Term Rewriting System R:

   [k, l, x, y]
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))

Termination of R to be shown.


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

   app(nil, k) -> k
   app(l, nil) -> l
   plus(0, y) -> y

where the Polynomial interpretation:
POL(nil) = 0
POL(plus(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(sum(x_1)) = x_1
POL(app(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 1
POL(cons(x_1, x_2)) = x_1 + x_2
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(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))

where the Polynomial interpretation:
POL(nil) = 0
POL(plus(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(sum(x_1)) = x_1
POL(app(x_1, x_2)) = 2*x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2
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: 

   plus(s(x), y) -> s(plus(x, y))

where the Polynomial interpretation:
POL(nil) = 0
POL(plus(x_1, x_2)) = 2*x_1 + x_2
POL(s(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(app(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2
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: 


   SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))
   SUM(app(l, cons(x, cons(y, k)))) -> SUM(cons(x, cons(y, k)))

Furthermore, R contains one SCC.


SCC1:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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

   SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))
no new Dependency Pairs
are created.

none
The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.55 seconds.