(* 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                                                     *)


Set Implicit Arguments.
Unset Strict Implicit.
Require Export Monoid_cat.
(** Title "The monoid of maps of a set into itself." *)
Section Def.
Variable E : SET.

Definition endo_comp : law_of_composition (Hom E E).
(* Goal: Carrier (law_of_composition (@Hom SET E E)) *)
unfold law_of_composition in |- *.
apply
 (Build_Map
    (Ap:=fun x : cart (Hom E E) (Hom E E) => comp_map_map (proj1 x) (proj2 x))).
(* Goal: @unit_l (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (Id E) *)
red in |- *.
(* Goal: forall (x y : Carrier (cart (@Hom SET E E) (@Hom SET E E))) (_ : @Equal (cart (@Hom SET E E) (@Hom SET E E)) x y), @Equal (MAP E E) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) x) (@proj2 (@Hom SET E E) (@Hom SET E E) x)) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) y) (@proj2 (@Hom SET E E) (@Hom SET E E) y)) *)
auto with algebra.
Defined.

Definition Endo_SET_sgroup : SGROUP.
(* Goal: Ob SGROUP *)
apply (Build_sgroup (sgroup_set:=Hom E E)).
(* Goal: sgroup_on (@Hom SET E E) *)
apply (Build_sgroup_on (E:=Hom E E) (sgroup_law_map:=endo_comp)).
(* Goal: @unit_l (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (Id E) *)
red in |- *.
(* Goal: forall x : Carrier (sgroup_set Endo_SET_sgroup), @Equal (sgroup_set Endo_SET_sgroup) (@Ap (cart (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup)) (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (@couple (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup) (Id E) x)) x *)
simpl in |- *.
(* Goal: forall (x y : Carrier (cart (@Hom SET E E) (@Hom SET E E))) (_ : @Equal (cart (@Hom SET E E) (@Hom SET E E)) x y), @Equal (MAP E E) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) x) (@proj2 (@Hom SET E E) (@Hom SET E E) x)) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) y) (@proj2 (@Hom SET E E) (@Hom SET E E) y)) *)
unfold Map_eq in |- *; auto with algebra.
Defined.

Definition Endo_SET : MONOID.
(* Goal: Ob MONOID *)
apply (Build_monoid (monoid_sgroup:=Endo_SET_sgroup)).
(* Goal: monoid_on Endo_SET_sgroup *)
apply (Build_monoid_on (A:=Endo_SET_sgroup) (monoid_unit:=Id E)).
(* Goal: @unit_l (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (Id E) *)
red in |- *.
(* Goal: forall x : Carrier (sgroup_set Endo_SET_sgroup), @Equal (sgroup_set Endo_SET_sgroup) (@Ap (cart (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup)) (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (@couple (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup) (Id E) x)) x *)
simpl in |- *.
(* Goal: forall (x y : Carrier (cart (@Hom SET E E) (@Hom SET E E))) (_ : @Equal (cart (@Hom SET E E) (@Hom SET E E)) x y), @Equal (MAP E E) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) x) (@proj2 (@Hom SET E E) (@Hom SET E E) x)) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) y) (@proj2 (@Hom SET E E) (@Hom SET E E) y)) *)
unfold Map_eq in |- *; auto with algebra.
(* Goal: @unit_l (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (Id E) *)
red in |- *.
(* Goal: forall x : Carrier (sgroup_set Endo_SET_sgroup), @Equal (sgroup_set Endo_SET_sgroup) (@Ap (cart (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup)) (sgroup_set Endo_SET_sgroup) (@sgroup_law_map (sgroup_set Endo_SET_sgroup) (sgroup_on_def Endo_SET_sgroup)) (@couple (sgroup_set Endo_SET_sgroup) (sgroup_set Endo_SET_sgroup) (Id E) x)) x *)
simpl in |- *.
(* Goal: forall (x y : Carrier (cart (@Hom SET E E) (@Hom SET E E))) (_ : @Equal (cart (@Hom SET E E) (@Hom SET E E)) x y), @Equal (MAP E E) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) x) (@proj2 (@Hom SET E E) (@Hom SET E E) x)) (@comp_map_map E E E (@proj1 (@Hom SET E E) (@Hom SET E E) y) (@proj2 (@Hom SET E E) (@Hom SET E E) y)) *)
unfold Map_eq in |- *; auto with algebra.
Defined.
End Def.