Section MAIN.
(* conseq adjoins an element x of X to finite functions f:(fin N)->X *)
(* This is done by {0->x; m+1->f(m)} : (fin N+1)->X, ie. the new element *)
(* is inserted at the front. That's why it's called conseq: it's the seq *)
(* variant of the cons operator as seen in lists *)
(** %\label{conseq}% *)
Definition conseq :
forall (A : Setoid) (n : Nat) (a : A) (v : seq n A), seq (S n) A.
intros.
apply
(Build_Map
(Ap:=fun m : fin (S n) =>
match m return (A:Type) with
| Build_finiteT x0 x1 =>
match x0 as x2 return (x2 < S n -> (A:Type)) with
| O => fun _ : 0 < S n => a
| S m' =>
fun HSm' : S m' < S n =>
Ap v (Build_finiteT (lt_S_n m' n HSm'))
end x1
end)).
red in |- *.
simpl in |- *.
intros x0 y.
case x0.
case y.
simpl in |- *.
intros x2 y1.
induction x2.
intros x2 y2.
induction x2.
intro; apply Refl.
intro H.
inversion H.
clear IHx2.
simple induction index.
intros h H.
inversion H.
intros.
clear H.
apply Ap_comp; auto with algebra.
Defined.
Notation "a ;; b" := (conseq a b) (at level 60, right associativity).
Variables (A : Setoid) (n : Nat) (a : A).
intros.
simpl in |- *.
red in |- *.
intro i'.
elim i'.
intro i.
simpl in |- *.
case i.
auto.
intros.
apply Ap_comp; auto with algebra.
Qed.
Hint Resolve cons_comp: algebra.
auto with algebra.
Qed.
(** %\label{head}% *)
Definition head (A : Setoid) (n : Nat) (v : seq (S n) A) :=
v (Build_finiteT (lt_O_Sn n)).
unfold head in |- *.
auto with algebra.
Qed.
Hint Resolve head_comp cons_first_element: algebra.
auto with algebra.
Qed.
auto with algebra.
Qed.
Hint Resolve head_unfolding1 head_unfolding2: algebra.
Hint Extern 0 (head _ =' _ in _) => unfold head in |- *: algebra.
Hint Extern 0 (_ =' head _ in _) => unfold head in |- *: algebra.
auto with algebra.
Qed.
Hint Resolve head_cons_inv: algebra.
intros.
unfold head in |- *.
apply Ap_comp; auto with algebra.
Qed.
Hint Resolve seq_S_O_contains_single_elt: algebra.
intros.
simpl in |- *.
red in |- *.
intro.
apply Trans with (head v).
apply seq_S_O_contains_single_elt; auto with algebra.
apply Trans with (head v'); auto with algebra.
Qed.
Hint Resolve seq_S_O_head_fixes_everything: algebra.

Lemma cons_later_elements :
forall (v : seq n A) (i : Nat) (Hi : S i < S n) (Hi' : i < n),
(a;; v) (Build_finiteT Hi) =' v (Build_finiteT Hi') in _.