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(* 02110-1301 USA                                                     *)


(*****************************************************************************)
(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
(*****************************************************************************)
(*                                                                           *)
(*          Category Product : Projection Functors                           *)
(*                                                                           *)
(*          Amokrane Saibi,  May 1994                                        *)
(*                                                                           *)
(*****************************************************************************)

Require Export PROD.
Require Export Functor. 

Set Implicit Arguments.
Unset Strict Implicit.

(* Projection Functors *)

Section prod_proj.

Variable A B : Category.

(* Fst *)

Lemma Fst_map_law : forall a b : PROD A B, Map_law (Hom_l (u:=a) (t:=b)).
Proof.
(* Goal: forall a b : Ob (PROD A B), @Map_law (@Prod_Hom A B a b) (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b) *)
unfold Map_law in |- *.
(* Goal: forall (a b : Ob (PROD A B)) (x y : Carrier (@Prod_Hom A B a b)) (_ : @Equal (@Prod_Hom A B a b) x y), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b x) (@Hom_r A B a b y) *)
intros a b; unfold Equal at 1 in |- *; simpl in |- *.
(* Goal: forall (x y : @Pmor A B a b) (_ : @Equal_Pmor A B a b x y), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b x) (@Hom_r A B a b y) *)
unfold Equal_Pmor in |- *; intros f g H.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b f) (@Hom_r A B a b g) *)
elim H; intros; assumption.
Qed.

Canonical Structure Fst_map (a b : PROD A B) :=
  Build_Map (Fst_map_law (a:=a) (b:=b)).

Lemma Fst_comp_law : Fcomp_law Fst_map.
Proof.
(* Goal: @Fcomp_law (PROD A B) B (@Ob_r A B) Snd_map *)
unfold Fcomp_law in |- *; simpl in |- *.
(* Goal: forall (a b c : POb A B) (f : @Pmor A B a b) (g : @Pmor A B b c), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
intros a b c f g.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
apply Refl.
Qed.

Lemma Fst_id_law : Fid_law Fst_map.
Proof.
(* Goal: @Fid_law (PROD A B) B (@Ob_r A B) Snd_map *)
unfold Fid_law in |- *; simpl in |- *.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
intro a; apply Refl.
Qed.
 
Canonical Structure Fst := Build_Functor Fst_comp_law Fst_id_law.

(* Snd *)

Lemma Snd_map_law : forall a b : PROD A B, Map_law (Hom_r (u:=a) (t:=b)).
Proof.
(* Goal: forall a b : Ob (PROD A B), @Map_law (@Prod_Hom A B a b) (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b) *)
unfold Map_law in |- *.
(* Goal: forall (a b : Ob (PROD A B)) (x y : Carrier (@Prod_Hom A B a b)) (_ : @Equal (@Prod_Hom A B a b) x y), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b x) (@Hom_r A B a b y) *)
intros a b; unfold Equal at 1 in |- *; simpl in |- *.
(* Goal: forall (x y : @Pmor A B a b) (_ : @Equal_Pmor A B a b x y), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b x) (@Hom_r A B a b y) *)
unfold Equal_Pmor in |- *; intros f g H.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B b)) (@Hom_r A B a b f) (@Hom_r A B a b g) *)
elim H; intros; assumption.
Qed.

Canonical Structure Snd_map (a b : PROD A B) :=
  Build_Map (Snd_map_law (a:=a) (b:=b)).

Lemma Snd_comp_law : Fcomp_law Snd_map.
Proof.
(* Goal: @Fcomp_law (PROD A B) B (@Ob_r A B) Snd_map *)
unfold Fcomp_law in |- *; simpl in |- *.
(* Goal: forall (a b c : POb A B) (f : @Pmor A B a b) (g : @Pmor A B b c), @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
intros a b c f g.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
apply Refl.
Qed.

Lemma Snd_id_law : Fid_law Snd_map.
Proof.
(* Goal: @Fid_law (PROD A B) B (@Ob_r A B) Snd_map *)
unfold Fid_law in |- *; simpl in |- *.
(* Goal: @Equal (@Hom B (@Ob_r A B a) (@Ob_r A B c)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) (@Comp B (@Ob_r A B a) (@Ob_r A B b) (@Ob_r A B c) (@Hom_r A B a b f) (@Hom_r A B b c g)) *)
intro a; apply Refl.
Qed.
 
Canonical Structure Snd := Build_Functor Snd_comp_law Snd_id_law.

End prod_proj.