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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	   A sufficient condition for a Comma to be Complete        	     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
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Require Export Pres_Limits.
Require Export Comma_proj.
Set Implicit Arguments.
Unset Strict Implicit.
Section comma_complete.
Variables (X A : Category) (G : Functor A X) (J : Category).
Hypothesis A_comp_for_J : forall F : Functor J A, Limit F.
Hypothesis G_pres_JA : forall F : Functor J A, Preserves_limits F G.
Variables (x : X) (F : Functor J (Comma G x)).
Let F' := F o_F Comma_proj G x.
Let l_F' := A_comp_for_J F'.
Let l_GF' := G_pres_JA l_F'.
(* construction d'un cone Delta(x)->F' o G *)
Definition Tc_tau (i : J) := Mor_com_ob (F i).

(* Specifications: *)
Lemma Tc_tau_cone_law : Cone_law (F:=F' o_F G) Tc_tau.
Definition Tc_cone := Build_Cone Tc_tau_cone_law.