Term Rewriting System R:

   [x]
   app(f, app(s, x)) -> app(f, app(app(g, x), x))
   app(app(g, 0), 1) -> app(s, 0)
   0 -> 1

Innermost Termination of R to be shown.


Removing the following rules from R which left hand sides contain non normal subterms

   app(app(g, 0), 1) -> app(s, 0)

R contains the following Dependency Pairs: 


   APP(f, app(s, x)) -> APP(f, app(app(g, x), x))
   APP(f, app(s, x)) -> APP(app(g, x), x)
   APP(f, app(s, x)) -> APP(g, x)

Furthermore, R contains one SCC.


SCC1:

APP(f, app(s, x)) -> APP(f, app(app(g, x), x))

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

Additionally, the following rule can be oriented: 

   app(f, app(s, x)) -> app(f, app(app(g, x), x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g) = 0
POL(s) = 1
POL(APP(x_1, x_2)) = x_2
POL(f) = 1
POL(app(x_1, x_2)) = x_1

 resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.464 seconds.