MAYBE

Problem:
 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(pred(),app(s(),x)) -> x
 app(app(minus(),x),0()) -> x
 app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
 app(app(gcd(),0()),y) -> y
 app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
 app(app(gcd(),app(s(),x)),app(s(),y)) ->
 app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
 app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
 app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
 app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
 app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(app(filter(),f),nil()) -> nil()
 app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
 app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
 app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)

Proof:
 DP Processor:
  DPs:
   app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
   app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
   app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
   app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
   app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
   app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
   app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
   app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
   app#(app(gcd(),app(s(),x)),app(s(),y)) ->
   app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
   app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(minus(),x)
   app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y)
   app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),x),y))
   app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
   app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
   app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(minus(),y)
   app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x)
   app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),y),x))
   app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
   app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
   app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
   app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
   app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
   app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
   app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
  TRS:
   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(pred(),app(s(),x)) -> x
   app(app(minus(),x),0()) -> x
   app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
   app(app(gcd(),0()),y) -> y
   app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
   app(app(gcd(),app(s(),x)),app(s(),y)) ->
   app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
   app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
   app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
   app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
   app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
   app(app(map(),f),nil()) -> nil()
   app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
   app(app(filter(),f),nil()) -> nil()
   app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
   app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
   app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
  EDG Processor:
   DPs:
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(minus(),x)
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),x),y))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(minus(),y)
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x)
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),y),x))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
   TRS:
    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(pred(),app(s(),x)) -> x
    app(app(minus(),x),0()) -> x
    app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
    app(app(gcd(),0()),y) -> y
    app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
    app(app(gcd(),app(s(),x)),app(s(),y)) ->
    app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
    app(app(map(),f),nil()) -> nil()
    app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
    app(app(filter(),f),nil()) -> nil()
    app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
    app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
    app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
   graph:
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(minus(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),x),y))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(minus(),y)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),y),x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(minus(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),x),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(minus(),y)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),y),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(cons(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(cons(),app(f,x)),app(app(map(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(filter(),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(cons(),x)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(cons(),x),app(app(filter(),f),xs))
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(filter(),f)
    app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x) ->
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(minus(),x)
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),x),y))
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(minus(),y)
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x)
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(gcd(),app(app(minus(),y),x))
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y) ->
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y) ->
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),y),x)),app(s(),x)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) -> app#(app(minus(),y),x) ->
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(le(),y)
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(le(),y),x)
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(if_gcd(),app(app(le(),y),x))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) -> app#(app(if_gcd(),app(app(le(),y),x)),app(s(),x))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
    app#(app(gcd(),app(app(minus(),x),y)),app(s(),y)) ->
    app#(app(gcd(),app(s(),x)),app(s(),y)) ->
    app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y) ->
    app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) -> app#(app(minus(),x),y) ->
    app#(app(minus(),x),app(s(),y)) -> app#(pred(),app(app(minus(),x),y))
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(filter2(),app(f,x))
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(filter2(),app(f,x)),f)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(filter2(),app(f,x)),f),x)
    app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs) ->
    app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(le(),x)
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y) ->
    app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
   SCC Processor:
    #sccs: 4
    #rules: 11
    #arcs: 112/1024
    DPs:
     app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(app(app(app(filter2(),app(f,x)),f),x),xs)
     app#(app(app(app(filter2(),false()),f),x),xs) -> app#(app(filter(),f),xs)
     app#(app(filter(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(app(app(filter2(),true()),f),x),xs) -> app#(app(filter(),f),xs)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(f,x)
     app#(app(map(),f),app(app(cons(),x),xs)) -> app#(app(map(),f),xs)
    TRS:
     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(pred(),app(s(),x)) -> x
     app(app(minus(),x),0()) -> x
     app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
     app(app(gcd(),0()),y) -> y
     app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
     app(app(gcd(),app(s(),x)),app(s(),y)) ->
     app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
     app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
     app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
     app(app(map(),f),nil()) -> nil()
     app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
     app(app(filter(),f),nil()) -> nil()
     app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
     app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
     app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
    Open
    
    DPs:
     app#(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
     app#(app(gcd(),app(app(minus(),y),x)),app(s(),x))
     app#(app(gcd(),app(s(),x)),app(s(),y)) ->
     app#(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
     app#(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
     app#(app(gcd(),app(app(minus(),x),y)),app(s(),y))
    TRS:
     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(pred(),app(s(),x)) -> x
     app(app(minus(),x),0()) -> x
     app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
     app(app(gcd(),0()),y) -> y
     app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
     app(app(gcd(),app(s(),x)),app(s(),y)) ->
     app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
     app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
     app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
     app(app(map(),f),nil()) -> nil()
     app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
     app(app(filter(),f),nil()) -> nil()
     app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
     app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
     app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
    Open
    
    DPs:
     app#(app(le(),app(s(),x)),app(s(),y)) -> app#(app(le(),x),y)
    TRS:
     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(pred(),app(s(),x)) -> x
     app(app(minus(),x),0()) -> x
     app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
     app(app(gcd(),0()),y) -> y
     app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
     app(app(gcd(),app(s(),x)),app(s(),y)) ->
     app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
     app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
     app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
     app(app(map(),f),nil()) -> nil()
     app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
     app(app(filter(),f),nil()) -> nil()
     app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
     app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
     app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
    Open
    
    DPs:
     app#(app(minus(),x),app(s(),y)) -> app#(app(minus(),x),y)
    TRS:
     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(pred(),app(s(),x)) -> x
     app(app(minus(),x),0()) -> x
     app(app(minus(),x),app(s(),y)) -> app(pred(),app(app(minus(),x),y))
     app(app(gcd(),0()),y) -> y
     app(app(gcd(),app(s(),x)),0()) -> app(s(),x)
     app(app(gcd(),app(s(),x)),app(s(),y)) ->
     app(app(app(if_gcd(),app(app(le(),y),x)),app(s(),x)),app(s(),y))
     app(app(app(if_gcd(),true()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),x),y)),app(s(),y))
     app(app(app(if_gcd(),false()),app(s(),x)),app(s(),y)) ->
     app(app(gcd(),app(app(minus(),y),x)),app(s(),x))
     app(app(map(),f),nil()) -> nil()
     app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
     app(app(filter(),f),nil()) -> nil()
     app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs)
     app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs))
     app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs)
    Open