Term Rewriting System R:

   [X, Y, Z, X1, X2]
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Furthermore, R contains one SCC.


SCC1:

ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n__s(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: 

   first(X1, X2) -> n__first(X1, X2)
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X
   activate(n__from(X)) -> from(activate(X))
   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(n__first(x_1, x_2)) = x_1 + x_2
POL(first(x_1, x_2)) = x_1 + x_2
POL(n__from(x_1)) = 1 + x_1
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(from(x_1)) = 1 + x_1
POL(0) = 0
POL(FIRST(x_1, x_2)) = x_2
POL(n__s(x_1)) = x_1
POL(cons(x_1, x_2)) = x_2

 resulting in one subcycle.


SCC1.Polo1:

ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__first(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: 

   first(X1, X2) -> n__first(X1, X2)
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   activate(X) -> X
   activate(n__from(X)) -> from(activate(X))
   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(n__first(x_1, x_2)) = 1 + x_1 + x_2
POL(first(x_1, x_2)) = 1 + x_1 + x_2
POL(n__from(x_1)) = 0
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(ACTIVATE(x_1)) = x_1
POL(from(x_1)) = 0
POL(0) = 0
POL(FIRST(x_1, x_2)) = x_1 + x_2
POL(n__s(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.38 seconds.