(*****************************************************************************)
(** * Classical Dedekind reals                                               *)
(*****************************************************************************)

(** ** Definition *)

Definition DReal : Set
  := { f : Q -> bool | isLowerCut f }.(** ** Induction principle *)

Fixpoint DRealQlim_rec (f : Q -> bool) (low : isLowerCut f) (n p : nat) { struct p }
  : f (proj1_sig (lowerCutBelow f low) + (Z.of_nat p # Pos.of_nat (S n)))%Q = false
    -> { q : Q | f q = true /\ f (q + (1 # Pos.of_nat (S n)))%Q = false }.

Lemma DRealAbstr_aux : forall x H, isLowerCut (fun q : Q => if sig_forall_dec (fun n : nat => seq x (- Z.of_nat n) <= q + 2 ^ (- Z.of_nat n)) (H q) then true else false).