MAYBE

Problem:
 app'(app'(minus(),x),0()) -> x
 app'(app'(minus(),app'(s(),x)),app'(s(),y)) -> app'(app'(minus(),x),y)
 app'(app'(quot(),0()),app'(s(),y)) -> 0()
 app'(app'(quot(),app'(s(),x)),app'(s(),y)) ->
 app'(s(),app'(app'(quot(),app'(app'(minus(),x),y)),app'(s(),y)))
 app'(app'(le(),0()),y) -> true()
 app'(app'(le(),app'(s(),x)),0()) -> false()
 app'(app'(le(),app'(s(),x)),app'(s(),y)) -> app'(app'(le(),x),y)
 app'(app'(app(),nil()),y) -> y
 app'(app'(app(),app'(app'(add(),n),x)),y) -> app'(app'(add(),n),app'(app'(app(),x),y))
 app'(app'(low(),n),nil()) -> nil()
 app'(app'(low(),n),app'(app'(add(),m),x)) ->
 app'(app'(app'(if_low(),app'(app'(le(),m),n)),n),app'(app'(add(),m),x))
 app'(app'(app'(if_low(),true()),n),app'(app'(add(),m),x)) ->
 app'(app'(add(),m),app'(app'(low(),n),x))
 app'(app'(app'(if_low(),false()),n),app'(app'(add(),m),x)) -> app'(app'(low(),n),x)
 app'(app'(high(),n),nil()) -> nil()
 app'(app'(high(),n),app'(app'(add(),m),x)) ->
 app'(app'(app'(if_high(),app'(app'(le(),m),n)),n),app'(app'(add(),m),x))
 app'(app'(app'(if_high(),true()),n),app'(app'(add(),m),x)) -> app'(app'(high(),n),x)
 app'(app'(app'(if_high(),false()),n),app'(app'(add(),m),x)) ->
 app'(app'(add(),m),app'(app'(high(),n),x))
 app'(quicksort(),nil()) -> nil()
 app'(quicksort(),app'(app'(add(),n),x)) ->
 app'(app'(app(),app'(quicksort(),app'(app'(low(),n),x))),app'(app'(add(),n),
                                                               app'(quicksort(),
                                                                    app'
                                                                    (app'(high(),n),x))))
 app'(app'(map(),f),nil()) -> nil()
 app'(app'(map(),f),app'(app'(add(),x),xs)) -> app'(app'(add(),app'(f,x)),app'(app'(map(),f),xs))
 app'(app'(filter(),f),nil()) -> nil()
 app'(app'(filter(),f),app'(app'(add(),x),xs)) ->
 app'(app'(app'(app'(filter2(),app'(f,x)),f),x),xs)
 app'(app'(app'(app'(filter2(),true()),f),x),xs) -> app'(app'(add(),x),app'(app'(filter(),f),xs))
 app'(app'(app'(app'(filter2(),false()),f),x),xs) -> app'(app'(filter(),f),xs)

Proof:
 Open