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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.

Lemma pointwise_comp :
forall (n : Nat) (B C D : Setoid) (op op' : Map (cart B C) D)
(f f' : seq n B) (g g' : seq n C),
op =' op' in MAP _ _ ->
f =' f' in _ -> g =' g' in _ -> pointwise op f g =' pointwise op' f' g' in _.