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(* 02110-1301 USA                                                     *)


(*****************************************************************************)
(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
(*****************************************************************************)
(*                                                                           *)
(*          Category Theory : Dual Category                                  *)
(*                                                                           *)
(*          Amokrane Saibi May 1994                                          *)
(*                                                                           *)
(*****************************************************************************)

Require Export Category.

Set Implicit Arguments.
Unset Strict Implicit.

Section d_cat.

Variable C : Category.

Definition DHom (a b : C) := b --> a.

(* construire un Comp_Type *)

Definition Comp_Darrow (a b c : C) (df : DHom a b) (dg : DHom b c) := dg o df.
                          
Lemma Comp_dual_congl : Congl_law Comp_Darrow.
Proof.
(* Goal: @Congl_law (Ob C) DHom Comp_Darrow *)
unfold Congl_law in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f g : Carrier (DHom a b)) (h : Carrier (DHom b c)) (_ : @Equal (DHom a b) f g), @Equal (DHom a c) (@Comp_Darrow a b c f h) (@Comp_Darrow a b c g h) *)
unfold Comp_Darrow in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f g : Carrier (DHom a b)) (h : Carrier (DHom b c)) (_ : @Equal (DHom a b) f g), @Equal (DHom a c) (@Comp C c b a h f) (@Comp C c b a h g) *)
intros a b c f g h eqf_g.
(* Goal: @Equal (DHom a c) (@Comp C c b a f h) (@Comp C c b a g h) *)
unfold DHom in |- *; apply Comp_r; apply eqf_g.
Qed.
                          
Lemma Comp_dual_congr : Congr_law Comp_Darrow.
Proof.
(* Goal: @Congr_law (Ob C) DHom Comp_Darrow *)
unfold Congr_law in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f g : Carrier (DHom a b)) (h : Carrier (DHom b c)) (_ : @Equal (DHom a b) f g), @Equal (DHom a c) (@Comp_Darrow a b c f h) (@Comp_Darrow a b c g h) *)
unfold Comp_Darrow in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f g : Carrier (DHom a b)) (h : Carrier (DHom b c)) (_ : @Equal (DHom a b) f g), @Equal (DHom a c) (@Comp C c b a h f) (@Comp C c b a h g) *)
intros a b c f g h eqf_g.
(* Goal: @Equal (DHom a c) (@Comp C c b a h f) (@Comp C c b a h g) *)
unfold DHom in |- *; apply Comp_l; apply eqf_g.
Qed.

Definition Comp_Dual := Build_Comp Comp_dual_congl Comp_dual_congr.
            
Lemma Assoc_Dual : Assoc_law Comp_Dual.
Proof.
(* Goal: @Assoc_law (Ob C) DHom Comp_Dual *)
unfold Assoc_law, DHom, Cat_comp, Ap2 in |- *; simpl in |- *. 
(* Goal: forall (a b : Ob C) (f : Carrier (@Hom C a b)), @Equal (@Hom C a b) f (@Comp_Darrow b a a f (@Id C a)) *)
unfold Comp_Darrow in |- *.
(* Goal: forall (a b c d : Ob C) (f : Carrier (@Hom C b a)) (g : Carrier (@Hom C c b)) (h : Carrier (@Hom C d c)), @Equal (@Hom C d a) (@Comp C d b a (@Comp C d c b h g) f) (@Comp C d c a h (@Comp C c b a g f)) *)
intros a b c d df dg dh.
(* Goal: @Equal (@Hom C d a) (@Comp C d b a (@Comp C d c b dh dg) df) (@Comp C d c a dh (@Comp C c b a dg df)) *)
apply Ass1.
Qed.

Lemma Idl_Dual : Idl_law (Hom:=DHom) Comp_Dual (Id (c:=C)).
Proof.
(* Goal: @Idl_law (Ob C) DHom Comp_Dual (@Id C) *)
unfold Idl_law, DHom, Cat_comp, Ap2 in |- *; simpl in |- *. 
(* Goal: forall (a b : Ob C) (f : Carrier (@Hom C a b)), @Equal (@Hom C a b) f (@Comp_Darrow b a a f (@Id C a)) *)
unfold Comp_Darrow in |- *.
(* Goal: forall (a b : Ob C) (f : Carrier (@Hom C a b)), @Equal (@Hom C a b) f (@Comp C a a b (@Id C a) f) *)
intros a b df.
(* Goal: @Equal (@Hom C b a) (@Comp C b a a df (@Id C a)) df *)
apply Idr1.
Qed.

Lemma Idr_Dual : Idr_law (Hom:=DHom) Comp_Dual (Id (c:=C)).
Proof.
(* Goal: @Idr_law (Ob C) DHom Comp_Dual (@Id C) *)
unfold Idr_law, DHom, Cat_comp, Ap2 in |- *; simpl in |- *. 
(* Goal: forall (a b : Ob C) (f : Carrier (@Hom C a b)), @Equal (@Hom C a b) f (@Comp_Darrow b a a f (@Id C a)) *)
unfold Comp_Darrow in |- *.
(* Goal: forall (a b : Ob C) (f : Carrier (@Hom C a b)), @Equal (@Hom C a b) f (@Comp C a a b (@Id C a) f) *)
intros a b df.
(* Goal: @Equal (@Hom C a b) df (@Comp C a a b (@Id C a) df) *)
apply Idl1.
Qed.

Canonical Structure Dual := Build_Category Assoc_Dual Idl_Dual Idr_Dual.

End d_cat.
