Term Rewriting System R:

   [y, n, x]
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

Termination of R to be shown.


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

   reverse(nil) -> nil

where the Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = 2 + x_1 + x_2
POL(shuffle(x_1)) = 2*x_1
POL(reverse(x_1)) = 1 + x_1
POL(app(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: 

   shuffle(nil) -> nil

where the Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = x_1 + x_2
POL(shuffle(x_1)) = 1 + x_1
POL(reverse(x_1)) = x_1
POL(app(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: 

   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

where the Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(shuffle(x_1)) = 2*x_1
POL(reverse(x_1)) = x_1
POL(app(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(nil, y) -> y

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 1
POL(reverse(x_1)) = 2*x_1
POL(app(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: 

   reverse(add(n, x)) -> app(reverse(x), add(n, nil))

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(reverse(x_1)) = 2*x_1
POL(app(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(add(n, x), y) -> add(n, app(x, y))

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(app(x_1, x_2)) = 2*x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.649 seconds.