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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                   Natural Isomorphism            		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export CatProperty.
Require Export CatFunct.
Set Implicit Arguments.
Unset Strict Implicit.
(* 1ere solution: def primitive *)
Section natiso_def.
Variable A B : Category.
Definition RNatIso_law (F G : Functor A B) (T : NT F G)
(T1 : NT G F) := T1 o_NTv T =_NT Id_CatFunct G.
Variable F G : Functor A B.
Definition AreNatIsos (T : NT F G) (T1 : NT G F) :=
RNatIso_law T T1 /\ RNatIso_law T1 T.
Structure > NatIso : Type :=
{NatIso_mor :> NT F G;
NatIso_inv : NT G F;
Prf_NatIso :> AreNatIsos NatIso_mor NatIso_inv}.

(* Specifications: *)
Lemma Idl_nat_inv :
forall i : NatIso, RNatIso_law (NatIso_mor i) (NatIso_inv i).
Lemma Idr_nat_inv :
forall i : NatIso, RNatIso_law (NatIso_inv i) (NatIso_mor i).