Term Rewriting System R:

   [x, y, z]
   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   null(.(x, y)) -> false
   ++(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: 

   rev(nil) -> nil
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y

where the Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(null(x_1)) = x_1
POL(car(x_1)) = 1 + x_1
POL(true) = 0
POL(rev(x_1)) = 1 + x_1
POL(cdr(x_1)) = 2 + x_1
POL(.(x_1, x_2)) = x_1 + x_2
POL(false) = 0
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), .(x, nil))
   null(.(x, y)) -> false

where the Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(null(x_1)) = x_1
POL(true) = 0
POL(rev(x_1)) = 2*x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2
POL(false) = 0
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: 

   ++(nil, y) -> y
   null(nil) -> true

where the Polynomial interpretation:
POL(nil) = 1
POL(++(x_1, x_2)) = x_1 + x_2
POL(null(x_1)) = x_1
POL(true) = 0
POL(.(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: 

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

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.434 seconds.