MAYBE

Problem:
 app'(app'(eq(),0()),0()) -> true()
 app'(app'(eq(),0()),app'(s(),x)) -> false()
 app'(app'(eq(),app'(s(),x)),0()) -> false()
 app'(app'(eq(),app'(s(),x)),app'(s(),y)) -> app'(app'(eq(),x),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'(min(),app'(app'(add(),n),nil())) -> n
 app'(min(),app'(app'(add(),n),app'(app'(add(),m),x))) ->
 app'(app'(if_min(),app'(app'(le(),n),m)),app'(app'(add(),n),app'(app'(add(),m),x)))
 app'(app'(if_min(),true()),app'(app'(add(),n),app'(app'(add(),m),x))) ->
 app'(min(),app'(app'(add(),n),x))
 app'(app'(if_min(),false()),app'(app'(add(),n),app'(app'(add(),m),x))) ->
 app'(min(),app'(app'(add(),m),x))
 app'(app'(rm(),n),nil()) -> nil()
 app'(app'(rm(),n),app'(app'(add(),m),x)) ->
 app'(app'(app'(if_rm(),app'(app'(eq(),n),m)),n),app'(app'(add(),m),x))
 app'(app'(app'(if_rm(),true()),n),app'(app'(add(),m),x)) -> app'(app'(rm(),n),x)
 app'(app'(app'(if_rm(),false()),n),app'(app'(add(),m),x)) ->
 app'(app'(add(),m),app'(app'(rm(),n),x))
 app'(app'(minsort(),nil()),nil()) -> nil()
 app'(app'(minsort(),app'(app'(add(),n),x)),y) ->
 app'(app'(app'(if_minsort(),app'(app'(eq(),n),app'(min(),app'(app'(add(),n),x)))),
           app'(app'(add(),n),x)),y)
 app'(app'(app'(if_minsort(),true()),app'(app'(add(),n),x)),y) ->
 app'(app'(add(),n),app'(app'(minsort(),app'(app'(app(),app'(app'(rm(),n),x)),y)),nil()))
 app'(app'(app'(if_minsort(),false()),app'(app'(add(),n),x)),y) ->
 app'(app'(minsort(),x),app'(app'(add(),n),y))
 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