Section first_definitions.
Variable A : Type.
Hypothesis Aeq_dec : forall x y:A, {x = y} + {x <> y}.
Definition set := list A.
Definition empty_set : set := nil.
Fixpoint set_add (a:A) (x:set) : 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) : 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. Invariant: any element should occur at most once in [x], see for instance [set_add]. We hence remove here only the first occurrence of [a] in [x]. *)
Fixpoint set_remove (a:A) (x:set) : set :=
match x with
| nil => empty_set
| a1 :: x1 =>
match Aeq_dec a a1 with
| left _ => x1
| right _ => a1 :: set_remove a x1
end
end.
Fixpoint set_inter (x:set) : set -> set :=
match x with
| nil => fun y => nil
| 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) : 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 y:set) : set :=
match x with
| nil => nil
| a1 :: x1 =>
if set_mem a1 y then set_diff x1 y else set_add a1 (set_diff x1 y)
end.
Definition set_In : A -> set -> Prop := In (A:=A).

Lemma set_mem_ind2 :
forall (B:Type) (P:B -> Prop) (y z:B) (a:A) (x:set),
(set_In a x -> P y) ->
(~ set_In a x -> P z) -> P (if set_mem a x then y else z).