(*=========================================================
                     SETS OF DARTS 

       FOR NOETHERIAN INDUCTIVE REASONINGS
==========================================================*)

Inductive set:Set:=
    Vs : set | Is : set -> dart -> set.(*=========================================================
                     SETS OF DARTS 

       FOR NOETHERIAN INDUCTIVE REASONINGS
==========================================================*)

Inductive set:Set:=
    Vs : set | Is : set -> dart -> set.(*=========================================================
                     SETS OF DARTS 

       FOR NOETHERIAN INDUCTIVE REASONINGS
==========================================================*)

Inductive set:Set:=
    Vs : set | Is : set -> dart -> set.Fixpoint exds(s:set)(z:dart){struct s}:Prop:=
  match s with
     Vs => False
   | Is s0 x => x=z \/ exds s0 z
  end.

(* Specifications: *)
Lemma exds_dec: forall(s:set)(z:dart), {exds s z}+{~exds s z}.
Lemma not_exds_Vs: forall s:set, (forall z:dart, ~exds s z) -> s = Vs.
Lemma not_exds_diff: forall(s:set)(z:dart), ~exds s z -> forall(t:dart), exds s t -> z <> t.