Term Rewriting System R:

   [X, Y]
   +(X, 0) -> X
   +(X, s(Y)) -> s(+(X, Y))
   f(0, s(0), X) -> f(X, +(X, X), X)
   g(X, Y) -> X
   g(X, Y) -> Y

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(X, s(Y)) -> +'(X, Y)
   F(0, s(0), X) -> F(X, +(X, X), X)
   F(0, s(0), X) -> +'(X, 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:

F(0, s(0), X) -> F(X, +(X, X), X)

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   F(0, s(0), X) -> F(X, +(X, X), X)
two new Dependency Pairs
are created:


   F(0, s(0), 0) -> F(0, 0, 0)
   F(0, s(0), s(Y')) -> F(s(Y'), s(+(s(Y'), Y')), s(Y'))

The transformation is resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.451 seconds.