Term Rewriting System R:

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

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(0, s(0), x) -> F(x, plus(x, x), x)
   F(0, s(0), x) -> PLUS(x, x)
   PLUS(x, s(y)) -> PLUS(x, y)

Furthermore, R contains two SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

F(0, s(0), x) -> F(x, plus(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, plus(x, x), x)
two new Dependency Pairs
are created:


   F(0, s(0), s(y')) -> F(s(y'), s(plus(s(y'), y')), s(y'))
   F(0, s(0), 0) -> F(0, 0, 0)

The transformation is resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.460 seconds.