(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** %\subsection*{ examples :  vecspace\_functionspace.v }%*)
(** - Another vectorspace, defined old-fashionedly. Fortunately, given vecspace_Fn, this file is hardly more than cut-n-paste *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export vecspaces_verybasic.
Require Export Map2.
Section Function_spaces.
Definition func_set (A B : Setoid) := MAP A B.
Let func_plus_fun :
forall (A : Setoid) (B : sgroup),
func_set A B -> func_set A B -> func_set A B.
intros A B f g.
simpl in |- *.
simpl in f.
simpl in g.
apply (Build_Map (A:=A) (B:=B) (Ap:=fun a : A => Ap f a +' Ap g a)).
red in |- *.
intros a a' Ha.
apply SGROUP_comp; auto with algebra.
Defined.
Definition func_plus :
forall (A : Setoid) (B : sgroup), law_of_composition (func_set A B).
intros.
simpl in |- *.
apply Map2_Mapcart.
apply (Build_Map2 (Ap2:=func_plus_fun (A:=A) (B:=B))).
red in |- *.
intros f f' g g' Hf Hg.
unfold func_plus_fun in |- *.
simpl in |- *.
red in |- *.
intro a.
simpl in |- *.
apply SGROUP_comp; auto with algebra.
Defined.

Lemma func_plus_associative :
forall (A : Setoid) (B : sgroup), associative (func_plus A B).