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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                      Equalizers                		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export CatProperty.
Set Implicit Arguments.
Unset Strict Implicit.
(* Equalizers *)
Section equaz_def.
Variables (C : Category) (a b : C) (I : Type) (k : I -> (a --> b)).
Section equaz_laws.
Variables (c : C) (e : c --> a).
Definition Equalizer_eq (r : C) (h : r --> a) :=
forall i j : I, h o k i =_S h o k j.
Definition Equalizer_law1 := Equalizer_eq e.
Variable E_diese : forall (r : C) (h : r --> a), Equalizer_eq h -> (r --> c).
Definition Equalizer_law2 :=
forall (r : C) (h : r --> a) (p : Equalizer_eq h), h =_S E_diese p o e.
Definition Equalizer_law3 :=
forall (r : C) (h : r --> a) (p : Equalizer_eq h) (l : r --> c),
h =_S l o e -> l =_S E_diese p.
End equaz_laws.
Structure Equalizer : Type :=
{E_ob : C;
E_mor : E_ob --> a;
Prf_E_law1 : Equalizer_law1 E_mor;
E_diese : forall (r : C) (h : r --> a), Equalizer_eq h -> (r --> E_ob);
Prf_E_law2 : Equalizer_law2 E_mor E_diese;
Prf_E_law3 : Equalizer_law3 E_mor E_diese}.
Variable f : Equalizer.
(* tout Equalizer est monic *)

(* Specifications: *)
Lemma E_monic : Monic_law (E_mor f).
(* un epic Equalizer est iso *)
Lemma Epic_Equalizer_id : Epic_law (E_mor f) -> Equalizer_eq (Id a).
Lemma Epic_Equalizer_iso :
forall p : Epic_law (E_mor f),
let f1 := E_diese f (Epic_Equalizer_id p) in AreIsos (E_mor f) f1.
Lemma Equalizer_iso :
forall (h1 : a --> E_ob f) (p : RIso_law (E_mor f) h1),
let f1 := E_diese f (Epic_Equalizer_id (RightInv_epic p)) in
AreIsos (E_mor f) f1.