Term Rewriting System R:

   [x, y, z]
   f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
   f(s(x), y, z) -> f(x, s(c(y)), c(z))
   f(c(x), x, y) -> c(y)
   g(x, y) -> x
   g(x, y) -> y

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(x, c(x), c(y)) -> F(y, y, f(y, x, y))
   F(x, c(x), c(y)) -> F(y, x, y)
   F(s(x), y, z) -> F(x, s(c(y)), c(z))

Furthermore, R contains two SCCs.


SCC1:

F(s(x), y, z) -> F(x, s(c(y)), c(z))

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(F(x_1, x_2, x_3)) = x_1
POL(c(x_1)) = 0

 resulting in no subcycles.


SCC2:

F(x, c(x), c(y)) -> F(y, x, y)

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(F(x_1, x_2, x_3)) = 1 + x_2
POL(c(x_1)) = 1 + x_1

 resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.507 seconds.