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(* 02110-1301 USA                                                     *)
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(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
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(*                                                                           *)
(*          Category Theory : Hom Equality                                   *)
(*                                                                           *)
(*          Amokrane Saibi May 1994                                          *)
(*                                                                           *)
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Require Export Category.
Set Implicit Arguments.
Unset Strict Implicit.
Inductive Equal_hom (C : Category) (a b : C) (f : a --> b) :
forall c d : C, (c --> d) -> Prop :=
Build_Equal_hom : forall g : a --> b, f =_S g -> Equal_hom f g.
Hint Resolve Build_Equal_hom.
Infix "=_H" := Equal_hom (at level 70).
Section equal_hom_equiv.
Variables (C : Category) (a b : C) (f : a --> b).

(* Specifications: *)
Lemma Equal_hom_refl : f =_H f.
Variables (c d : C) (g : c --> d).
Lemma Equal_hom_sym : f =_H g -> g =_H f.
Variables (i j : C) (h : i --> j).
Lemma Equal_hom_trans : f =_H g -> g =_H h -> f =_H h.
End equal_hom_equiv.