Term Rewriting System R:

   [X, XS, X1, X2]
   active(zeros) -> mark(cons(0, zeros))
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   tail(ok(X)) -> ok(tail(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tail(X)) -> tail(proper(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Termination of R to be shown.


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

   active(tail(cons(X, XS))) -> mark(XS)

where the Polynomial interpretation:
POL(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(tail(x_1)) = 1 + x_1
POL(top(x_1)) = 1 + x_1
POL(mark(x_1)) = x_1
POL(zeros) = 0
POL(ok(x_1)) = x_1
POL(0) = 0
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.


R contains the following Dependency Pairs: 


   ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
   ACTIVE(zeros) -> CONS(0, zeros)
   ACTIVE(tail(X)) -> TAIL(active(X))
   ACTIVE(tail(X)) -> ACTIVE(X)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
   TAIL(mark(X)) -> TAIL(X)
   TAIL(ok(X)) -> TAIL(X)
   PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
   PROPER(cons(X1, X2)) -> PROPER(X1)
   PROPER(cons(X1, X2)) -> PROPER(X2)
   PROPER(tail(X)) -> TAIL(proper(X))
   PROPER(tail(X)) -> PROPER(X)
   TOP(mark(X)) -> TOP(proper(X))
   TOP(mark(X)) -> PROPER(X)
   TOP(ok(X)) -> TOP(active(X))
   TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains five SCCs.


SCC1:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)

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(ok(x_1)) = 1 + x_1
POL(CONS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)



 This transformation is resulting in no new subcycles.


SCC2:

TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(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(TAIL(x_1)) = 1 + x_1
POL(mark(x_1)) = 1 + x_1
POL(ok(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   TAIL(ok(X)) -> TAIL(X)
   TAIL(mark(X)) -> TAIL(X)



 This transformation is resulting in no new subcycles.


SCC3:

ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)

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(tail(x_1)) = 1 + x_1
POL(ACTIVE(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ACTIVE(tail(X)) -> ACTIVE(X)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)



 This transformation is resulting in no new subcycles.


SCC4:

PROPER(tail(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)

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(PROPER(x_1)) = 1 + x_1
POL(tail(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   PROPER(tail(X)) -> PROPER(X)
   PROPER(cons(X1, X2)) -> PROPER(X2)
   PROPER(cons(X1, X2)) -> PROPER(X1)



 This transformation is resulting in no new subcycles.


SCC5:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(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(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(tail(x_1)) = x_1
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))


 This transformation is resulting in one new subcycle:


SCC5.MRR1:

TOP(mark(X)) -> TOP(proper(X))
TOP(ok(X)) -> TOP(active(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: 

   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(tail(X)) -> tail(proper(X))
   proper(0) -> ok(0)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(zeros) -> mark(cons(0, zeros))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   tail(mark(X)) -> mark(tail(X))
   tail(ok(X)) -> ok(tail(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(tail(x_1)) = x_1
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(zeros) = 1
POL(ok(x_1)) = x_1
POL(0) = 0
POL(cons(x_1, x_2)) = x_1

 resulting in one subcycle.


SCC5.MRR1.Polo1:

TOP(ok(X)) -> TOP(active(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(active(x_1)) = x_1
POL(tail(x_1)) = x_1
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(zeros) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(tail(X)) -> tail(proper(X))
   proper(0) -> ok(0)


 This transformation is resulting in one new subcycle:


SCC5.MRR1.Polo1.MRR1:

TOP(ok(X)) -> TOP(active(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(active(x_1)) = x_1
POL(tail(x_1)) = x_1
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 1 + x_1
POL(zeros) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   TOP(ok(X)) -> TOP(active(X))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.545 seconds.