(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)
 
(*i $Id$ i*)

Require Rbase.
Require Rseries.
Require Alembert.
Require Binome.
Require Rtrigo_def.
Require Rtrigo_alt.
Require Export Cauchy_prod.

Lemma minus_sum : (An,Bn:nat->R;N:nat) (sum_f_R0 [i:nat]``(An i)-(Bn i)`` N)==``(sum_f_R0 An N)-(sum_f_R0 Bn N)``. 
Intros; Induction N. 
Simpl; Ring. 
Do 3 Rewrite tech5; Rewrite HrecN; Ring. 
Qed. 

Lemma sum_decomposition : (An:nat->R;N:nat) (Rplus (sum_f_R0 [l:nat](An (mult (2) l)) (S N)) (sum_f_R0 [l:nat](An (S (mult (2) l))) N))==(sum_f_R0 An (mult (2) (S N))).
Intros.
Induction N.
Simpl; Ring.
Rewrite tech5.
Rewrite (tech5 [l:nat](An (S (mult (2) l))) N).
Replace (mult (2) (S (S N))) with (S (S (mult (2) (S N)))).
Rewrite (tech5 An (S (mult (2) (S N)))).
Rewrite (tech5 An (mult (2) (S N))).
Rewrite <- HrecN.
Ring.
Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR;Repeat Rewrite S_INR.
Ring.
Qed.

Definition A1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))`` N). 
 
Definition B1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))`` N). 
 
Definition C1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))`` N). 
 
Definition Reste1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (mult (S (S O)) (S (plus l k)))))*(pow x (mult (S (S O)) (S (plus l k))))*(pow (-1) (minus N l))/(INR (fact (mult (S (S O)) (minus N l))))*(pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k))) (pred N)).

Definition Reste2 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))*(pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))*(pow (-1) (minus N l))/(INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))*(pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k))) (pred N)).

