#[global]
Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal : fset.
(** ** Cardinal and set operators *)

(* Specifications: *)
Lemma empty_cardinal : cardinal empty = 0.
#[global]
Hint Immediate empty_cardinal cardinal_1 : set.
Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.
#[global]
Hint Resolve singleton_cardinal: set.
Lemma diff_inter_cardinal :
forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .
Lemma union_cardinal:
forall s s', (forall x, ~(In x s/\In x s')) ->
cardinal (union s s')=cardinal s+cardinal s'.
Lemma subset_cardinal :
forall s s', s[<=]s' -> cardinal s <= cardinal s' .
Lemma subset_cardinal_lt :
forall s s' x, s[<=]s' -> In x s' -> ~In x s -> cardinal s < cardinal s'.
Theorem union_inter_cardinal :
forall s s', cardinal (union s s') + cardinal (inter s s')  = cardinal s  + cardinal s' .
Lemma union_cardinal_inter :
forall s s', cardinal (union s s') = cardinal s + cardinal s' - cardinal (inter s s').
Lemma union_cardinal_le :
forall s s', cardinal (union s s') <= cardinal s  + cardinal s'.
Lemma add_cardinal_1 :
forall s x, In x s -> cardinal (add x s) = cardinal s.
Lemma add_cardinal_2 :
forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).
Lemma remove_cardinal_1 :
forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.
Lemma remove_cardinal_2 :
forall s x, ~In x s -> cardinal (remove x s) = cardinal s.