Term Rewriting System R:

   [X, Y, X1, X2]
   a__nats -> a__adx(a__zeros)
   a__nats -> nats
   a__zeros -> cons(0, zeros)
   a__zeros -> zeros
   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__incr(X) -> incr(X)
   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))
   a__adx(X) -> adx(X)
   a__hd(cons(X, Y)) -> mark(X)
   a__hd(X) -> hd(X)
   a__tl(cons(X, Y)) -> mark(Y)
   a__tl(X) -> tl(X)
   mark(nats) -> a__nats
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros) -> a__zeros
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(tl(X)) -> a__tl(mark(X))
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(0) -> 0
   mark(s(X)) -> s(X)

Termination of R to be shown.


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

   a__nats -> a__adx(a__zeros)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(nats) = 1
POL(tl(x_1)) = x_1
POL(hd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(a__hd(x_1)) = x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__nats -> nats

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(nats) = 1
POL(tl(x_1)) = 2*x_1
POL(mark(x_1)) = 2*x_1
POL(hd(x_1)) = 2*x_1
POL(a__tl(x_1)) = 2*x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(a__hd(x_1)) = 2*x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__zeros -> cons(0, zeros)
   a__zeros -> zeros

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(nats) = 0
POL(tl(x_1)) = 2*x_1
POL(mark(x_1)) = 2*x_1
POL(hd(x_1)) = 2*x_1
POL(a__tl(x_1)) = 2*x_1
POL(zeros) = 1
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(a__hd(x_1)) = 2*x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 2
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__incr(cons(X, Y)) -> cons(s(X), incr(Y))
   a__incr(X) -> incr(X)
   a__hd(cons(X, Y)) -> mark(X)
   mark(cons(X1, X2)) -> cons(X1, X2)
   a__tl(cons(X, Y)) -> mark(Y)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = 2*x_1
POL(nats) = 0
POL(mark(x_1)) = 2*x_1
POL(hd(x_1)) = 2*x_1
POL(tl(x_1)) = 2*x_1
POL(a__tl(x_1)) = 2*x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = 2 + x_1
POL(cons(x_1, x_2)) = 2 + x_1 + x_2
POL(a__hd(x_1)) = 2*x_1
POL(adx(x_1)) = 2*x_1
POL(a__zeros) = 0
POL(incr(x_1)) = 1 + x_1
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y)))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = 2*x_1
POL(nats) = 0
POL(hd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
POL(a__hd(x_1)) = x_1
POL(adx(x_1)) = 2*x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__adx(X) -> adx(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = 2 + x_1
POL(nats) = 0
POL(hd(x_1)) = x_1
POL(mark(x_1)) = 2*x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(adx(x_1)) = 1 + x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__hd(X) -> hd(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(nats) = 0
POL(mark(x_1)) = 2*x_1
POL(hd(x_1)) = 1 + x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(a__hd(x_1)) = 2 + x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(nats) -> a__nats

where the Polynomial interpretation:
POL(a__adx(x_1)) = x_1
POL(s(x_1)) = x_1
POL(nats) = 1
POL(mark(x_1)) = x_1
POL(hd(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(adx(X)) -> a__adx(mark(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(hd(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(adx(x_1)) = 1 + x_1
POL(a__zeros) = 0
POL(incr(x_1)) = x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(zeros) -> a__zeros

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(hd(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(zeros) = 1
POL(0) = 0
POL(incr(x_1)) = x_1
POL(a__incr(x_1)) = x_1
POL(a__zeros) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(incr(X)) -> a__incr(mark(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(hd(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(0) = 0
POL(incr(x_1)) = 1 + x_1
POL(a__incr(x_1)) = x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(hd(X)) -> a__hd(mark(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__hd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(tl(x_1)) = x_1
POL(hd(x_1)) = 1 + x_1
POL(a__tl(x_1)) = x_1
POL(0) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(tl(X)) -> a__tl(mark(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(mark(x_1)) = 2*x_1
POL(tl(x_1)) = 1 + x_1
POL(a__tl(x_1)) = 1 + x_1
POL(0) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   mark(0) -> 0
   mark(s(X)) -> s(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = x_1
POL(0) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


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

   a__tl(X) -> tl(X)

where the Polynomial interpretation:
POL(tl(x_1)) = x_1
POL(a__tl(x_1)) = 1 + x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.910 seconds.