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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  cast\_between\_subsets.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Map_embed.
Require Export algebra_omissions.
(** - Various casting functions between equal but non-convertible types *)
(** - This one maps elements of a subset to the same subset under a different name %\label{mapbetweenequalsubsets}% *)
Definition map_between_equal_subsets :
forall (A : Setoid) (X Y : part_set A), X =' Y in _ -> X -> Y.
intros A X Y H x.
inversion_clear x.
apply (Build_subtype (E:=A) (P:=Y) (subtype_elt:=subtype_elt)).
simpl in H.
red in H.
generalize (H subtype_elt).
intro H'.
inversion_clear H'.
apply H0; auto with algebra.
Defined.
intros.
elim x.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_map_between_equal_subsets: algebra.
simpl in |- *.
unfold subtype_image_equal in |- *; simpl in |- *.
intros.
apply Trans with (subtype_elt (map_between_equal_subsets H x));
auto with algebra.
apply Trans with (subtype_elt (map_between_equal_subsets H' x'));
auto with algebra.
Qed.
(** - This one turns $f:A\to X$ into $A\to Y$ whenever $X='Y$ *)
(** %\label{Maptoequalsubsets}% *)
Definition Map_to_equal_subsets :
forall (A B : Setoid) (X Y : part_set A), X =' Y in _ -> MAP B X -> MAP B Y.
intros.
apply (Build_Map (Ap:=fun b : B => map_between_equal_subsets H (X0 b))).
red in |- *; simpl in |- *; red in |- *.
intros.
unfold map_between_equal_subsets in |- *.
generalize (Map_compatible_prf X0 H0).
case (X0 x); case (X0 y).
simpl in |- *.
auto.
Defined.
intros.
simpl in |- *.
case (M b).
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_Map_to_equal_subsets: algebra.
intros.
simpl in |- *.
red in |- *.
intros b.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_Map_to_equal_subsets: algebra.
unfold Map_to_equal_subsets in |- *; simpl in |- *; unfold Map_eq in |- *;
simpl in |- *; unfold subtype_image_equal in |- *;
simpl in |- *.
intros.
apply subtype_elt_comp.
apply map_between_equal_subsets_inj with Y H H'; auto with algebra.
simpl in |- *; red in |- *; simpl in |- *.
auto.
Qed.
(** - if $\forall b\in B, f(b)\in W\subset A$ for $f:B\to A$ then $f$ can be seen as $f:B\to W$. This is done by cast_map_to_subset. *)
Definition cast_to_subset_fun :
forall (A B : Setoid) (v : MAP B A) (W : part_set A),
(forall i : B, in_part (v i) W) -> (B -> W:Type).
intros A B v.
elim v.
intros vseq vprf; intros.
generalize X; clear X.
simpl in |- *.
exact (fun i : B => Build_subtype (H i)).
Defined.
intros A B v.
elim v.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve cast_doesn't_change: algebra.
(** %\label{castmaptosubset}% *)
Definition cast_map_to_subset :
forall (A B : Setoid) (v : MAP B A) (W : part_set A),
(forall i : B, in_part (v i) W) -> MAP B W.
intros.
cut (fun_compatible (cast_to_subset_fun H)).
intro.
exact (Build_Map H0).
red in |- *.
simpl in |- *.
red in |- *.
intros.
apply Trans with (v x); auto with algebra.
apply Trans with (v y); auto with algebra.
Defined.
intros.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve cast_map_to_subset_doesn't_change: algebra.
intros.
simpl in |- *.
red in |- *.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve Map_embed_cast_map_to_subset_inv: algebra.
intros.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_cast_map_to_subset: algebra.
intros.
exists (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve seq_castable: algebra.
intros.
exists (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve subset_seq_castable: algebra.
destruct P.
intros.
red in Pred_compatible_prf.
simpl in |- *.
simpl in H0.
apply Pred_compatible_prf with v; auto with algebra.
Qed.
Hint Resolve cast_seq_nice: algebra.
intros.
auto with algebra.
Qed.
Hint Resolve cast_subset_seq_nice: algebra.
intros.
generalize H0; clear H0; elim P.
intros Pf pc H0.
simpl in |- *.
simpl in H0.
auto.
Qed.

Lemma cast_respects_all_elt_predicates :
forall (A D : Setoid) (B C : part_set A) (v : MAP D B)
(P : Predicate A) (H : forall i : D, in_part (Map_embed v i) C),
(forall i : D, Pred_fun P (Map_embed v i)) ->
forall j : D, Pred_fun P (Map_embed (cast_map_to_subset H) j).