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(* 02110-1301 USA                                                     *)
(** dec. Some utilities for proving decidability of predicates.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** Checks that all nats before n are P *)
Fixpoint allbefore (P : nat -> Prop) (n : nat) {struct n} : Prop :=
match n with
| O => True
| S x => allbefore P x /\ P x
end.
(** Checks that some nat before n is P *)
Fixpoint exbefore (P : nat -> Prop) (n : nat) {struct n} : Prop :=
match n with
| O => False
| S x => exbefore P x \/ P x
end.
(** some decidable relations on nat *)
(** relations on Z *)
(** the connectives preserve decidability *)
(** bounded quantifiers preserve decidability *)
(** De Morgan's law holds for decidable P if the quantifiers are bounded *)

Theorem decDeMorgan :
forall (N : nat) (P : nat -> Prop),
(forall n : nat, P n \/ ~ P n) ->
((exists x : nat, x < N /\ P x) <-> ~ (forall x : nat, x < N -> ~ P x)).