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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 :  Map\_embed.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export concat.
(** - Any element of $A\subset E$ is of course also an element of $E$ itself. Map_embed takes setoid functions $X\to A$ to setoid functions $X\to E$ by embedding  %\label{Mapembed}% *)
Definition Map_embed :
forall (X E : Setoid) (A : part_set E) (f : Map X A), MAP X E.
intros.
apply (Build_Map (Ap:=fun n : X => A (f n))).
red in |- *.
intros x y.
case f.
intros.
red in Map_compatible_prf.
generalize Map_compatible_prf.
intro H0.
generalize (H0 x y).
intros.
simpl in |- *.
apply H1.
assumption.
Defined.
intros.
unfold Map_embed in |- *; simpl in |- *; red in |- *; simpl in |- *.
intuition.
Qed.
Hint Resolve Map_embed_comp: algebra.
intros.
simpl in |- *.
red in |- *.
intro x; case x; intro i; case i.
simpl in |- *.
auto with algebra.
intros.
simpl in |- *.
apply Refl.
Qed.
Hint Resolve Map_embed_cons: algebra.
auto with algebra.
Qed.
Hint Resolve cons_Map_embed: algebra.
simple induction n.
simpl in |- *.
red in |- *.
auto with algebra.
intros.
simpl in |- *.
red in |- *.
simpl in |- *.
intro i.
case i.
auto with algebra.
Qed.
Hint Resolve Map_embed_Seqtl: algebra.
auto with algebra.
Qed.
Hint Resolve Seqtl_Map_embed: algebra.
simple induction n.
intros.
unfold concat in |- *.
unfold nat_rect in |- *.
apply Refl.
intros.
(* Change =' to Map_eq *)
change (Map_eq (Map_embed (f ++ g)) (Map_embed f ++ Map_embed g)) in |- *.
red in |- *.
intros.
apply Trans with (hdtl (Map_embed (f ++ g)) x); auto with algebra.
case x; intro i; case i.
intro l.
unfold hdtl in |- *.
unfold head in |- *.
unfold concat in |- *.
unfold nat_rect in |- *.
simpl in |- *.
apply Refl.
intros.
unfold hdtl in |- *.
unfold head in |- *.
set (l := in_range_prf) in *.
apply Trans with (Seqtl (Map_embed (f ++ g)) (Build_finiteT (lt_S_n _ _ l)));
auto with algebra.
fold (n0 + m) in |- *.
apply Sym.
apply
Trans
with (Seqtl (Map_embed f ++ Map_embed g) (Build_finiteT (lt_S_n _ _ l))).
fold (n0 + m) in |- *.
apply Sym.
apply (Seqtl_to_seq (Map_embed f ++ Map_embed g)); auto with algebra.
apply Ap_comp; auto with algebra.
apply Trans with (Seqtl (Map_embed f) ++ Map_embed g); auto with algebra.
apply Trans with (Map_embed (Seqtl (f ++ g))).
2: apply (Map_embed_Seqtl (f ++ g)); auto with algebra.
apply Trans with (Map_embed (Seqtl f ++ g)).
2: apply Map_embed_comp; auto with algebra.
apply Trans with (Map_embed (Seqtl f) ++ Map_embed g).
apply
(concat_comp (f:=Seqtl (Map_embed f)) (f':=Map_embed (Seqtl f))
(g:=Map_embed g) (g':=Map_embed g)); auto with algebra.
apply Sym.
change
(Map_embed (Seqtl f ++ g) =' Map_embed (Seqtl f) ++ Map_embed g in seq _ _)
in |- *.
apply H; auto with algebra.
Qed.
Hint Resolve Map_embed_concat: algebra.
auto with algebra.
Qed.
Hint Resolve concat_Map_embed.

Lemma Map_embed_prop :
forall (A D : Setoid) (B : part_set A) (v : MAP D B) (i : D),
in_part (Map_embed v i) B.