Definition Reste [x,y:R] : nat -> R := [N:nat]``(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 : (x,y:R;n:nat) (lt O n) -> ``(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.
Rewrite (cauchy_finite [k:nat]
        ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*
        (pow x (mult (S (S O)) k))`` [k:nat]
      ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*
      (pow y (mult (S (S O)) k))`` (S n)).
Rewrite (cauchy_finite [k:nat]
      ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*
      (pow x (plus (mult (S (S O)) k) (S O)))`` [k:nat]
      ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*
      (pow y (plus (mult (S (S O)) k) (S O)))`` n H).
Unfold Reste.
Replace (sum_f_R0
   [k:nat]
      (sum_f_R0
        [l:nat]
         ``(pow ( -1) (S (plus l k)))/
         (INR (fact (mult (S (S O)) (S (plus l k)))))*
         (pow x (mult (S (S O)) (S (plus l k))))*
         ((pow ( -1) (minus (S n) l))/
         (INR (fact (mult (S (S O)) (minus (S n) l))))*
         (pow y (mult (S (S O)) (minus (S n) l))))``
        (pred (minus (S n) k))) (pred (S n))) with (Reste1 x y (S n)).
Replace (sum_f_R0
   [k:nat]
      (sum_f_R0
        [l:nat]
         ``(pow ( -1) (S (plus l k)))/
         (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))*
         (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))*
         ((pow ( -1) (minus n l))/
         (INR (fact (plus (mult (S (S O)) (minus n l)) (S O))))*
         (pow y (plus (mult (S (S O)) (minus n l)) (S O))))``
        (pred (minus n k))) (pred n)) with (Reste2 x y n).
Ring.
Replace (sum_f_R0
   [k:nat]
      (sum_f_R0
        [p:nat]
         ``(pow ( -1) p)/(INR (fact (mult (S (S O)) p)))*
         (pow x (mult (S (S O)) p))*((pow ( -1) (minus k p))/
         (INR (fact (mult (S (S O)) (minus k p))))*
         (pow y (mult (S (S O)) (minus k p))))`` k) (S n)) with (sum_f_R0 [k:nat](Rmult ``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) k) (mult (S (S O)) l))*(pow x (mult (S (S O)) l))*(pow y (mult (S (S O)) (minus k l)))`` k)) (S n)).
Pose sin_nnn := [n:nat]Cases n of O => R0 | (S p) => (Rmult ``(pow (-1) (S p))/(INR (fact (mult (S (S O)) (S p))))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) (S p)) (S (mult (S (S O)) l)))*(pow x (S (mult (S (S O)) l)))*(pow y (S (mult (S (S O)) (minus p l))))`` p)) end.
Replace (Ropp (sum_f_R0
       [k:nat]
          (sum_f_R0
            [p:nat]
             ``(pow ( -1) p)/
             (INR (fact (plus (mult (S (S O)) p) (S O))))*
             (pow x (plus (mult (S (S O)) p) (S O)))*
             ((pow ( -1) (minus k p))/
             (INR (fact (plus (mult (S (S O)) (minus k p)) (S O))))*
             (pow y (plus (mult (S (S O)) (minus k p)) (S O))))`` k)
       n)) with (sum_f_R0 sin_nnn (S n)).
Rewrite <- sum_plus.
Unfold C1.
Apply sum_eq; Intros.
Induction i.
Simpl.
Rewrite Rplus_Ol.
Replace (C O O) with R1.
Unfold Rdiv; Rewrite Rinv_R1.
Ring.
Unfold C.
Rewrite <- minus_n_n.
Simpl.
Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rinv_R1; Ring.
Unfold sin_nnn.
Rewrite <- Rmult_Rplus_distr.
Apply Rmult_mult_r.
Rewrite binome.
Pose Wn := [i0:nat]``(C (mult (S (S O)) (S i)) i0)*(pow x i0)*
         (pow y (minus (mult (S (S O)) (S i)) i0))``.
Replace (sum_f_R0
   [l:nat]
      ``(C (mult (S (S O)) (S i)) (mult (S (S O)) l))*
      (pow x (mult (S (S O)) l))*
      (pow y (mult (S (S O)) (minus (S i) l)))`` (S i)) with (sum_f_R0 [l:nat](Wn (mult (2) l)) (S i)).
Replace (sum_f_R0
     [l:nat]
        ``(C (mult (S (S O)) (S i)) (S (mult (S (S O)) l)))*
        (pow x (S (mult (S (S O)) l)))*
        (pow y (S (mult (S (S O)) (minus i l))))`` i) with (sum_f_R0 [l:nat](Wn (S (mult (2) l))) i).
Rewrite Rplus_sym.
Apply sum_decomposition.
Apply sum_eq; Intros.
Unfold Wn.
Apply Rmult_mult_r.
Replace (minus (mult (2) (S i)) (S (mult (2) i0))) with (S (mult (2) (minus 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 (mult (2) (S i)) with (S (S (mult (2) i))).
Apply le_n_S.
Apply le_trans with (mult (2) i).
Apply mult_le; 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.
Apply Rmult_mult_r.
Replace (minus (mult (2) (S i)) (mult (2) i0)) with (mult (2) (minus (S i) i0)).
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 mult_le; Assumption.
Assumption.
Rewrite <- (Ropp_Ropp (sum_f_R0 sin_nnn (S n))).
Apply eq_Ropp.
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 O) with R0.
Rewrite Rmult_Ol; Rewrite Rplus_Ol.
Replace (pred (S n)) with n; [Idtac | Reflexivity].
Apply sum_eq; Intros.
Rewrite Rmult_sym.
Unfold sin_nnn.
Rewrite scal_sum.
Rewrite scal_sum.
Apply sum_eq; Intros.
Unfold Rdiv.
Repeat Rewrite Rmult_assoc.
Rewrite (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``).
Repeat Rewrite <- Rmult_assoc.
Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``).
Repeat Rewrite <- Rmult_assoc.
Replace ``/(INR (fact (mult (S (S O)) (S i))))*
   (C (mult (S (S O)) (S i)) (S (mult (S (S O)) i0)))`` with ``/(INR (fact (plus (mult (S (S O)) i0) (S O))))*/(INR (fact (plus (mult (S (S O)) (minus i i0)) (S O))))``.
Replace (S (mult (2) i0)) with (plus (mult (2) i0) (1)); [Idtac | Ring].
Replace (S (mult (2) (minus i i0))) with (plus (mult (2) (minus i i0)) (1)); [Idtac | Ring].
Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i0)*(pow (-1) (minus i i0))``.
Ring.
Simpl.
Pattern 2 i; Replace i with (plus i0 (minus i i0)).
Rewrite pow_add.
Ring.
Symmetry; Apply le_plus_minus; Assumption.
Unfold C.
Unfold Rdiv; Repeat Rewrite <- Rmult_assoc.
Rewrite <- Rinv_l_sym.
Rewrite Rmult_1l.
Rewrite Rinv_Rmult.
Replace (S (mult (S (S O)) i0)) with (plus (mult (2) i0) (1)); [Apply Rmult_mult_r | Ring].
Replace (minus (mult (2) (S i)) (plus (mult (2) i0) (1))) with (plus (mult (2) (minus i i0)) (1)).
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 (plus (mult (2) i0) (1)) with (S (mult (2) i0)).
Replace (mult (2) (S i)) with (S (S (mult (2) i))).
Apply le_n_S.
Apply le_trans with (mult (2) i).
Apply mult_le; 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.
Repeat Rewrite <- Rmult_assoc.
Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) i)))``).
Repeat Rewrite <- Rmult_assoc.
Replace ``/(INR (fact (mult (S (S O)) i)))*
   (C (mult (S (S O)) i) (mult (S (S O)) i0))`` with ``/(INR (fact (mult (S (S O)) i0)))*/(INR (fact (mult (S (S O)) (minus i i0))))``.
Replace ``(pow (-1) i)`` with ``(pow (-1) i0)*(pow (-1) (minus i i0))``.
Ring.
Pattern 2 i; Replace i with (plus i0 (minus i i0)).
Rewrite pow_add.
Ring.
Symmetry; Apply le_plus_minus; Assumption.
Unfold C.
Unfold Rdiv; Repeat Rewrite <- Rmult_assoc.
Rewrite <- Rinv_l_sym.
Rewrite Rmult_1l.
Rewrite Rinv_Rmult.
Replace (minus (mult (2) i) (mult (2) i0)) with (mult (2) (minus i i0)).
Reflexivity.
Apply INR_eq.
Rewrite mult_INR; Repeat Rewrite minus_INR.
Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring.
Apply mult_le; Assumption.
Assumption.
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 : (x:R;i:nat) (pow x (mult (2) i))==(pow ``x*x`` i). 
Intros. 
Assert H := (pow_Rsqr x i).
Unfold Rsqr in H; Exact H.
Qed. 
 
