Term Rewriting System R:

   [x]
   app(g, app(h, app(g, x))) -> app(g, x)
   app(g, app(g, x)) -> app(g, app(h, app(g, x)))
   app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))

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

Additionally, the following rules can be oriented: 

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


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

 resulting in no subcycles.


SCC2:

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

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

Additionally, the following rules can be oriented: 

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


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.604 seconds.