Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Reals.Abstract.ConstructiveLimits

Require Import QArith Qabs.
Require Import ConstructiveReals.
Require Import ConstructiveAbs.
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.
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.
#[global]
Instance CR_cv_morph
: forall {R : ConstructiveReals} (un : nat -> CRcarrier R), CMorphisms.Proper
(CMorphisms.respectful (CReq R) CRelationClasses.iffT) (CR_cv R un).
Lemma Un_cv_nat_real : forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (l : CRcarrier R),
CR_cv R un l
-> forall eps : CRcarrier R,
0 < eps
-> { p : nat & forall i:nat, le p i -> CRabs R (un i - l) < eps }.
Lemma Un_cv_real_nat : forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (l : CRcarrier R),
(forall eps : CRcarrier R,
0 < eps
-> { p : nat & forall i:nat, le p i -> CRabs R (un i - l) < eps })
-> CR_cv R un l.
Lemma CR_cv_minus :
forall {R : ConstructiveReals}
(An Bn:nat -> CRcarrier R) (l1 l2:CRcarrier R),
CR_cv R An l1 -> CR_cv R Bn l2
-> CR_cv R (fun i:nat => An i - Bn i) (l1 - l2).
Lemma CR_cv_nonneg :
forall {R : ConstructiveReals} (An:nat -> CRcarrier R) (l:CRcarrier R),
CR_cv R An l
-> (forall n:nat, 0 <= An n)
-> 0 <= l.
Lemma CR_cv_scale : forall {R : ConstructiveReals} (u : nat -> CRcarrier R)
(a : CRcarrier R) (s : CRcarrier R),
CR_cv R u s -> CR_cv R (fun n => u n * a) (s * a).
Lemma CR_cv_const : forall {R : ConstructiveReals} (a : CRcarrier R),
CR_cv R (fun n => a) a.
Lemma Rcv_cauchy_mod : forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (l : CRcarrier R),
CR_cv R un l -> CR_cauchy R un.
Lemma CR_growing_transit : forall {R : ConstructiveReals} (un : nat -> CRcarrier R),
(forall n:nat, un n <= un (S n))
-> forall n p : nat, le n p -> un n <= un p.
Lemma growing_ineq :
forall {R : ConstructiveReals} (Un:nat -> CRcarrier R) (l:CRcarrier R),
(forall n:nat, Un n <= Un (S n))
-> CR_cv R Un l -> forall n:nat, Un n <= l.
Lemma CR_cv_open_below
: forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (m l : CRcarrier R),
CR_cv R un l
-> m < l
-> { n : nat & forall i:nat, le n i -> m < un i }.
Lemma CR_cv_open_above
: forall {R : ConstructiveReals}
(un : nat -> CRcarrier R) (m l : CRcarrier R),
CR_cv R un l
-> l < m
-> { n : nat & forall i:nat, le n i -> un i < m }.
Lemma CR_cv_bound_down : forall {R : ConstructiveReals}
(u : nat -> CRcarrier R) (A l : CRcarrier R) (N : nat),
(forall n:nat, le N n -> A <= u n)
-> CR_cv R u l
-> A <= l.
Lemma CR_cv_bound_up : forall {R : ConstructiveReals}
(u : nat -> CRcarrier R) (A l : CRcarrier R) (N : nat),
(forall n:nat, le N n -> u n <= A)
-> CR_cv R u l
-> l <= A.
Lemma CR_cv_le : forall {R : ConstructiveReals}
(u v : nat -> CRcarrier R) (a b : CRcarrier R),
(forall n:nat, u n <= v n)
-> CR_cv R u a
-> CR_cv R v b
-> a <= b.
Lemma CR_cv_abs_cont : forall {R : ConstructiveReals}
(u : nat -> CRcarrier R) (s : CRcarrier R),
CR_cv R u s
-> CR_cv R (fun n => CRabs R (u n)) (CRabs R s).
Lemma CR_cv_dist_cont : forall {R : ConstructiveReals}
(u : nat -> CRcarrier R) (a s : CRcarrier R),
CR_cv R u s
-> CR_cv R (fun n => CRabs R (a - u n)) (CRabs R (a - s)).
Lemma CR_cv_shift :
forall {R : ConstructiveReals} f k l,
CR_cv R (fun n => f (n + k)%nat) l -> CR_cv R f l.
Lemma CR_cv_shift' :
forall {R : ConstructiveReals} f k l,
CR_cv R f l -> CR_cv R (fun n => f (n + k)%nat) l.