Term Rewriting System R:

   [X, Y, X1, X2, Z]
   and(true, X) -> activate(X)
   and(false, Y) -> false
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   add(0, X) -> activate(X)
   add(s(X), Y) -> s(n__add(activate(X), activate(Y)))
   add(X1, X2) -> n__add(X1, X2)
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> cons(activate(X), n__from(n__s(activate(X))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__s(X)) -> s(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ADD(s(X), Y) -> S(n__add(activate(X), activate(Y)))
   ADD(s(X), Y) -> ACTIVATE(X)
   ADD(s(X), Y) -> ACTIVATE(Y)
   ADD(0, X) -> ACTIVATE(X)
   ACTIVATE(n__from(X)) -> FROM(X)
   ACTIVATE(n__s(X)) -> S(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
   FROM(X) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   AND(true, X) -> ACTIVATE(X)
   IF(false, X, Y) -> ACTIVATE(Y)
   IF(true, X, Y) -> ACTIVATE(X)

Furthermore, R contains one SCC.


SCC1:

ADD(0, X) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
ADD(s(X), Y) -> ACTIVATE(Y)
ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
FROM(X) -> ACTIVATE(X)
ACTIVATE(n__from(X)) -> FROM(X)
ADD(s(X), Y) -> ACTIVATE(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(n__first(x_1, x_2)) = x_1 + x_2
POL(n__from(x_1)) = x_1
POL(s(x_1)) = x_1
POL(ADD(x_1, x_2)) = x_1 + x_2
POL(FROM(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(0) = 0
POL(FIRST(x_1, x_2)) = x_1 + x_2
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   ADD(0, X) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
   ADD(s(X), Y) -> ACTIVATE(Y)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ACTIVATE(n__from(X)) -> FROM(X)
   ADD(s(X), Y) -> ACTIVATE(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.974 seconds.