Term Rewriting System R:

   [X, Y, Z, X1, X2]
   active(first(0, X)) -> mark(nil)
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   first(mark(X1), X2) -> mark(first(X1, X2))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   s(mark(X)) -> mark(s(X))
   s(ok(X)) -> ok(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
   ACTIVE(from(X)) -> FROM(active(X))
   ACTIVE(from(X)) -> ACTIVE(X)
   ACTIVE(from(X)) -> CONS(X, from(s(X)))
   ACTIVE(from(X)) -> FROM(s(X))
   ACTIVE(from(X)) -> S(X)
   ACTIVE(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
   ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
   ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
   ACTIVE(first(X1, X2)) -> ACTIVE(X1)
   ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
   ACTIVE(first(X1, X2)) -> ACTIVE(X2)
   ACTIVE(s(X)) -> S(active(X))
   ACTIVE(s(X)) -> ACTIVE(X)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
   TOP(mark(X)) -> TOP(proper(X))
   TOP(mark(X)) -> PROPER(X)
   TOP(ok(X)) -> TOP(active(X))
   TOP(ok(X)) -> ACTIVE(X)
   PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
   PROPER(first(X1, X2)) -> PROPER(X1)
   PROPER(first(X1, X2)) -> PROPER(X2)
   PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
   PROPER(cons(X1, X2)) -> PROPER(X1)
   PROPER(cons(X1, X2)) -> PROPER(X2)
   PROPER(from(X)) -> FROM(proper(X))
   PROPER(from(X)) -> PROPER(X)
   PROPER(s(X)) -> S(proper(X))
   PROPER(s(X)) -> PROPER(X)
   FIRST(X1, mark(X2)) -> FIRST(X1, X2)
   FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
   FIRST(mark(X1), X2) -> FIRST(X1, X2)
   FROM(mark(X)) -> FROM(X)
   FROM(ok(X)) -> FROM(X)
   S(ok(X)) -> S(X)
   S(mark(X)) -> S(X)

Furthermore, R contains seven 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:

FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(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(FROM(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:

   FROM(ok(X)) -> FROM(X)
   FROM(mark(X)) -> FROM(X)



 This transformation is resulting in no new subcycles.


SCC3:

S(mark(X)) -> S(X)
S(ok(X)) -> S(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(ok(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   S(mark(X)) -> S(X)
   S(ok(X)) -> S(X)



 This transformation is resulting in no new subcycles.


SCC4:

FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(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(FIRST(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC5:

ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(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(first(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1
POL(from(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(s(X)) -> ACTIVE(X)
   ACTIVE(first(X1, X2)) -> ACTIVE(X2)
   ACTIVE(first(X1, X2)) -> ACTIVE(X1)
   ACTIVE(from(X)) -> ACTIVE(X)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)



 This transformation is resulting in no new subcycles.


SCC6:

PROPER(s(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(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(first(x_1, x_2)) = 1 + x_1 + x_2
POL(PROPER(x_1)) = 1 + x_1
POL(s(x_1)) = 1 + x_1
POL(from(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   PROPER(s(X)) -> PROPER(X)
   PROPER(from(X)) -> PROPER(X)
   PROPER(cons(X1, X2)) -> PROPER(X2)
   PROPER(cons(X1, X2)) -> PROPER(X1)
   PROPER(first(X1, X2)) -> PROPER(X2)
   PROPER(first(X1, X2)) -> PROPER(X1)



 This transformation is resulting in no new subcycles.


SCC7:

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

   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(0, X)) -> mark(nil)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   proper(nil) -> ok(nil)
   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   s(mark(X)) -> mark(s(X))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(first(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(proper(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(TOP(x_1)) = 1 + x_1
POL(mark(x_1)) = 1 + x_1
POL(from(x_1)) = 1 + x_1
POL(ok(x_1)) = x_1
POL(0) = 1
POL(cons(x_1, x_2)) = x_1

 resulting in one subcycle.


SCC7.Polo1:

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: 

   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
   active(first(X1, X2)) -> first(active(X1), X2)
   active(first(0, X)) -> mark(nil)
   active(first(X1, X2)) -> first(X1, active(X2))
   active(s(X)) -> s(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   s(mark(X)) -> mark(s(X))
   first(X1, mark(X2)) -> mark(first(X1, X2))
   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
   first(mark(X1), X2) -> mark(first(X1, X2))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(first(x_1, x_2)) = x_1
POL(nil) = 0
POL(s(x_1)) = x_1
POL(TOP(x_1)) = 1 + x_1
POL(mark(x_1)) = 0
POL(from(x_1)) = x_1
POL(ok(x_1)) = 1 + x_1
POL(0) = 0
POL(cons(x_1, x_2)) = x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 5.679 seconds.