Imported Upstream version 8.0pl1upstream/8.0pl1

1 files changed, 420 insertions, 0 deletions

diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+ 
+(*i $Id: Cos_rel.v,v 1.12.2.1 2004/07/16 19:31:10 herbelin Exp $ i*)
+
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import SeqSeries.
+Require Import Rtrigo_def.
+Open Local 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). 
+intros.
+unfold A1, B1 in |- *.
+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 in |- *.
+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).
+ring.
+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)).
+set
+ (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).
+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 (sum_f_R0 sin_nnn (S n)).
+rewrite <- sum_plus.
+unfold C1 in |- *.
+apply sum_eq; intros.
+induction  i as [| i Hreci].
+simpl in |- *.
+rewrite Rplus_0_l.
+replace (C 0 0) with 1.
+unfold Rdiv in |- *; rewrite Rinv_1.
+ring.
+unfold C in |- *.
+rewrite <- minus_n_n.
+simpl in |- *.
+unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rinv_1; ring.
+unfold sin_nnn in |- *.
+rewrite <- Rmult_plus_distr_l.
+apply Rmult_eq_compat_l.
+rewrite binomial.
+set (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).
+rewrite Rplus_comm.
+apply sum_decomposition.
+apply sum_eq; intros.
+unfold Wn in |- *.
+apply Rmult_eq_compat_l.
+replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))).
+reflexivity.
+apply INR_eq.
+rewrite S_INR; rewrite mult_INR.
+repeat rewrite minus_INR.
+rewrite mult_INR; repeat rewrite S_INR.
+rewrite mult_INR; repeat rewrite S_INR; ring.
+replace (2 * S i)%nat with (S (S (2 * i))).
+apply le_n_S.
+apply le_trans with (2 * i)%nat.
+apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
+apply le_n_Sn.
+apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR;
+ ring.
+assumption.
+apply sum_eq; intros.
+unfold Wn in |- *.
+apply Rmult_eq_compat_l.
+replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat.
+reflexivity.
+apply INR_eq.
+rewrite mult_INR.
+repeat rewrite minus_INR.
+rewrite mult_INR; repeat rewrite S_INR.
+rewrite mult_INR; repeat rewrite S_INR; ring.
+apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
+assumption.
+rewrite <- (Ropp_involutive (sum_f_R0 sin_nnn (S n))).
+apply Ropp_eq_compat.
+replace (- sum_f_R0 sin_nnn (S n)) with (-1 * sum_f_R0 sin_nnn (S n));
+ [ idtac | ring ].
+rewrite scal_sum.
+rewrite decomp_sum.
+replace (sin_nnn 0%nat) with 0.
+rewrite Rmult_0_l; rewrite Rplus_0_l.
+replace (pred (S n)) with n; [ idtac | reflexivity ].
+apply sum_eq; intros.
+rewrite Rmult_comm.
+unfold sin_nnn in |- *.
+rewrite scal_sum.
+rewrite scal_sum.
+apply sum_eq; intros.
+unfold Rdiv in |- *.
+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 in |- *.
+pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat.
+rewrite pow_add.
+ring.
+symmetry  in |- *; apply le_plus_minus; assumption.
+unfold C in |- *.
+unfold Rdiv in |- *; 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.
+apply INR_eq.
+rewrite plus_INR; rewrite mult_INR; repeat rewrite minus_INR.
+rewrite plus_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; ring.
+replace (2 * i0 + 1)%nat with (S (2 * i0)).
+replace (2 * S i)%nat with (S (S (2 * i))).
+apply le_n_S.
+apply le_trans with (2 * i)%nat.
+apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
+apply le_n_Sn.
+apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR;
+ ring.
+apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
+ repeat rewrite S_INR; ring.
+assumption.
+apply INR_fact_neq_0.
+apply INR_fact_neq_0.
+apply INR_fact_neq_0.
+reflexivity.
+apply lt_O_Sn.
+apply sum_eq; intros.
+rewrite scal_sum.
+apply sum_eq; intros.
+unfold Rdiv in |- *.
+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 in |- *; replace i with (i0 + (i - i0))%nat.
+rewrite pow_add.
+ring.
+symmetry  in |- *; apply le_plus_minus; assumption.
+unfold C in |- *.
+unfold Rdiv in |- *; 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.
+apply INR_eq.
+rewrite mult_INR; repeat rewrite minus_INR.
+do 2 rewrite mult_INR; repeat rewrite S_INR; ring.
+apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
+assumption.
+apply INR_fact_neq_0.
+apply INR_fact_neq_0.
+apply INR_fact_neq_0.
+unfold Reste2 in |- *; apply sum_eq; intros.
+apply sum_eq; intros.
+unfold Rdiv in |- *; ring. 
+unfold Reste1 in |- *; apply sum_eq; intros.
+apply sum_eq; intros.
+unfold Rdiv in |- *; ring.
+apply lt_O_Sn.
+Qed.
+
+Lemma pow_sqr : forall (x:R) (i:nat), x ^ (2 * i) = (x * x) ^ i. 
+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). 
+intro. 
+assert (H := exist_cos (x * x)). 
+elim H; intros. 
+assert (p_i := p). 
+unfold cos_in in p. 
+unfold cos_n, infinit_sum in p. 
+unfold R_dist in p. 
+cut (cos x = x0). 
+intro. 
+rewrite H0. 
+unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
+elim (p eps H1); intros. 
+exists x1; intros. 
+unfold A1 in |- *. 
+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 H2; assumption. 
+apply sum_eq. 
+intros. 
+replace ((x * x) ^ i) with (x ^ (2 * i)). 
+reflexivity. 
+apply pow_sqr. 
+unfold cos in |- *. 
+case (exist_cos (Rsqr x)). 
+unfold Rsqr in |- *; intros. 
+unfold cos_in in p_i. 
+unfold cos_in in c. 
+apply uniqueness_sum with (fun i:nat => cos_n i * (x * x) ^ i); assumption. 
+Qed. 
+ 
+Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)). 
+intros. 
+assert (H := exist_cos ((x + y) * (x + y))). 
+elim H; intros. 
+assert (p_i := p). 
+unfold cos_in in p. 
+unfold cos_n, infinit_sum in p. 
+unfold R_dist in p. 
+cut (cos (x + y) = x0). 
+intro. 
+rewrite H0. 
+unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
+elim (p eps H1); intros. 
+exists x1; intros. 
+unfold C1 in |- *. 
+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 H2; assumption. 
+apply sum_eq. 
+intros. 
+replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)). 
+reflexivity. 
+apply pow_sqr. 
+unfold cos in |- *. 
+case (exist_cos (Rsqr (x + y))). 
+unfold Rsqr in |- *; intros. 
+unfold cos_in in p_i. 
+unfold cos_in in c. 
+apply uniqueness_sum with (fun i:nat => cos_n i * ((x + y) * (x + y)) ^ i);
+ assumption. 
+Qed. 
+ 
+Lemma B1_cvg : forall x:R, Un_cv (B1 x) (sin x). 
+intro. 
+case (Req_dec x 0); intro. 
+rewrite H. 
+rewrite sin_0. 
+unfold B1 in |- *. 
+unfold Un_cv in |- *; unfold R_dist in |- *; 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 in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. 
+induction  n as [| n Hrecn]. 
+simpl in |- *; ring. 
+rewrite tech5; rewrite <- Hrecn. 
+simpl in |- *; ring. 
+unfold ge in |- *; apply le_O_n. 
+assert (H0 := exist_sin (x * x)). 
+elim H0; intros. 
+assert (p_i := p). 
+unfold sin_in in p. 
+unfold sin_n, infinit_sum in p. 
+unfold R_dist in p. 
+cut (sin x = x * x0). 
+intro. 
+rewrite H1. 
+unfold Un_cv in |- *; unfold R_dist in |- *; intros. 
+cut (0 < eps / Rabs x);
+ [ intro
+ | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+    [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ]. 
+elim (p (eps / Rabs x) H3); intros. 
+exists x1; intros. 
+unfold B1 in |- *. 
+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. 
+rewrite <- Rinv_l_sym. 
+rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H4; apply H4;
+ assumption. 
+apply Rabs_no_R0; assumption. 
+rewrite scal_sum. 
+apply sum_eq. 
+intros. 
+rewrite pow_add. 
+rewrite pow_sqr. 
+simpl in |- *. 
+ring. 
+unfold sin in |- *. 
+case (exist_sin (Rsqr x)). 
+unfold Rsqr in |- *; intros. 
+unfold sin_in in p_i. 
+unfold sin_in in s. 
+assert
+ (H1 := uniqueness_sum (fun i:nat => sin_n i * (x * x) ^ i) x0 x1 p_i s). 
+rewrite H1; reflexivity. 
+Qed. 
\ No newline at end of file