Term Rewriting System R:

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

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

G(c(x, s(y))) -> G(c(s(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(G(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(c(x_1, x_2)) = 1 + x_2

 resulting in no subcycles.


SCC2:

F(c(s(x), y)) -> F(c(x, s(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(s(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1
POL(c(x_1, x_2)) = x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.508 seconds.