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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  pointwise.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export concat.
From Algebra Require Export Cartesian.
(** - The pointwise operation of a binary operator: suppose $f:A\to B, g:A\to C, \circ:B\times C\to D$, then (pointwise $\circ$ f g) = $a\mapsto f(a)\circ g(a): A\to D$. This is used almost solely to define linear combinations: we have a sequence $\vec a$ of scalars and a sequence $\vec v$ of vectors, then (sum (pointwise (mX) a v)) = $\sum_{i=0}^{n-1} a_i v_i$.  %\label{pointwise}% *)
Definition pointwise :
forall (A B C D : Setoid) (op : Map (cart B C) D)
(f : Map A B) (g : Map A C), MAP A D.
intros.
apply (Build_Map (Ap:=fun a : A => op (couple (f a) (g a)))).
red in |- *.
intros.
apply Ap_comp; auto with algebra.
Defined.
intros.
intro.
unfold pointwise in |- *.
simpl in |- *.
apply Ap_comp; auto with algebra.
Qed.
intros.
apply (pointwise_comp_general H H0 H1).
Qed.
Hint Resolve pointwise_comp: algebra.
intros.
unfold pointwise in |- *.
simpl in |- *.
apply Refl.
Qed.
Hint Resolve pointwise_hd: algebra.
intros.
apply split_hd_tl_equality; auto with algebra.
Qed.
Hint Resolve pointwise_cons: algebra.
intros.
apply split_hd_tl_equality; auto with algebra.
unfold pointwise in |- *; intro; simpl in |- *.
destruct x; (apply Ap_comp; auto with algebra).
Qed.
Hint Resolve pointwise_hd_tl: algebra.
induction n.
simpl in |- *.
red in |- *.
auto with algebra.
intros.
apply
Trans
with
(Seqtl (op (couple (head v) (head w));; pointwise op (Seqtl v) (Seqtl w)));
auto with algebra.
Qed.
Hint Resolve pointwise_Seqtl: algebra.

Lemma pointwise_concat :
forall (A B C : Setoid) (n m : Nat) (v : seq n A)
(w : seq m A) (x : seq n B) (y : seq m B) (op : MAP (cart A B) C),
pointwise op (v ++ w) (x ++ y) =' pointwise op v x ++ pointwise op w y
in seq _ _.