Definition set := list A.Definition empty_set := nil (A:=A).Fixpoint set_add (a : A) (x : set) {struct x} : set :=
    match x with
    | nil => a :: nil
    | a1 :: x1 =>
        match Aeq_dec a a1 with
        | left _ => a1 :: x1
        | right _ => a1 :: set_add a x1
        end
    end.Fixpoint set_mem (a : A) (x : set) {struct x} : bool :=
    match x with
    | nil => false
    | a1 :: x1 =>
        match Aeq_dec a a1 with
        | left _ => true
        | right _ => set_mem a x1
        end
    end.(* If a belongs to x, removes a from x. If not, does nothing *) 
  Fixpoint set_remove (a : A) (x : set) {struct x} : set :=
    match x with
    | nil => empty_set
    | a1 :: x1 =>
        match Aeq_dec a a1 with
        | left _ => set_remove a x1
        | right _ => a1 :: set_remove a x1
        end
    end.Fixpoint set_inter (x : set) : set -> set :=
    match x with
    | nil => fun y => nil (A:=A)
    | a1 :: x1 =>
        fun y =>
        if set_mem a1 y then a1 :: set_inter x1 y else set_inter x1 y
    end.Fixpoint set_union (x y : set) {struct y} : set :=
    match y with
    | nil => x
    | a1 :: y1 => set_add a1 (set_union x y1)
    end.(* returns the set of all els of x that does not belong to y *) 
  Fixpoint set_diff (x : set) : set -> set :=
    fun y =>
    match x with
    | nil => nil (A:=A)
    | a1 :: x1 =>
        if set_mem a1 y then set_diff x1 y else set_add a1 (set_diff x1 y)
    end.(**  
  Inductive set_In : A -> set -> Prop := 
    set_In_singl : (a:A)(x:set)(set_In a (cons a (nil A)))  
  | set_In_add : (a,b:A)(x:set)(set_In a x)->(set_In a (set_add b x)) 
  . 
  **) 
 
  Definition set_In : A -> set -> Prop := In (A:=A).

(* Specifications: *)
Lemma Set_remove1 : forall (B : set) (x : A), ~ set_In x (set_remove x B).
Lemma set_In_dec : forall (a : A) (x : set), {set_In a x} + {~ set_In a x}.
Lemma set_mem_ind : forall (B : Set) (P : B -> Prop) (y z : B) (a : A) (x : set), (set_In a x -> P y) -> P z -> P (if set_mem a x then y else z).
Lemma set_mem_correct1 : forall (a : A) (x : set), set_mem a x = true -> set_In a x.
Lemma set_mem_correct2 : forall (a : A) (x : set), set_In a x -> set_mem a x = true.
Lemma set_mem_complete1 : forall (a : A) (x : set), set_mem a x = false -> ~ set_In a x.
Lemma set_mem_complete2 : forall (a : A) (x : set), ~ set_In a x -> set_mem a x = false.
Lemma set_add_intro1 : forall (a b : A) (x : set), set_In a x -> set_In a (set_add b x).
Lemma set_add_intro2 : forall (a b : A) (x : set), a = b -> set_In a (set_add b x).
Lemma set_add_intro : forall (a b : A) (x : set), a = b \/ set_In a x -> set_In a (set_add b x).
Lemma set_add_elim : forall (a b : A) (x : set), set_In a (set_add b x) -> a = b \/ set_In a x.
Lemma set_add_not_empty : forall (a : A) (x : set), set_add a x <> empty_set.
Lemma set_union_intro1 : forall (a : A) (x y : set), set_In a x -> set_In a (set_union x y).
Lemma set_union_intro2 : forall (a : A) (x y : set), set_In a y -> set_In a (set_union x y).
Lemma set_union_intro : forall (a : A) (x y : set), set_In a x \/ set_In a y -> set_In a (set_union x y).
Lemma set_union_elim : forall (a : A) (x y : set), set_In a (set_union x y) -> set_In a x \/ set_In a y.
Lemma set_inter_intro : forall (a : A) (x y : set), set_In a x -> set_In a y -> set_In a (set_inter x y).
Lemma set_inter_elim1 : forall (a : A) (x y : set), set_In a (set_inter x y) -> set_In a x.
Lemma set_inter_elim2 : forall (a : A) (x y : set), set_In a (set_inter x y) -> set_In a y.
Lemma set_inter_elim : forall (a : A) (x y : set), set_In a (set_inter x y) -> set_In a x /\ set_In a y.
Lemma set_diff_intro : forall (a : A) (x y : set), set_In a x -> ~ set_In a y -> set_In a (set_diff x y).
Lemma set_diff_elim1 : forall (a : A) (x y : set), set_In a (set_diff x y) -> set_In a x.