  2·x1  
  POL(a__adx(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  2·x1  
  POL(nats)=  1  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(a__incr(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:

aincr(X) -> incr(X)
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
ahd(cons(X, Y)) -> mark(X)
mark(cons(X1, X2)) -> cons(X1, X2)
aadx(X) -> adx(X)
azeros -> zeros
atl(cons(X, Y)) -> mark(Y)

where the Polynomial interpretation:
  POL(a__nats)=  0  
  POL(adx(x1))=  1 + 2·x1  
  POL(a__zeros)=  2  
  POL(incr(x1))=  1 + x1  
  POL(a__hd(x1))=  2·x1  
  POL(mark(x1))=  2·x1  
  POL(tl(x1))=  2·x1  
  POL(a__tl(x1))=  2·x1  
  POL(a__adx(x1))=  2 + 2·x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(hd(x1))=  2·x1  
  POL(nats)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  1  
  POL(a__incr(x1))=  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
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

azeros -> cons(0, zeros)
mark(s(X)) -> s(X)
mark(0) -> 0

where the Polynomial interpretation:
  POL(a__nats)=  0  
  POL(adx(x1))=  x1  
  POL(a__zeros)=  1  
  POL(incr(x1))=  x1  
  POL(a__hd(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(a__adx(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(a__incr(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:

anats -> nats

where the Polynomial interpretation:
  POL(a__nats)=  1  
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(incr(x1))=  x1  
  POL(a__hd(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(a__adx(x1))=  x1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  0  
  POL(zeros)=  0  
  POL(a__incr(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:

ahd(X) -> hd(X)

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  1 + x1  
  POL(incr(x1))=  x1  
  POL(a__hd(x1))=  2 + x1  
  POL(mark(x1))=  2·x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(zeros)=  0  
  POL(a__adx(x1))=  x1  
  POL(a__incr(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(incr(X)) -> aincr(mark(X))

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(incr(x1))=  1 + x1  
  POL(a__hd(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(zeros)=  0  
  POL(a__adx(x1))=  x1  
  POL(a__incr(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:

aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))

where the Polynomial interpretation:
  POL(adx(x1))=  1 + x1  
  POL(a__zeros)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(a__hd(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(mark(x1))=  2·x1  
  POL(zeros)=  0  
  POL(a__adx(x1))=  2 + x1  
  POL(a__incr(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
             ...
               →TRS9
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

atl(X) -> tl(X)

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(hd(x1))=  x1  
  POL(a__hd(x1))=  x1  
  POL(mark(x1))=  2·x1  
  POL(tl(x1))=  1 + x1  
  POL(a__tl(x1))=  2 + x1  
  POL(zeros)=  0  
  POL(a__adx(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(hd(X)) -> ahd(mark(X))

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(hd(x1))=  1 + x1  
  POL(a__hd(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(a__tl(x1))=  x1  
  POL(zeros)=  0  
  POL(a__adx(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
             ...
               →TRS11
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

mark(tl(X)) -> atl(mark(X))

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(mark(x1))=  x1  
  POL(zeros)=  0  
  POL(tl(x1))=  1 + x1  
  POL(a__tl(x1))=  x1  
  POL(a__adx(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
             ...
               →TRS12
Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

mark(zeros) -> azeros

where the Polynomial interpretation:
  POL(adx(x1))=  x1  
  POL(a__zeros)=  0  
  POL(mark(x1))=  x1  
  POL(zeros)=  1  
  POL(a__adx(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:

mark(adx(X)) -> aadx(mark(X))

where the Polynomial interpretation:
  POL(adx(x1))=  1 + x1  
  POL(mark(x1))=  x1  
  POL(a__adx(x1))=  x1  
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:
0:01 minutes