Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Cos_rel.v Sprache: Coq

Original von: Coq^©

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <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)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Require Import OmegaTactic.
Local Open Scope R_scope.

Definition A1 (x:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) N.

Definition B1 (x:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
    N.

Definition C1 (x y:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) N.

Definition Reste1 (x y:R) (N:nat) : R :=
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
            x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
            y ^ (2 * (N - l))) (pred (N - k))) (pred N).

Definition Reste2 (x y:R) (N:nat) : R :=
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
            x ^ (2 * S (l + k) + 1) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
            y ^ (2 * (N - l) + 1)) (pred (N - k))) (
    pred N).

Definition Reste (x y:R) (N:nat) : R := Reste2 x y N - Reste1 x y (S N).

(* Here is the main result that will be used to prove that (cos (x+y))=(cos x)(cos y)-(sin x)(sin y) *)
Theorem cos_plus_form :
forall (x y:R) (n:nat),
   (0 < n)%nat ->
   A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n).
Proof.
intros.
unfold A1, B1.
rewrite
(cauchy_finite (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k))
    (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * y ^ (2 * k)) (
    S n)).
rewrite
(cauchy_finite
    (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
    (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * y ^ (2 * k + 1)) n H)
.
unfold Reste.
replace
(sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
            x ^ (2 * S (l + k)) *
            ((-1) ^ (S n - l) / INR (fact (2 * (S n - l))) *
             y ^ (2 * (S n - l)))) (pred (S n - k))) (
    pred (S n))) with (Reste1 x y (S n)).
replace
(sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
            x ^ (2 * S (l + k) + 1) *
            ((-1) ^ (n - l) / INR (fact (2 * (n - l) + 1)) *
             y ^ (2 * (n - l) + 1))) (pred (n - k))) (
    pred n)) with (Reste2 x y n).
replace
(sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun p:nat =>
            (-1) ^ p / INR (fact (2 * p)) * x ^ (2 * p) *
            ((-1) ^ (k - p) / INR (fact (2 * (k - p))) * y ^ (2 * (k - p))))
         k) (S n)) with
(sum_f_R0
    (fun k:nat =>
       (-1) ^ k / INR (fact (2 * k)) *
       sum_f_R0
         (fun l:nat => C (2 * k) (2 * l) * x ^ (2 * l) * y ^ (2 * (k - l))) k)
    (S n)).
pose
(sin_nnn :=
  fun n:nat =>
    match n with
    | O => 0
    | S p =>
        (-1) ^ S p / INR (fact (2 * S p)) *
        sum_f_R0
          (fun l:nat =>
             C (2 * S p) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (p - l))) p
    end).
ring_simplify.
unfold Rminus.
replace
(* (-   old ring compat *)
(-
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun p:nat =>
            (-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
            ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
             y ^ (2 * (k - p) + 1))) k) n) with (sum_f_R0 sin_nnn (S n)).
rewrite <- sum_plus.
unfold C1.
apply sum_eq; intros.
induction  i as [| i Hreci].
simpl.
unfold C; simpl.
field; discrR.
unfold sin_nnn.
rewrite <- Rmult_plus_distr_l.
apply Rmult_eq_compat_l.
rewrite binomial.
pose (Wn := fun i0:nat => C (2 * S i) i0 * x ^ i0 * y ^ (2 * S i - i0)).
replace
(sum_f_R0
    (fun l:nat => C (2 * S i) (2 * l) * x ^ (2 * l) * y ^ (2 * (S i - l)))
    (S i)) with (sum_f_R0 (fun l:nat => Wn (2 * l)%nat) (S i)).
replace
(sum_f_R0
    (fun l:nat =>
       C (2 * S i) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (i - l))) i) with
(sum_f_R0 (fun l:nat => Wn (S (2 * l))) i).
apply sum_decomposition.
apply sum_eq; intros.
unfold Wn.
apply Rmult_eq_compat_l.
replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))).
reflexivity.
omega.
apply sum_eq; intros.
unfold Wn.
apply Rmult_eq_compat_l.
replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat.
reflexivity.
omega.
replace
(-
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun p:nat =>
            (-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
            ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
             y ^ (2 * (k - p) + 1))) k) n) with
(-1 *
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun p:nat =>
            (-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) *
            ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
             y ^ (2 * (k - p) + 1))) k) n);[idtac|ring].
rewrite scal_sum.
rewrite decomp_sum.
replace (sin_nnn 0%nat) with 0.
rewrite Rplus_0_l.
change (pred (S n)) with n.
   (* replace (pred (S n)) with n; [ idtac | reflexivity ]. *)
apply sum_eq; intros.
rewrite Rmult_comm.
unfold sin_nnn.
rewrite scal_sum.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv.
(*repeat rewrite Rmult_assoc.*)
(* rewrite (Rmult_comm (/ INR (fact (2 * S i)))). *)
repeat rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * S i)))).
repeat rewrite <- Rmult_assoc.
replace (/ INR (fact (2 * S i)) * C (2 * S i) (S (2 * i0))) with
(/ INR (fact (2 * i0 + 1)) * / INR (fact (2 * (i - i0) + 1))).
replace (S (2 * i0)) with (2 * i0 + 1)%nat; [ idtac | ring ].
replace (S (2 * (i - i0))) with (2 * (i - i0) + 1)%nat; [ idtac | ring ].
replace ((-1) ^ S i) with (-1 * (-1) ^ i0 * (-1) ^ (i - i0)).
ring.
simpl.
pattern i at 2; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry ; apply le_plus_minus; assumption.
unfold C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
rewrite Rinv_mult_distr.
replace (S (2 * i0)) with (2 * i0 + 1)%nat;
[ apply Rmult_eq_compat_l | ring ].
replace (2 * S i - (2 * i0 + 1))%nat with (2 * (i - i0) + 1)%nat.
reflexivity.
omega.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
reflexivity.
apply lt_O_Sn.
(* ring. *)
apply sum_eq; intros.
rewrite scal_sum.
apply sum_eq; intros.
unfold Rdiv.
repeat rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * i)))).
repeat rewrite <- Rmult_assoc.
replace (/ INR (fact (2 * i)) * C (2 * i) (2 * i0)) with
(/ INR (fact (2 * i0)) * / INR (fact (2 * (i - i0)))).
replace ((-1) ^ i) with ((-1) ^ i0 * (-1) ^ (i - i0)).
ring.
pattern i at 2; replace i with (i0 + (i - i0))%nat.
rewrite pow_add.
ring.
symmetry ; apply le_plus_minus; assumption.
unfold C.
unfold Rdiv; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
rewrite Rinv_mult_distr.
replace (2 * i - 2 * i0)%nat with (2 * (i - i0))%nat.
reflexivity.
omega.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
unfold Reste2; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv; ring.
unfold Reste1; apply sum_eq; intros.
apply sum_eq; intros.
unfold Rdiv; ring.
apply lt_O_Sn.
Qed.

