Term Rewriting System R:

   [x, y]
   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   g(s(x), y) -> g(x, s(+(y, x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

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(s(x_1)) = 1 + x_1
POL(G(x_1, x_2)) = x_1
POL(+(x_1, x_2)) = 0
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.659 seconds.