Term Rewriting System R:

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

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: 


   F(s(x)) -> MINUS(s(x), g(f(x)))
   F(s(x)) -> G(f(x))
   F(s(x)) -> F(x)
   MINUS(s(x), s(y)) -> MINUS(x, y)
   G(s(x)) -> MINUS(s(x), f(g(x)))
   G(s(x)) -> F(g(x))
   G(s(x)) -> G(x)

Furthermore, R contains two SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

   MINUS(s(x), s(y)) -> MINUS(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

G(s(x)) -> G(x)
F(s(x)) -> F(x)
G(s(x)) -> F(g(x))
F(s(x)) -> G(f(x))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   g(0) -> 0
   g(s(x)) -> minus(s(x), f(g(x)))
   f(0) -> s(0)
   f(s(x)) -> minus(s(x), g(f(x)))
   minus(s(x), s(y)) -> minus(x, y)
   minus(x, 0) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(G(x_1)) = x_1
POL(minus(x_1, x_2)) = x_1
POL(F(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.828 seconds.