Term Rewriting System R:

   [X, X1, X2]
   a__f(0) -> cons(0, f(s(0)))
   a__f(s(0)) -> a__f(a__p(s(0)))
   a__f(X) -> f(X)
   a__p(s(0)) -> 0
   a__p(X) -> p(X)
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))

Termination of R to be shown.


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

   a__f(0) -> cons(0, f(s(0)))
   a__f(X) -> f(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__f(x_1)) = 2 + x_1
POL(mark(x_1)) = 2*x_1
POL(f(x_1)) = 1 + x_1
POL(0) = 0
POL(p(x_1)) = x_1
POL(a__p(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_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__p(s(0)) -> 0

where the Polynomial interpretation:
POL(s(x_1)) = 2*x_1
POL(a__f(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(f(x_1)) = x_1
POL(0) = 1
POL(p(x_1)) = x_1
POL(a__p(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_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: 

   mark(f(X)) -> a__f(mark(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__f(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(0) = 0
POL(p(x_1)) = x_1
POL(a__p(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_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: 

   mark(0) -> 0

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__f(x_1)) = x_1
POL(mark(x_1)) = 2*x_1
POL(0) = 1
POL(p(x_1)) = x_1
POL(a__p(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_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: 

   mark(cons(X1, X2)) -> cons(mark(X1), X2)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(a__f(x_1)) = x_1
POL(mark(x_1)) = 2*x_1
POL(0) = 0
POL(p(x_1)) = x_1
POL(a__p(x_1)) = x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_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: 

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

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



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


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   MARK(p(X)) -> A__P(mark(X))
   MARK(p(X)) -> MARK(X)
   A__F(s(0)) -> A__F(a__p(s(0)))
   A__F(s(0)) -> A__P(s(0))

Furthermore, R contains one SCC.


SCC1:

MARK(p(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(MARK(x_1)) = 1 + x_1
POL(p(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   MARK(p(X)) -> MARK(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.929 seconds.