Term Rewriting System R:

   [x, y]
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(x, s(y)) -> +'(x, y)
   NP(pair(x, y)) -> +'(x, y)
   SP(pair(x, y)) -> +'(x, y)
   G(s(x)) -> NP(g(x))
   G(s(x)) -> G(x)
   FIB(s(s(x))) -> SP(g(x))
   FIB(s(s(x))) -> G(x)

Furthermore, R contains two SCCs.


SCC1:

+'(x, s(y)) -> +'(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(+'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   +'(x, s(y)) -> +'(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

G(s(x)) -> G(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(G(x_1)) = 1 + x_1
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   G(s(x)) -> G(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.596 seconds.