Term Rewriting System R:

   [Z, X, Y, X1, X2]
   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(X1, X2) -> n__fst(X1, X2)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   add(X1, X2) -> n__add(X1, X2)
   len(nil) -> 0
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   len(X) -> n__len(X)
   s(X) -> n__s(X)
   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(X)
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(n__len(X)) -> len(activate(X))
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__add(X1, X2)) -> ADD(activate(X1), activate(X2))
   ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__len(X)) -> LEN(activate(X))
   ACTIVATE(n__len(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> S(X)
   ACTIVATE(n__fst(X1, X2)) -> FST(activate(X1), activate(X2))
   ACTIVATE(n__fst(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__fst(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ADD(s(X), Y) -> S(n__add(activate(X), Y))
   ADD(s(X), Y) -> ACTIVATE(X)
   LEN(cons(X, Z)) -> S(n__len(activate(Z)))
   LEN(cons(X, Z)) -> ACTIVATE(Z)
   FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Furthermore, R contains one SCC.


SCC1:

FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__fst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__fst(X1, X2)) -> ACTIVATE(X1)
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n__fst(X1, X2)) -> FST(activate(X1), activate(X2))
ACTIVATE(n__len(X)) -> ACTIVATE(X)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n__len(X)) -> LEN(activate(X))
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) -> ADD(activate(X1), activate(X2))

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

Additionally, the following rules can be oriented: 

   add(X1, X2) -> n__add(X1, X2)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(n__len(X)) -> len(activate(X))
   activate(n__s(X)) -> s(X)
   activate(X) -> X
   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(0, Z) -> nil
   fst(X1, X2) -> n__fst(X1, X2)
   len(nil) -> 0
   len(X) -> n__len(X)
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   s(X) -> n__s(X)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__fst(x_1, x_2)) = 1 + x_1 + x_2
POL(fst(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_2
POL(n__from(x_1)) = x_1
POL(ADD(x_1, x_2)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(FST(x_1, x_2)) = 1 + x_1 + x_2
POL(from(x_1)) = x_1
POL(LEN(x_1)) = x_1
POL(len(x_1)) = x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(n__s(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1:

ACTIVATE(n__len(X)) -> ACTIVATE(X)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n__len(X)) -> LEN(activate(X))
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n__from(X)) -> ACTIVATE(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: 

   add(X1, X2) -> n__add(X1, X2)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(n__len(X)) -> len(activate(X))
   activate(n__s(X)) -> s(X)
   activate(X) -> X
   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(0, Z) -> nil
   fst(X1, X2) -> n__fst(X1, X2)
   len(nil) -> 0
   len(X) -> n__len(X)
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   s(X) -> n__s(X)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__fst(x_1, x_2)) = 0
POL(fst(x_1, x_2)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = x_2
POL(n__from(x_1)) = x_1
POL(ADD(x_1, x_2)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(LEN(x_1)) = x_1
POL(from(x_1)) = x_1
POL(len(x_1)) = 1 + x_1
POL(n__len(x_1)) = 1 + x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(n__s(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1.Polo1:

ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) -> ADD(activate(X1), activate(X2))

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

Additionally, the following rules can be oriented: 

   add(X1, X2) -> n__add(X1, X2)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(n__len(X)) -> len(activate(X))
   activate(n__s(X)) -> s(X)
   activate(X) -> X
   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(0, Z) -> nil
   fst(X1, X2) -> n__fst(X1, X2)
   len(nil) -> 0
   len(X) -> n__len(X)
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   s(X) -> n__s(X)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__fst(x_1, x_2)) = 0
POL(fst(x_1, x_2)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = 0
POL(n__from(x_1)) = 1 + x_1
POL(ADD(x_1, x_2)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(from(x_1)) = 1 + x_1
POL(len(x_1)) = 0
POL(n__len(x_1)) = 0
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(n__s(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1.Polo1.Polo1:

ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) -> ACTIVATE(X2)

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

Additionally, the following rules can be oriented: 

   add(X1, X2) -> n__add(X1, X2)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
   activate(n__len(X)) -> len(activate(X))
   activate(n__s(X)) -> s(X)
   activate(X) -> X
   activate(n__fst(X1, X2)) -> fst(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(0, Z) -> nil
   fst(X1, X2) -> n__fst(X1, X2)
   len(nil) -> 0
   len(X) -> n__len(X)
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   s(X) -> n__s(X)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__fst(x_1, x_2)) = 0
POL(fst(x_1, x_2)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = 0
POL(n__from(x_1)) = 0
POL(ADD(x_1, x_2)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(from(x_1)) = 0
POL(len(x_1)) = 0
POL(n__len(x_1)) = 0
POL(n__add(x_1, x_2)) = 1 + x_1 + x_2
POL(n__s(x_1)) = x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 3.875 seconds.