Term Rewriting System R:

   [x, y, z]
   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

Termination of R to be shown.


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

   flatten(nil) -> nil
   ++(x, nil) -> x
   ++(nil, y) -> y

where the Polynomial interpretation:
POL(nil) = 1
POL(++(x_1, x_2)) = x_1 + x_2
POL(flatten(x_1)) = 2*x_1
POL(rev(x_1)) = x_1
POL(unit(x_1)) = x_1
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: 

   flatten(unit(x)) -> flatten(x)
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))

where the Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(flatten(x_1)) = x_1
POL(rev(x_1)) = x_1
POL(unit(x_1)) = 1 + x_1
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: 

   flatten(++(x, y)) -> ++(flatten(x), flatten(y))

where the Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = 1 + x_1 + x_2
POL(flatten(x_1)) = 2*x_1
POL(rev(x_1)) = x_1
POL(unit(x_1)) = x_1
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: 

   flatten(flatten(x)) -> flatten(x)

where the Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(flatten(x_1)) = 1 + x_1
POL(rev(x_1)) = x_1
POL(unit(x_1)) = x_1
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: 

   rev(nil) -> nil

where the Polynomial interpretation:
POL(nil) = 1
POL(++(x_1, x_2)) = x_1 + x_2
POL(rev(x_1)) = 2*x_1
POL(unit(x_1)) = x_1
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: 

   rev(unit(x)) -> unit(x)

where the Polynomial interpretation:
POL(++(x_1, x_2)) = x_1 + x_2
POL(rev(x_1)) = 2*x_1
POL(unit(x_1)) = 1 + x_1
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: 

   rev(++(x, y)) -> ++(rev(y), rev(x))

where the Polynomial interpretation:
POL(++(x_1, x_2)) = 1 + x_1 + x_2
POL(rev(x_1)) = 2*x_1
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: 

   ++(++(x, y), z) -> ++(x, ++(y, z))
   rev(rev(x)) -> x

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.517 seconds.