Lemma pow_sqr : forall (x:R) (i:nat), x ^ (2 * i) = (x * x) ^ i.
Proof.
intros.
assert (H := pow_Rsqr x i).
unfold Rsqr in H; exact H.
Qed.

Lemma A1_cvg : forall x:R, Un_cv (A1 x) (cos x).
Proof.
intro.
unfold cos; destruct (exist_cos (Rsqr x)) as (x0,p).
unfold cos_in, cos_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
destruct (p eps H) as (x1,H0).
exists x1; intros.
unfold A1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) n) with
(sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * (x * x) ^ i) n).
apply H0; assumption.
apply sum_eq.
intros.
replace ((x * x) ^ i) with (x ^ (2 * i)).
reflexivity.
apply pow_sqr.
Qed.

Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)).
Proof.
intros.
unfold cos.
destruct (exist_cos (Rsqr (x + y))) as (x0,p).
unfold cos_in, cos_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
destruct (p eps H) as (x1,H0).
exists x1; intros.
unfold C1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) n)
with
(sum_f_R0
    (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * ((x + y) * (x + y)) ^ i) n).
apply H0; assumption.
apply sum_eq.
intros.
replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)).
reflexivity.
apply pow_sqr.
Qed.

Lemma B1_cvg : forall x:R, Un_cv (B1 x) (sin x).
Proof.
intro.
case (Req_dec x 0); intro.
rewrite H.
rewrite sin_0.
unfold B1.
unfold Un_cv; unfold R_dist; intros; exists 0%nat; intros.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k + 1))
    n) with 0.
unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
induction  n as [| n Hrecn].
simpl; ring.
rewrite tech5; rewrite <- Hrecn.
simpl; ring.
unfold ge; apply le_O_n.
unfold sin. destruct (exist_sin (Rsqr x)) as (x0,p).
unfold sin_in, sin_n, infinite_sum, R_dist in p.
unfold Un_cv, R_dist; intros.
cut (0 < eps / Rabs x);
[ intro
| unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ].
destruct (p (eps / Rabs x) H1) as (x1,H2).
exists x1; intros.
unfold B1.
replace
(sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
    n) with
(x *
  sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n).
replace
(x *
  sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n -
  x * x0) with
(x *
  (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n -
   x0)); [ idtac | ring ].
rewrite Rabs_mult.
apply Rmult_lt_reg_l with (/ Rabs x).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- Rmult_assoc, <- Rinv_l_sym, Rmult_1_l, <- (Rmult_comm eps). apply H2;
assumption.
apply Rabs_no_R0; assumption.
rewrite scal_sum.
apply sum_eq.
intros.
rewrite pow_add.
rewrite pow_sqr.
simpl.
ring.
Qed.

¤ Dauer der Verarbeitung: 0.18 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.