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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  finite\_subsets.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
From Algebra Require Export Diff.
From Algebra Require Export Singleton.
Require Export cast_between_subsets.
Require Export empty.
Require Export Classical_Prop.
(** - A set is finite if we can list its elements (possibly with duplicates in the list)  %\label{isfiniteset}% *)
Definition is_finite_set (A : Setoid) :=
exists n : Nat, (exists v : seq n A, full A =' seq_set v in _).
(** - For subsets, the following notion is more useful as it removes one one layer of taking subsets: *)
Definition is_finite_subset (A : Setoid) (B : part_set A) :=
exists n : Nat, (exists v : seq n A, B =' seq_set v in _).
red in |- *.
intros.
red in H0.
inversion H0.
inversion H1.
exists x.
exists x0.
apply Trans with B; auto with algebra.
Qed.
(** - There is no corresponding is_finite_set_comp, for there is no 'master' setoid wherein two Setoids can be compared. *)
Hint Resolve is_finite_subset_comp: algebra.
intros.
red in |- *; red in H.
inversion_clear H; inversion_clear H0.
exists x.
assert (forall i : fin x, in_part (x0 i) B).
simpl in H; red in H; simpl in H.
intros.
elim (H (x0 i)).
intros p q; (apply q; auto with algebra).
exists i; auto with algebra.
exists (cast_map_to_subset H0).
simpl in |- *; red in |- *; simpl in |- *; split; intro; auto.
unfold subtype_image_equal in |- *.
simpl in H; red in H; simpl in H.
elim (H (subtype_elt x1)).
intros.
assert (in_part (subtype_elt x1) B).
auto with algebra.
elim (H2 H4).
intros.
exists x2.
apply Trans with (x0 x2); auto with algebra.
Qed.
intros; red in |- *; red in H.
inversion_clear H; inversion_clear H0.
exists x; exists (Map_embed x0).
destruct x0; simpl in H; red in H; simpl in H; simpl in |- *; red in |- *;
simpl in |- *.
split; intros.
red in H0; elim (H (Build_subtype H0)).
intros p _; generalize (p I); clear p; intro.
inversion_clear H1.
red in H2.
exists x1.
simpl in H2.
auto.
inversion_clear H0.
apply in_part_comp_l with (subtype_elt (Ap x1)); auto with algebra.
Qed.
intros.
red in |- *.
exists n.
exists v.
auto with algebra.
Qed.
(** - This one is surprisingly hard to prove (or did I not think hard enough?) *)

Lemma included_reflects_is_finite_subset :
forall (A : Setoid) (B C : part_set A),
is_finite_subset C -> included B C -> is_finite_subset B.