Term Rewriting System R:

   [X, XS, N, X1, X2]
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   take(X1, X2) -> n__take(X1, X2)
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
   ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
   SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
   2ND(cons(X, XS)) -> HEAD(activate(XS))
   2ND(cons(X, XS)) -> ACTIVATE(XS)

Furthermore, R contains two SCCs.


SCC1:

ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2))
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)

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

Additionally, the following rules can be oriented: 

   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   activate(n__s(X)) -> s(activate(X))
   activate(n__from(X)) -> from(activate(X))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   take(X1, X2) -> n__take(X1, X2)
   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__from(x_1)) = 1 + x_1
POL(nil) = 0
POL(TAKE(x_1, x_2)) = x_2
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(take(x_1, x_2)) = x_1 + x_2
POL(ACTIVATE(x_1)) = x_1
POL(n__take(x_1, x_2)) = x_1 + x_2
POL(from(x_1)) = 1 + x_1
POL(0) = 0
POL(n__s(x_1)) = x_1
POL(cons(x_1, x_2)) = x_2

 resulting in one subcycle.


SCC1.Polo1:

ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n__take(X1, X2)) -> TAKE(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: 

   activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
   activate(X) -> X
   activate(n__s(X)) -> s(activate(X))
   activate(n__from(X)) -> from(activate(X))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   take(X1, X2) -> n__take(X1, X2)
   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__from(x_1)) = 0
POL(nil) = 0
POL(TAKE(x_1, x_2)) = x_1 + x_2
POL(activate(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(take(x_1, x_2)) = 1 + x_1 + x_2
POL(ACTIVATE(x_1)) = x_1
POL(n__take(x_1, x_2)) = 1 + x_1 + x_2
POL(from(x_1)) = 0
POL(0) = 0
POL(n__s(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = x_2

 resulting in no subcycles.


SCC2:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(n__from(x_1)) = 0
POL(nil) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(take(x_1, x_2)) = 0
POL(n__take(x_1, x_2)) = 0
POL(SEL(x_1, x_2)) = x_1
POL(from(x_1)) = 0
POL(0) = 0
POL(n__s(x_1)) = 0
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.218 seconds.