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(* 02110-1301 USA                                                     *)


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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	Diagonal Functor (used in the definition of Cartesian) 		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export PROD.
Require Export Functor.

Set Implicit Arguments.
Unset Strict Implicit.

Section diag.

Variable C : Category.

Definition Diag_ob (c : C) := Build_POb c c.

 Section diag_map_def.

 Variable a b : C.
 
 Definition Diag_mor (f : a --> b) :=
   Build_Pmor (u:=Diag_ob a) (t:=Diag_ob b) f f. 

 Lemma Diag_map_law : Map_law Diag_mor.
 Proof.
(* Goal: @Map_law (@Hom C a b) (@Prod_Hom C C (Diag_ob a) (Diag_ob b)) Diag_mor *)
 unfold Map_law, Diag_mor in |- *; simpl in |- *.
(* Goal: forall (x y : Carrier (@Hom C a b)) (_ : @Equal (@Hom C a b) x y), @Equal_Pmor C C (Diag_ob a) (Diag_ob b) (@Build_Pmor C C (Diag_ob a) (Diag_ob b) x x) (@Build_Pmor C C (Diag_ob a) (Diag_ob b) y y) *)
 intros f g H; unfold Equal_Pmor in |- *; simpl in |- *.
(* Goal: and (@Equal (@Hom C a b) f g) (@Equal (@Hom C a b) f g) *)
 split; trivial.
 Qed.

 Canonical Structure Diag_map := Build_Map Diag_map_law.

 End diag_map_def.

Lemma Diag_comp_law : Fcomp_law Diag_map.
Proof.
(* Goal: @Fcomp_law C (PROD C C) Diag_ob Diag_map *)
unfold Fcomp_law, Diag_map, Diag_mor in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f : Carrier (@Hom C a b)) (g : Carrier (@Hom C b c)), @Equal_Pmor C C (Diag_ob a) (Diag_ob c) (@Build_Pmor C C (Diag_ob a) (Diag_ob c) (@Comp C a b c f g) (@Comp C a b c f g)) (@Comp (PROD C C) (Diag_ob a) (Diag_ob b) (Diag_ob c) (@Build_Pmor C C (Diag_ob a) (Diag_ob b) f f) (@Build_Pmor C C (Diag_ob b) (Diag_ob c) g g)) *)
unfold Equal_Pmor in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f : Carrier (@Hom C a b)) (g : Carrier (@Hom C b c)), and (@Equal (@Hom C a c) (@Comp C a b c f g) (@Comp C a b c f g)) (@Equal (@Hom C a c) (@Comp C a b c f g) (@Comp C a b c f g)) *)
intros a b c f g; split; apply Refl.
Qed.

Lemma Diag_id_law : Fid_law Diag_map.
Proof.
(* Goal: @Fid_law C (PROD C C) Diag_ob Diag_map *)
unfold Fid_law, Diag_map, Diag_mor in |- *; simpl in |- *.
(* Goal: forall a : Ob C, @Equal_Pmor C C (Diag_ob a) (Diag_ob a) (@Build_Pmor C C (Diag_ob a) (Diag_ob a) (@Id C a) (@Id C a)) (@Id_PROD C C (Diag_ob a)) *)
unfold Equal_Pmor, Id_PROD in |- *; simpl in |- *.
(* Goal: forall a : Ob C, and (@Equal (@Hom C a a) (@Id C a) (@Id C a)) (@Equal (@Hom C a a) (@Id C a) (@Id C a)) *)
intro a; split; apply Refl.
Qed.

Canonical Structure Diag := Build_Functor Diag_comp_law Diag_id_law. 

End diag.

