Term Rewriting System R:

   [x, y]
   app(intlist, nil) -> nil
   app(intlist, app(app(cons, x), y)) -> app(app(cons, app(s, x)), app(intlist, y))
   app(app(int, 0), 0) -> app(app(cons, 0), nil)
   app(app(int, 0), app(s, y)) -> app(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))
   app(app(int, app(s, x)), 0) -> nil
   app(app(int, app(s, x)), app(s, y)) -> app(intlist, app(app(int, x), y))

Innermost Termination of R to be shown.


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

   app(app(int, 0), 0) -> app(app(cons, 0), nil)
   app(app(int, app(s, x)), 0) -> nil

where the Polynomial interpretation:
POL(nil) = 0
POL(s) = 0
POL(intlist) = 0
POL(int) = 1
POL(app(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(cons) = 0
was used. 



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


R contains the following Dependency Pairs: 


   APP(app(int, app(s, x)), app(s, y)) -> APP(intlist, app(app(int, x), y))
   APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, x), y)
   APP(app(int, app(s, x)), app(s, y)) -> APP(int, x)
   APP(intlist, app(app(cons, x), y)) -> APP(app(cons, app(s, x)), app(intlist, y))
   APP(intlist, app(app(cons, x), y)) -> APP(cons, app(s, x))
   APP(intlist, app(app(cons, x), y)) -> APP(s, x)
   APP(intlist, app(app(cons, x), y)) -> APP(intlist, y)
   APP(app(int, 0), app(s, y)) -> APP(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))
   APP(app(int, 0), app(s, y)) -> APP(cons, 0)
   APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
   APP(app(int, 0), app(s, y)) -> APP(int, app(s, 0))
   APP(app(int, 0), app(s, y)) -> APP(s, 0)

Furthermore, R contains one SCC.


SCC1:

APP(intlist, app(app(cons, x), y)) -> APP(intlist, y)
APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
APP(app(int, 0), app(s, y)) -> APP(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))
APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, x), y)
APP(intlist, app(app(cons, x), y)) -> APP(app(cons, app(s, x)), app(intlist, y))
APP(app(int, app(s, x)), app(s, y)) -> APP(intlist, app(app(int, x), y))

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(intlist, nil) -> nil
   app(app(int, app(s, x)), app(s, y)) -> app(intlist, app(app(int, x), y))
   app(intlist, app(app(cons, x), y)) -> app(app(cons, app(s, x)), app(intlist, y))
   app(app(int, 0), app(s, y)) -> app(app(cons, 0), app(app(int, app(s, 0)), app(s, y)))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(s) = 0
POL(APP(x_1, x_2)) = x_1
POL(intlist) = 0
POL(int) = 1
POL(app(x_1, x_2)) = x_1
POL(0) = 0
POL(cons) = 0

 resulting in two subcycles.


SCC1.Polo1:

APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))
APP(app(int, app(s, x)), app(s, y)) -> APP(app(int, x), y)

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(s) = 0
POL(APP(x_1, x_2)) = x_2
POL(intlist) = 0
POL(int) = 0
POL(app(x_1, x_2)) = 1 + x_2
POL(0) = 0
POL(cons) = 0

 resulting in one subcycle.


SCC1.Polo1.Polo1:

APP(app(int, 0), app(s, y)) -> APP(app(int, app(s, 0)), app(s, y))

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

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

none
The transformation is resulting in no subcycles.


SCC1.Polo2:

APP(intlist, app(app(cons, x), y)) -> APP(intlist, 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(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(intlist) = 1
POL(app(x_1, x_2)) = 1 + x_1 + x_2
POL(cons) = 1

The following Dependency Pairs can be deleted:

   APP(intlist, app(app(cons, x), y)) -> APP(intlist, y)



 This transformation is resulting in no new subcycles.




Innermost Termination of R successfully shown.


Duration: 1.644 seconds.