Set Implicit Arguments.
Unset Strict Implicit.
Require Export concat.

Definition Map_embed :
  ∀ (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.

Lemma Map_embed_comp :
 ∀ (X E : Setoid) (A : part_set E) (f g : Map X A),
 f =' g in MAP _ _ → Map_embed f =' Map_embed g in _.
intros.
unfold Map_embed in |- *; simpl in |- *; red in |- *; simpl in |- ×.
intuition.
Qed.

Hint Resolve Map_embed_comp: algebra.

Lemma Map_embed_cons :
 ∀ (E : Setoid) (A : part_set E) (a : A) (n : Nat) (f : seq n A),
 Map_embed (a;; f) =' subtype_elt a;; Map_embed f in seq _ _.
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.

Lemma cons_Map_embed :
 ∀ (E : Setoid) (A : part_set E) (a : A) (n : Nat) (f : seq n A),
 subtype_elt a;; Map_embed f =' Map_embed (a;; f) in seq _ _.
auto with algebra.
Qed.

Hint Resolve cons_Map_embed: algebra.

Lemma Map_embed_Seqtl :
 ∀ (E : Setoid) (A : part_set E) (n : Nat) (f : seq n A),
 Map_embed (Seqtl f) =' Seqtl (Map_embed f) in seq _ _.
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.

Lemma Seqtl_Map_embed :
 ∀ (E : Setoid) (A : part_set E) (n : Nat) (f : seq n A),
 Seqtl (Map_embed f) =' Map_embed (Seqtl f) in seq _ _.
auto with algebra.
Qed.

Hint Resolve Seqtl_Map_embed: algebra.

Lemma Map_embed_concat :
 ∀ (E : Setoid) (A : part_set E) (n m : Nat) (f : seq n A) (g : seq m A),
 Map_embed (f ++ g) =' Map_embed f ++ Map_embed g in seq _ _.
simple induction n.
intros.
unfold concat in |- ×.
unfold nat_rect in |- ×.
apply Refl.
intros.
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.

Lemma concat_Map_embed :
 ∀ (E : Setoid) (A : part_set E) (n m : Nat) (f : seq n A) (g : seq m A),
 Map_embed f ++ Map_embed g =' Map_embed (f ++ g) in seq _ _.
auto with algebra.
Qed.

Hint Resolve concat_Map_embed.

Lemma Map_embed_prop :
 ∀ (A D : Setoid) (B : part_set A) (v : MAP D B) (i : D),
 in_part (Map_embed v i) B.
simpl in |- ×.
intros.
destruct (v i).
simpl in |- ×.
red in |- *; auto with algebra.
Qed.