Term Rewriting System R:

   [x, y]
   app(f, app(s, x)) -> app(f, x)
   app(g, app(app(cons, 0), y)) -> app(g, y)
   app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
   app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   app(f, app(s, x)) -> app(f, x)
   app(g, app(app(cons, 0), y)) -> app(g, y)

where the Polynomial interpretation:
POL(g) = 0
POL(s) = 1
POL(h) = 1
POL(f) = 0
POL(app(x_1, x_2)) = x_1 + x_2
POL(0) = 1
POL(cons) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   app(g, app(app(cons, app(s, x)), y)) -> app(s, x)

where the Polynomial interpretation:
POL(g) = 0
POL(s) = 1
POL(h) = 2
POL(app(x_1, x_2)) = 2*x_1 + x_2
POL(cons) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   APP(h, app(app(cons, x), y)) -> APP(h, app(g, app(app(cons, x), y)))
   APP(h, app(app(cons, x), y)) -> APP(g, app(app(cons, x), y))

Furthermore, R contains one SCC.


SCC1:

APP(h, app(app(cons, x), y)) -> APP(h, app(g, app(app(cons, x), y)))

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

   APP(h, app(app(cons, x), y)) -> APP(h, app(g, app(app(cons, x), y)))
no new Dependency Pairs
are created.

none
The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.488 seconds.