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(* as published by the Free Software Foundation; either version 2.1   *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* 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.

Lemma subtype_elt_eats_Map_to_equal_subsets :
forall (A B : Setoid) (X Y : part_set A) (H : X =' Y in _)
(b : B) (M : Map B X),
subtype_elt (Map_to_equal_subsets H M b) =' subtype_elt (M b) in _.