(* the number of different elements in X is finite *)
Inductive cardinality {X} (P:X->Prop): nat -> Prop:=
|Cardinality n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j -> f(i)<>f(j)) ->
(forall i, i<n -> P (f i)) ->
(forall x, P x -> exists i, i < n /\ f i = x) ->
cardinality P n.
Inductive cardinality' {X} (P:X->Prop): nat -> Prop:=
|Cardinality' n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j  -> f(i)<>f(j)) ->
(forall i, i<n -> P (f i)) ->
cardinality' P n.

(* Specifications: *)
Lemma repetition_dec: forall (f:nat -> X) x n,  (exists i, i<n /\ f(i) = x) \/ (forall i, i<n -> f i <> x).
Lemma finite_cardinality:
forall (P:X->Prop) (EM: forall x, P x \/ ~ P x), finite P -> exists n, cardinality P n.
(* If every element in the filter, is finitely branching then finite safety safeN implies infinite safetyy safe.*)
Lemma filtered_konigsafe:
forall (x : X),
(forall P : Prop, P \/ ~ P) ->
(forall x0 : X, Filter x0 -> finite (R x0)) ->
Filter x ->
(forall n : nat, safeN X R n x) -> konig.safe X R x.