Section Bool.
Variable A : Type.
Variable f : A -> bool.
(** find whether a boolean function can be satisfied by an elements of the list. *)
Fixpoint existsb (l:list A) : bool :=
match l with
| nil => false
| a::l => f a || existsb l
end.
(** find whether a boolean function is satisfied by all the elements of a list. *)
Fixpoint forallb (l:list A) : bool :=
match l with
| nil => true
| a::l => f a && forallb l
end.
(** [filter] *)
Fixpoint filter (l:list A) : list A :=
match l with
| nil => nil
| x :: l => if f x then x::(filter l) else filter l
end.
(** [find] *)
Fixpoint find (l:list A) : option A :=
match l with
| nil => None
| x :: tl => if f x then Some x else find tl
end.
(** [partition] *)
Fixpoint partition (l:list A) : list A * list A :=
match l with
| nil => (nil, nil)
| x :: tl => let (g,d) := partition tl in
if f x then (x::g,d) else (g,x::d)
end.

Theorem elements_in_partition l l1 l2:
partition l = (l1, l2) ->
forall x:A, In x l <-> In x l1 \/ In x l2.