(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2019       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope ConstructiveReals.
(** Definitions and basic properties of limits of real sequences and series.  WARNING: this file is experimental and likely to change in future releases. *)

(* Specifications: *)
Lemma CR_cv_extens
: forall {R : ConstructiveReals} (xn yn : nat -> CRcarrier R) (l : CRcarrier R),
(forall n:nat, xn n == yn n)
-> CR_cv R xn l
-> CR_cv R yn l.
Lemma CR_cv_opp : forall {R : ConstructiveReals} (xn : nat -> CRcarrier R) (l : CRcarrier R),
CR_cv R xn l
-> CR_cv R (fun n => - xn n) (- l).
Lemma CR_cv_plus : forall {R : ConstructiveReals} (xn yn : nat -> CRcarrier R) (a b : CRcarrier R),
CR_cv R xn a
-> CR_cv R yn b
-> CR_cv R (fun n => xn n + yn n) (a + b).
Lemma CR_cv_unique : forall {R : ConstructiveReals} (xn : nat -> CRcarrier R)
(a b : CRcarrier R),
CR_cv R xn a
-> CR_cv R xn b
-> a == b.
Lemma CR_cv_eq : forall {R : ConstructiveReals}
(v u : nat -> CRcarrier R) (s : CRcarrier R),
(forall n:nat, u n == v n)
-> CR_cv R u s
-> CR_cv R v s.
Lemma CR_cauchy_eq : forall {R : ConstructiveReals}
(un vn : nat -> CRcarrier R),
(forall n:nat, un n == vn n)
-> CR_cauchy R un
-> CR_cauchy R vn.
Lemma CR_cv_proper : forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (a b : CRcarrier R),
CR_cv R un a
-> a == b
-> CR_cv R un b.