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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 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*)
Require Rbase.
Require Rfunctions.
Require SeqSeries.
Require Rtrigo_def.
V7only [ Import nat_scope. Import Z_scope. Import R_scope. ].
Open Local Scope R_scope.
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 binomial.
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 unicity_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 unicity_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 := (unicity_sum [i:nat]``(sin_n i)*(pow (x*x) i)`` x0 x1 p_i s).
Rewrite H1; Reflexivity.
Qed.
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