Term Rewriting System R:

   [N, X, Y, X1, X2, XS]
   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   fib1(X1, X2) -> n__fib1(X1, X2)
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ADD(s(X), Y) -> ADD(X, Y)
   SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
   SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
   ACTIVATE(n__fib1(X1, X2)) -> FIB1(X1, X2)
   FIB1(X, Y) -> ADD(X, Y)
   FIB(N) -> SEL(N, fib1(s(0), s(0)))
   FIB(N) -> FIB1(s(0), s(0))

Furthermore, R contains two SCCs.


SCC1:

ADD(s(X), Y) -> ADD(X, Y)

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(ADD(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ADD(s(X), Y) -> ADD(X, Y)



 This transformation is resulting in no new 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(add(x_1, x_2)) = 0
POL(n__fib1(x_1, x_2)) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(fib1(x_1, x_2)) = 0
POL(SEL(x_1, x_2)) = x_1
POL(0) = 0
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.723 seconds.