Term Rewriting System R:

   [x, y]
   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

F(s(x)) -> F(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(s(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   F(s(x)) -> F(x)



 This transformation is resulting in no new subcycles.


SCC2:

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

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(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))
   if(true, x, y) -> x
   if(false, x, y) -> y


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1, x_2)) = 0
POL(s(x_1)) = 0
POL(1) = 0
POL(G(x_1, x_2)) = x_2
POL(true) = 0
POL(f(x_1)) = 0
POL(c(x_1)) = 1 + x_1
POL(if(x_1, x_2, x_3)) = x_2 + x_3
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.694 seconds.