Term Rewriting System R:

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

Innermost Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

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

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

   APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(app(f, y), y), app(g, x))
no new Dependency Pairs
are created.

none
The transformation is resulting in one subcycle:


SCC1.Nar1:

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

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

   APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(f, y), y)
no new Dependency Pairs
are created.

none
The transformation is resulting in no subcycles.


SCC2:

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

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(g) = 1
POL(s) = 1
POL(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(app(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Innermost Termination of R successfully shown.


Duration: 0.499 seconds.