Term Rewriting System R:

   [X, L, X1, X2]
   a__incr(nil) -> nil
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__incr(X) -> incr(X)
   a__adx(nil) -> nil
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   a__adx(X) -> adx(X)
   a__nats -> a__adx(a__zeros)
   a__nats -> nats
   a__zeros -> cons(0, zeros)
   a__zeros -> zeros
   a__head(cons(X, L)) -> mark(X)
   a__head(X) -> head(X)
   a__tail(cons(X, L)) -> mark(L)
   a__tail(X) -> tail(X)
   mark(incr(X)) -> a__incr(mark(X))
   mark(adx(X)) -> a__adx(mark(X))
   mark(nats) -> a__nats
   mark(zeros) -> a__zeros
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0

Termination of R to be shown.


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

   a__adx(nil) -> nil

where the Polynomial interpretation:
POL(nil) = 1
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = 2*x_1
POL(a__tail(x_1)) = x_1
POL(nats) = 0
POL(mark(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(head(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(tail(x_1)) = x_1
POL(a__head(x_1)) = x_1
POL(adx(x_1)) = 2*x_1
POL(a__zeros) = 0
POL(incr(x_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__nats -> a__adx(a__zeros)

where the Polynomial interpretation:
POL(nil) = 0
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(a__tail(x_1)) = x_1
POL(nats) = 1
POL(mark(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(head(x_1)) = x_1
POL(tail(x_1)) = x_1
POL(a__head(x_1)) = x_1
POL(adx(x_1)) = x_1
POL(a__zeros) = 0
POL(incr(x_1)) = x_1
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__head(cons(X, L)) -> mark(X)

where the Polynomial interpretation:
POL(nil) = 0
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(a__tail(x_1)) = x_1
POL(nats) = 0
POL(mark(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(head(x_1)) = 1 + x_1
POL(tail(x_1)) = x_1
POL(a__head(x_1)) = 1 + x_1
POL(adx(x_1)) = x_1
POL(a__zeros) = 0
POL(incr(x_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__tail(cons(X, L)) -> mark(L)

where the Polynomial interpretation:
POL(nil) = 0
POL(s(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(a__tail(x_1)) = 1 + x_1
POL(nats) = 0
POL(mark(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(head(x_1)) = x_1
POL(tail(x_1)) = 1 + x_1
POL(a__head(x_1)) = x_1
POL(adx(x_1)) = x_1
POL(a__zeros) = 0
POL(incr(x_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.


R contains the following Dependency Pairs: 


   MARK(nats) -> A__NATS
   MARK(s(X)) -> MARK(X)
   MARK(head(X)) -> A__HEAD(mark(X))
   MARK(head(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(incr(X)) -> A__INCR(mark(X))
   MARK(incr(X)) -> MARK(X)
   MARK(tail(X)) -> A__TAIL(mark(X))
   MARK(tail(X)) -> MARK(X)
   MARK(zeros) -> A__ZEROS
   MARK(adx(X)) -> A__ADX(mark(X))
   MARK(adx(X)) -> MARK(X)
   A__INCR(cons(X, L)) -> MARK(X)
   A__ADX(cons(X, L)) -> A__INCR(cons(mark(X), adx(L)))
   A__ADX(cons(X, L)) -> MARK(X)

Furthermore, R contains one SCC.


SCC1:

MARK(adx(X)) -> MARK(X)
A__ADX(cons(X, L)) -> MARK(X)
A__ADX(cons(X, L)) -> A__INCR(cons(mark(X), adx(L)))
MARK(adx(X)) -> A__ADX(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
A__INCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> A__INCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(head(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(A__INCR(x_1)) = x_1
POL(a__adx(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(a__tail(x_1)) = x_1
POL(nats) = 0
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(zeros) = 0
POL(A__ADX(x_1)) = 1 + x_1
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(head(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(tail(x_1)) = x_1
POL(a__head(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

The following Dependency Pairs can be deleted:

   MARK(adx(X)) -> MARK(X)
   A__ADX(cons(X, L)) -> MARK(X)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

MARK(adx(X)) -> A__ADX(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(incr(X)) -> A__INCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(head(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)
A__INCR(cons(X, L)) -> MARK(X)
A__ADX(cons(X, L)) -> A__INCR(cons(mark(X), adx(L)))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(A__INCR(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(s(x_1)) = x_1
POL(a__tail(x_1)) = x_1
POL(nats) = 0
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(zeros) = 0
POL(A__ADX(x_1)) = x_1
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(head(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(tail(x_1)) = x_1
POL(a__head(x_1)) = 1 + x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0

The following Dependency Pairs can be deleted:

   MARK(head(X)) -> MARK(X)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1:

MARK(tail(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(incr(X)) -> A__INCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
A__INCR(cons(X, L)) -> MARK(X)
A__ADX(cons(X, L)) -> A__INCR(cons(mark(X), adx(L)))
MARK(adx(X)) -> A__ADX(mark(X))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(A__INCR(x_1)) = x_1
POL(a__adx(x_1)) = x_1
POL(s(x_1)) = x_1
POL(a__tail(x_1)) = 1 + x_1
POL(nats) = 0
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(zeros) = 0
POL(A__ADX(x_1)) = x_1
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(head(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(tail(x_1)) = 1 + x_1
POL(a__head(x_1)) = x_1
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 0
POL(a__nats) = 0

The following Dependency Pairs can be deleted:

   MARK(tail(X)) -> MARK(X)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1.MRR1:

A__ADX(cons(X, L)) -> A__INCR(cons(mark(X), adx(L)))
MARK(adx(X)) -> A__ADX(mark(X))
A__INCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> A__INCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   a__adx(X) -> adx(X)
   a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L)))
   mark(nats) -> a__nats
   mark(s(X)) -> s(mark(X))
   mark(head(X)) -> a__head(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(incr(X)) -> a__incr(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(zeros) -> a__zeros
   mark(adx(X)) -> a__adx(mark(X))
   mark(0) -> 0
   a__incr(X) -> incr(X)
   a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L))
   a__incr(nil) -> nil
   a__nats -> nats
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)
   a__zeros -> cons(0, zeros)
   a__zeros -> zeros


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(A__INCR(x_1)) = x_1
POL(nil) = 0
POL(a__adx(x_1)) = x_1
POL(s(x_1)) = x_1
POL(a__tail(x_1)) = 0
POL(nats) = 0
POL(MARK(x_1)) = 1 + x_1
POL(mark(x_1)) = x_1
POL(zeros) = 1
POL(A__ADX(x_1)) = 1 + x_1
POL(0) = 0
POL(a__incr(x_1)) = x_1
POL(cons(x_1, x_2)) = 1 + x_1
POL(head(x_1)) = 0
POL(tail(x_1)) = 0
POL(a__head(x_1)) = 0
POL(adx(x_1)) = x_1
POL(incr(x_1)) = x_1
POL(a__zeros) = 1
POL(a__nats) = 0

 resulting in one subcycle.


SCC1.MRR1.MRR1.MRR1.Polo1:

MARK(incr(X)) -> MARK(X)
MARK(s(X)) -> MARK(X)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(MARK(x_1)) = 1 + x_1
POL(incr(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   MARK(incr(X)) -> MARK(X)
   MARK(s(X)) -> MARK(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 5.648 seconds.