Term Rewriting System R:

   [x, y]
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(s(s(x))) -> P(h(g(x)))
   F(s(s(x))) -> H(g(x))
   F(s(s(x))) -> G(x)
   F(s(s(x))) -> +'(p(g(x)), q(g(x)))
   F(s(s(x))) -> P(g(x))
   F(s(s(x))) -> Q(g(x))
   +'(x, s(y)) -> +'(x, y)
   H(x) -> +'(p(x), q(x))
   H(x) -> P(x)
   H(x) -> Q(x)
   G(s(x)) -> H(g(x))
   G(s(x)) -> G(x)
   G(s(x)) -> +'(p(g(x)), q(g(x)))
   G(s(x)) -> P(g(x))
   G(s(x)) -> Q(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: 0.767 seconds.