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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	               Category with Binary Products       		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export Category.
Require Export Map2.
Set Implicit Arguments.
Unset Strict Implicit.
(* cate'gorie avec produits *)
Section bp_def.
Variables (C : Category) (a b : C).
Section bp_laws.
Variables (Obj_prod : C) (Proj1_prod : Obj_prod --> a)
(Proj2_prod : Obj_prod --> b).
Variable Op : forall c : C, Map2 (c --> a) (c --> b) (c --> Obj_prod).
Definition Together_prod (c : C) (f : c --> a) (g : c --> b) := Op c f g.
Definition Eq1_prod_law :=
forall (c : C) (f : c --> a) (g : c --> b),
Together_prod f g o Proj1_prod =_S f.
Definition Eq2_prod_law :=
forall (c : C) (f : c --> a) (g : c --> b),
Together_prod f g o Proj2_prod =_S g.
Definition Unique_together_law :=
forall (c : C) (h : c --> Obj_prod),
Together_prod (h o Proj1_prod) (h o Proj2_prod) =_S h.
End bp_laws.
Structure BinProd : Type :=
{Obj_prod : C;
Proj1_prod : Obj_prod --> a;
Proj2_prod : Obj_prod --> b;
Op_together : forall c : C, Map2 (c --> a) (c --> b) (c --> Obj_prod);
Prf_eq1_prod : Eq1_prod_law Proj1_prod Op_together;
Prf_eq2_prod : Eq2_prod_law Proj2_prod Op_together;
Prf_unique_together :
Unique_together_law Proj1_prod Proj2_prod Op_together}.
Definition Together (C1 : BinProd) (c : C) (f : c --> a)
(g : c --> b) := Together_prod (Op_together C1) f g.
End bp_def.
Definition HasBinProd (C : Category) := forall a b : C, BinProd a b.
Section hasbinprod_proj.
Variables (C : Category) (C1 : HasBinProd C) (a b : C).
Definition H_obj_prod := Obj_prod (C1 a b).
Definition H_proj1_prod := Proj1_prod (C1 a b).
Definition H_proj2_prod := Proj2_prod (C1 a b).
Definition H_together (c : C) (f : c --> a) (g : c --> b) :=
Together (C1 a b) f g.
End hasbinprod_proj.
(* qques defs et res utils *)

Lemma Eq3_prod :
forall (C : Category) (a b : C) (C1 : BinProd a b)
(c : C) (f : c --> a) (g : c --> b) (c' : C) (k : c' --> c),
k o Together C1 f g =_S Together C1 (k o f) (k o g).