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(*                                                                    *)
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(*                                                                    *)
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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                       SubSetoid                  		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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(* (SubSetoid A P) setoid of elements of A with the property P *)
Require Export Map.
Set Implicit Arguments.
Unset Strict Implicit.
Section sub_setoid.
Variable U : Setoid.
Definition Reg_law (A : U -> Prop) := forall x y : U, x =_S y -> A x -> A y.
Structure > Setoid_pred : Type :=
{Pred :> U -> Prop; Prf_reg :> Reg_law Pred}.
Variable A : Setoid_pred.
Structure SubType : Type :=  {Elt_sub : U; Prf_constr : A Elt_sub}.
Definition Equal_SubType (a b : SubType) := Elt_sub a =_S Elt_sub b.

(* Specifications: *)
Lemma Equal_SubType_equiv : Equivalence Equal_SubType.
Canonical Structure SubSetoid : Setoid := Equal_SubType_equiv.