Term Rewriting System R:

   [Z, X, Y, X1, X2]
   active(fst(0, Z)) -> mark(nil)
   active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(len(nil)) -> mark(0)
   active(len(cons(X, Z))) -> mark(s(len(Z)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(fst(X1, X2)) -> fst(active(X1), X2)
   active(fst(X1, X2)) -> fst(X1, active(X2))
   active(from(X)) -> from(active(X))
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   active(len(X)) -> len(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   fst(mark(X1), X2) -> mark(fst(X1, X2))
   fst(X1, mark(X2)) -> mark(fst(X1, X2))
   fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   len(mark(X)) -> mark(len(X))
   len(ok(X)) -> ok(len(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(fst(X1, X2)) -> fst(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   proper(len(X)) -> len(proper(X))
   s(ok(X)) -> ok(s(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: 


   PROPER(len(X)) -> LEN(proper(X))
   PROPER(len(X)) -> PROPER(X)
   PROPER(fst(X1, X2)) -> FST(proper(X1), proper(X2))
   PROPER(fst(X1, X2)) -> PROPER(X1)
   PROPER(fst(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(add(X1, X2)) -> ADD(proper(X1), proper(X2))
   PROPER(add(X1, X2)) -> PROPER(X1)
   PROPER(add(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)
   LEN(mark(X)) -> LEN(X)
   LEN(ok(X)) -> LEN(X)
   ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
   ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
   ACTIVE(add(X1, X2)) -> ACTIVE(X1)
   ACTIVE(fst(X1, X2)) -> FST(active(X1), X2)
   ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
   ACTIVE(add(s(X), Y)) -> S(add(X, Y))
   ACTIVE(add(s(X), Y)) -> ADD(X, Y)
   ACTIVE(fst(s(X), cons(Y, Z))) -> CONS(Y, fst(X, Z))
   ACTIVE(fst(s(X), cons(Y, Z))) -> FST(X, Z)
   ACTIVE(len(cons(X, Z))) -> S(len(Z))
   ACTIVE(len(cons(X, Z))) -> LEN(Z)
   ACTIVE(len(X)) -> LEN(active(X))
   ACTIVE(len(X)) -> ACTIVE(X)
   ACTIVE(from(X)) -> FROM(active(X))
   ACTIVE(from(X)) -> ACTIVE(X)
   ACTIVE(fst(X1, X2)) -> FST(X1, active(X2))
   ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
   ACTIVE(from(X)) -> CONS(X, from(s(X)))
   ACTIVE(from(X)) -> FROM(s(X))
   ACTIVE(from(X)) -> S(X)
   ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
   ACTIVE(add(X1, X2)) -> ACTIVE(X2)
   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)
   ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
   ADD(mark(X1), X2) -> ADD(X1, X2)
   ADD(X1, mark(X2)) -> ADD(X1, X2)
   FROM(mark(X)) -> FROM(X)
   FROM(ok(X)) -> FROM(X)
   FST(ok(X1), ok(X2)) -> FST(X1, X2)
   FST(mark(X1), X2) -> FST(X1, X2)
   FST(X1, mark(X2)) -> FST(X1, X2)
   S(ok(X)) -> S(X)

Furthermore, R contains nine SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

   LEN(ok(X)) -> LEN(X)
   LEN(mark(X)) -> LEN(X)



 This transformation is resulting in no new subcycles.


SCC2:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

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.


SCC4:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC5:

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.


SCC6:

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(ok(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC7:

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



 This transformation is resulting in no new subcycles.


SCC8:

ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(len(X)) -> ACTIVE(X)
ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
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(add(x_1, x_2)) = 1 + x_1 + x_2
POL(from(x_1)) = 1 + x_1
POL(ACTIVE(x_1)) = 1 + x_1
POL(fst(x_1, x_2)) = 1 + x_1 + x_2
POL(len(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ACTIVE(add(X1, X2)) -> ACTIVE(X2)
   ACTIVE(fst(X1, X2)) -> ACTIVE(X2)
   ACTIVE(from(X)) -> ACTIVE(X)
   ACTIVE(len(X)) -> ACTIVE(X)
   ACTIVE(fst(X1, X2)) -> ACTIVE(X1)
   ACTIVE(add(X1, X2)) -> ACTIVE(X1)
   ACTIVE(cons(X1, X2)) -> ACTIVE(X1)



 This transformation is resulting in no new subcycles.


SCC9:

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


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

 resulting in one subcycle.


SCC9.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(add(X1, X2)) -> add(active(X1), X2)
   active(fst(X1, X2)) -> fst(active(X1), X2)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(fst(s(X), cons(Y, Z))) -> mark(cons(Y, fst(X, Z)))
   active(len(cons(X, Z))) -> mark(s(len(Z)))
   active(len(X)) -> len(active(X))
   active(add(0, X)) -> mark(X)
   active(from(X)) -> from(active(X))
   active(fst(X1, X2)) -> fst(X1, active(X2))
   active(len(nil)) -> mark(0)
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(fst(0, Z)) -> mark(nil)
   active(add(X1, X2)) -> add(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   fst(ok(X1), ok(X2)) -> ok(fst(X1, X2))
   fst(mark(X1), X2) -> mark(fst(X1, X2))
   fst(X1, mark(X2)) -> mark(fst(X1, X2))
   len(mark(X)) -> mark(len(X))
   len(ok(X)) -> ok(len(X))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 28.66 seconds.