Lemma A1_cvg : (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; Unfold R_dist; Intros. 
Elim (p eps H1); Intros. 
Exists x1; Intros. 
Unfold A1. 
Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow (x*x) i)``) n). 
Apply H2; Assumption. 
Apply sum_eq. 
Intros. 
Replace ``(pow (x*x) i)`` with ``(pow x (mult (S (S O)) i))``. 
Reflexivity. 
Apply pow_sqr. 
Unfold cos. 
Case (exist_cos (Rsqr x)). 
Unfold Rsqr; Intros. 
Unfold cos_in in p_i. 
Unfold cos_in in c. 
Apply unicite_sum with [i:nat]``(cos_n i)*(pow (x*x) i)``; Assumption. 
Qed. 
 
Lemma C1_cvg : (x,y:R) (Un_cv (C1 x y) (cos (Rplus 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; Unfold R_dist; Intros. 
Elim (p eps H1); Intros. 
Exists x1; Intros. 
Unfold C1. 
Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow ((x+y)*(x+y)) i)``) n). 
Apply H2; Assumption. 
Apply sum_eq. 
Intros. 
Replace ``(pow ((x+y)*(x+y)) i)`` with ``(pow (x+y) (mult (S (S O)) i))``. 
Reflexivity. 
Apply pow_sqr. 
Unfold cos. 
Case (exist_cos (Rsqr ``x+y``)). 
Unfold Rsqr; Intros. 
Unfold cos_in in p_i. 
Unfold cos_in in c. 
Apply unicite_sum with [i:nat]``(cos_n i)*(pow ((x+y)*(x+y)) i)``; Assumption. 
Qed. 
 
Lemma B1_cvg : (x:R) (Un_cv (B1 x) (sin x)). 
Intro. 
Case (Req_EM x R0); Intro. 
Rewrite H. 
Rewrite sin_0. 
Unfold B1. 
Unfold Un_cv; Unfold R_dist; Intros; Exists O; Intros. 
Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (plus (mult (S (S O)) k) (S O)))``) n) with R0. 
Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. 
Induction n. 
Simpl; Ring. 
Rewrite tech5; Rewrite <- Hrecn. 
Simpl; Ring. 
Unfold ge; 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; Unfold R_dist; Intros. 
Cut ``0<eps/(Rabsolu x)``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]]. 
Elim (p ``eps/(Rabsolu x)`` H3); Intros. 
Exists x1; Intros. 
Unfold B1. 
Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))``) n) with (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)). 
Replace (Rminus (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)) (Rmult x x0)) with (Rmult x (Rminus (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n) x0)); [Idtac | Ring]. 
Rewrite Rabsolu_mult. 
Apply Rlt_monotony_contra with ``/(Rabsolu x)``. 
Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. 
Rewrite <- Rmult_assoc. 
Rewrite <- Rinv_l_sym. 
Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H4; Apply H4; Assumption. 
Apply Rabsolu_no_R0; Assumption. 
Rewrite scal_sum. 
Apply sum_eq. 
Intros. 
Rewrite pow_add. 
Rewrite pow_sqr. 
Simpl. 
Ring. 
Unfold sin. 
Case (exist_sin (Rsqr x)). 
Unfold Rsqr; Intros. 
Unfold sin_in in p_i. 
Unfold sin_in in s. 
Assert H1 := (unicite_sum [i:nat]``(sin_n i)*(pow (x*x) i)`` x0 x1 p_i s). 
Rewrite H1; Reflexivity. 
Qed.