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diff --git a/theories7/Reals/Rtrigo_reg.v b/theories7/Reals/Rtrigo_reg.v new file mode 100644 index 00000000..02e40caf --- /dev/null +++ b/theories7/Reals/Rtrigo_reg.v @@ -0,0 +1,497 @@ +(************************************************************************) +(* 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_reg.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require PSeries_reg. +V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +Lemma CVN_R_cos : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). +Unfold CVN_R; Intros. +Cut (r::R)<>``0``. +Intro hyp_r; Unfold CVN_r. +Apply Specif.existT with [n:nat]``/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))``. +Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (mult (S (S O)) k)))*(pow r (mult (S (S O)) k))``) n) l)). +Intro; Elim X; Intros. +Apply existTT with x. +Split. +Apply p. +Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult. +Rewrite pow_1_abs; Rewrite Rmult_1l. +Cut ``0</(INR (fact (mult (S (S O)) n)))``. +Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). +Apply Rle_monotony. +Left; Apply H1. +Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. +Rewrite Rabsolu_Rabsolu. +Unfold Boule in H0; Rewrite minus_R0 in H0. +Left; Apply H0. +Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Alembert_C2. +Intro; