path: root/theories/Reals/Rtrigo_def.v

(************************************************************************)
(*  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: Rtrigo_def.v 6295 2004-11-12 16:40:39Z gregoire $ i*)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_fun.
Require Import Max.
Open Local Scope R_scope.

(*****************************)
(* Definition of exponential *)
(*****************************)
Definition exp_in (x l:R) : Prop :=
  infinit_sum (fun i:nat => / INR (fact i) * x ^ i) l.

Lemma exp_cof_no_R0 : forall n:nat, / INR (fact n) <> 0.
intro.
apply Rinv_neq_0_compat.
apply INR_fact_neq_0.
Qed.

Lemma exist_exp : forall x:R, sigT (fun l:R => exp_in x l).
intro;
 generalize
  (Alembert_C3 (fun n:nat => / INR (fact n)) x exp_cof_no_R0 Alembert_exp).
unfold Pser, exp_in in |- *.
trivial.
Defined.

Definition exp (x:R) : R := projT1 (exist_exp x).

Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0.
intros; apply pow_ne_zero.
red in |- *; intro; rewrite H0 in H; elim (lt_irrefl _ H).
Qed.

(*i Calculus of $e^0$ *)
Lemma exist_exp0 : sigT (fun l:R => exp_in 0 l).
apply existT with 1.
unfold exp_in in |- *; unfold infinit_sum in |- *; intros.
exists 0%nat.
intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1.
unfold R_dist in |- *; replace (1 - 1) with 0;
 [ rewrite Rabs_R0; assumption | ring ].
induction  n as [| n Hrecn].
simpl in |- *; rewrite Rinv_1; ring.
rewrite tech5.
rewrite <- Hrecn.
simpl in |- *.
ring.
unfold ge in |- *; apply le_O_n.
Defined.

Lemma exp_0 : exp 0 = 1. 
cut (exp_in 0 (exp 0)).
cut (exp_in 0 1).
unfold exp_in in |- *; intros; eapply uniqueness_sum.
apply H0.
apply H.
exact (projT2 exist_exp0).
exact (projT2 (exist_exp 0)).
Qed.

(**************************************)
(* Definition of hyperbolic functions *)
(**************************************)
Definition cosh (x:R) : R := (exp x + exp (- x)) / 2.
Definition sinh (x:R) : R := (exp x - exp (- x)) / 2.
Definition tanh (x:R) : R := sinh x / cosh x.

Lemma cosh_0 : cosh 0 = 1.
unfold cosh in |- *; rewrite Ropp_0; rewrite exp_0.
unfold Rdiv in |- *; rewrite <- Rinv_r_sym; [ reflexivity | discrR ].
Qed.

Lemma sinh_0 : sinh 0 = 0.
unfold sinh in |- *; rewrite Ropp_0; rewrite exp_0.
unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; apply Rmult_0_l.
Qed.

Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).

Lemma simpl_cos_n :
 forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)). 
intro; unfold cos_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
rewrite Rinv_involutive.
replace
 ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1))) *
  (/ (-1) ^ n * INR (fact (2 * n)))) with
 ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1))) * INR (fact (2 * n)) *
  (-1) ^ 1); [ idtac | ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r.
replace (2 * (n + 1))%nat with (S (S (2 * n))); [ idtac | ring ].
do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
 repeat rewrite Rinv_mult_distr; try (apply not_O_INR; discriminate).
rewrite <- (Rmult_comm (-1)).
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
replace (S (2 * n)) with (2 * n + 1)%nat; [ idtac | ring ].
rewrite mult_INR; rewrite Rinv_mult_distr.
ring.
apply not_O_INR; discriminate.
replace (2 * n + 1)%nat with (S (2 * n));
 [ apply not_O_INR; discriminate | ring ].
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
apply pow_nonzero; discrR.
apply INR_fact_neq_0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.

Lemma archimed_cor1 :
 forall eps:R, 0 < eps ->  exists N : nat, / INR N < eps /\ (0 < N)%nat.
intros; cut (/ eps < IZR (up (/ eps))).
intro; cut (0 <= up (/ eps))%Z.
intro; assert (H2 := IZN _ H1); elim H2; intros; exists (max x 1).
split.
cut (0 < IZR (Z_of_nat x)).
intro; rewrite INR_IZR_INZ; apply Rle_lt_trans with (/ IZR (Z_of_nat x)).
apply Rmult_le_reg_l with (IZR (Z_of_nat x)).
assumption.
rewrite <- Rinv_r_sym;
 [ idtac | red in |- *; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ].
apply Rmult_le_reg_l with (IZR (Z_of_nat (max x 1))).
apply Rlt_le_trans with (IZR (Z_of_nat x)).
assumption.
repeat rewrite <- INR_IZR_INZ; apply le_INR; apply le_max_l.
rewrite Rmult_1_r; rewrite (Rmult_comm (IZR (Z_of_nat (max x 1))));
 rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; repeat rewrite <- INR_IZR_INZ; apply le_INR;
 apply le_max_l.
rewrite <- INR_IZR_INZ; apply not_O_INR.
red in |- *; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat;
 [ intro | apply lt_O_Sn ]; assert (H8 := lt_le_trans _ _ _ H7 H6);
 rewrite H5 in H8; elim (lt_irrefl _ H8).
pattern eps at 1 in |- *; rewrite <- Rinv_involutive.
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat; [ apply Rinv_0_lt_compat; assumption | assumption ].
rewrite H3 in H0; assumption.
red in |- *; intro; rewrite H5 in H; elim (Rlt_irrefl _ H).
apply Rlt_trans with (/ eps).
apply Rinv_0_lt_compat; assumption.
rewrite H3 in H0; assumption.
apply lt_le_trans with 1%nat; [ apply lt_O_Sn | apply le_max_r ].
apply le_IZR; replace (IZR 0) with 0; [ idtac | reflexivity ]; left;
 apply Rlt_trans with (/ eps);
 [ apply Rinv_0_lt_compat; assumption | assumption ].
assert (H0 := archimed (/ eps)).
elim H0; intros; assumption.
Qed.

Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0.
unfold Un_cv in |- *; intros.
assert (H0 := archimed_cor1 eps H).
elim H0; intros; exists x.
intros; rewrite simpl_cos_n; unfold R_dist in |- *; unfold Rminus in |- *;
 rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; 
 rewrite Rabs_Ropp; rewrite Rabs_right.
rewrite mult_INR; rewrite Rinv_mult_distr.
cut (/ INR (2 * S n) < 1).
intro; cut (/ INR (2 * n + 1) < eps).
intro; rewrite <- (Rmult_1_l eps).
apply Rmult_gt_0_lt_compat; try assumption.
change (0 < / INR (2 * n + 1)) in |- *; apply Rinv_0_lt_compat;
 apply lt_INR_0.
replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
apply Rlt_0_1.
cut (x < 2 * n + 1)%nat.
intro; assert (H5 := lt_INR _ _ H4).
apply Rlt_trans with (/ INR x).
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
apply lt_INR_0.
elim H1; intros; assumption.
apply lt_INR_0; replace (2 * n + 1)%nat with (S (2 * n));
 [ apply lt_O_Sn | ring ].
assumption.
elim H1; intros; assumption.
apply lt_le_trans with (S n).
unfold ge in H2; apply le_lt_n_Sm; assumption.
replace (2 * n + 1)%nat with (S (2 * n)); [ idtac | ring ].
apply le_n_S; apply le_n_2n.
apply Rmult_lt_reg_l with (INR (2 * S n)).
apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ idtac | ring ].
ring.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ].
replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_n_S; apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ ring | ring ].
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
replace (2 * n + 1)%nat with (S (2 * n));
 [ apply not_O_INR; discriminate | ring ].
apply Rle_ge; left; apply Rinv_0_lt_compat.
apply lt_INR_0.
replace (2 * S n * (2 * n + 1))%nat with (S (S (4 * (n * n) + 6 * n))).
apply lt_O_Sn.
apply INR_eq.
repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR;
 rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
 replace (INR 0) with 0; [ ring | reflexivity ].
Qed.

Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0.
intro; unfold cos_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat.
apply INR_fact_neq_0.
Qed.

(**********)
Definition cos_in (x l:R) : Prop :=
  infinit_sum (fun i:nat => cos_n i * x ^ i) l.

(**********)
Lemma exist_cos : forall x:R, sigT (fun l:R => cos_in x l).
intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos).
unfold Pser, cos_in in |- *; trivial.
Qed.

(* Definition of cosinus *)
(*************************)
Definition cos (x:R) : R :=
  match exist_cos (Rsqr x) with
  | existT a b => a
  end.


Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).

Lemma simpl_sin_n :
 forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)). 
intro; unfold sin_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
rewrite Rinv_involutive.
replace
 ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1) + 1)) *
  (/ (-1) ^ n * INR (fact (2 * n + 1)))) with
 ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1) + 1)) *
  INR (fact (2 * n + 1)) * (-1) ^ 1); [ idtac | ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r;
 replace (2 * (n + 1) + 1)%nat with (S (S (2 * n + 1))).
do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
 repeat rewrite Rinv_mult_distr.
rewrite <- (Rmult_comm (-1)); repeat rewrite Rmult_assoc;
 rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; replace (S (2 * n + 1)) with (2 * (n + 1))%nat.
repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.
ring.
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
apply not_O_INR; discriminate.
apply prod_neq_R0.
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
apply not_O_INR; discriminate.
replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ].
rewrite mult_plus_distr_l; cut (forall n:nat, S n = (n + 1)%nat).
intros; rewrite (H (2 * n + 1)%nat).
ring.
intros; ring.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
cut (forall n:nat, S (S n) = (n + 2)%nat);
 [ intros; rewrite (H (2 * n + 1)%nat); ring | intros; ring ].
apply pow_nonzero; discrR.
apply INR_fact_neq_0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.

Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0.
unfold Un_cv in |- *; intros; assert (H0 := archimed_cor1 eps H).
elim H0; intros; exists x.
intros; rewrite simpl_sin_n; unfold R_dist in |- *; unfold Rminus in |- *;
 rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; 
 rewrite Rabs_Ropp; rewrite Rabs_right.
rewrite mult_INR; rewrite Rinv_mult_distr.
cut (/ INR (2 * S n) < 1).
intro; cut (/ INR (2 * S n + 1) < eps).
intro; rewrite <- (Rmult_1_l eps); rewrite (Rmult_comm (/ INR (2 * S n + 1)));
 apply Rmult_gt_0_lt_compat; try assumption.
change (0 < / INR (2 * S n + 1)) in |- *; apply Rinv_0_lt_compat;
 apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n));
 [ apply lt_O_Sn | ring ].
apply Rlt_0_1.
cut (x < 2 * S n + 1)%nat.
intro; assert (H5 := lt_INR _ _ H4); apply Rlt_trans with (/ INR x).
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
apply lt_INR_0; elim H1; intros; assumption.
apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n));
 [ apply lt_O_Sn | ring ].
assumption.
elim H1; intros; assumption.
apply lt_le_trans with (S n).
unfold ge in H2; apply le_lt_n_Sm; assumption.
replace (2 * S n + 1)%nat with (S (2 * S n)); [ idtac | ring ].
apply le_S; apply le_n_2n.
apply Rmult_lt_reg_l with (INR (2 * S n)).
apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n)));
 [ apply lt_O_Sn | replace (S n) with (n + 1)%nat; [ idtac | ring ]; ring ].
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ].
replace (2 * S n)%nat with (S (S (2 * n))).
apply lt_n_S; apply lt_O_Sn.
replace (S n) with (n + 1)%nat; [ ring | ring ].
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
left; change (0 < / INR ((2 * S n + 1) * (2 * S n))) in |- *;
 apply Rinv_0_lt_compat.
apply lt_INR_0.
replace ((2 * S n + 1) * (2 * S n))%nat with
 (S (S (S (S (S (S (4 * (n * n) + 10 * n))))))).
apply lt_O_Sn.
apply INR_eq; repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR;
 rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
 replace (INR 0) with 0; [ ring | reflexivity ].
Qed.

Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.
intro; unfold sin_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0.
apply pow_nonzero; discrR.
apply Rinv_neq_0_compat; apply INR_fact_neq_0.
Qed.

(**********)
Definition sin_in (x l:R) : Prop :=
  infinit_sum (fun i:nat => sin_n i * x ^ i) l.

(**********)
Lemma exist_sin : forall x:R, sigT (fun l:R => sin_in x l).
intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin).
unfold Pser, sin_n in |- *; trivial.
Qed.

(***********************)
(* Definition of sinus *)
Definition sin (x:R) : R :=
  match exist_sin (Rsqr x) with
  | existT a b => x * a
  end.

(*********************************************)
(*                 PROPERTIES                *)
(*********************************************)

Lemma cos_sym : forall x:R, cos x = cos (- x).
intros; unfold cos in |- *; replace (Rsqr (- x)) with (Rsqr x).
reflexivity.
apply Rsqr_neg.
Qed.

Lemma sin_antisym : forall x:R, sin (- x) = - sin x.
intro; unfold sin in |- *; replace (Rsqr (- x)) with (Rsqr x);
 [ idtac | apply Rsqr_neg ]. 
case (exist_sin (Rsqr x)); intros; ring.
Qed.

Lemma sin_0 : sin 0 = 0.
unfold sin in |- *; case (exist_sin (Rsqr 0)).
intros; ring.
Qed.

Lemma exist_cos0 : sigT (fun l:R => cos_in 0 l).
apply existT with 1.
unfold cos_in in |- *; unfold infinit_sum in |- *; intros; exists 0%nat.
intros.
unfold R_dist in |- *.
induction  n as [| n Hrecn].
unfold cos_n in |- *; simpl in |- *.
unfold Rdiv in |- *; rewrite Rinv_1.
do 2 rewrite Rmult_1_r.
unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
rewrite tech5.
replace (cos_n (S n) * 0 ^ S n) with 0.
rewrite Rplus_0_r.
apply Hrecn; unfold ge in |- *; apply le_O_n.
simpl in |- *; ring.
Defined.

(* Calculus of (cos 0) *)
Lemma cos_0 : cos 0 = 1.
cut (cos_in 0 (cos 0)).
cut (cos_in 0 1).
unfold cos_in in |- *; intros; eapply uniqueness_sum.
apply H0.
apply H.
exact (projT2 exist_cos0).
assert (H := projT2 (exist_cos (Rsqr 0))); unfold cos in |- *;
 pattern 0 at 1 in |- *; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ].
Qed.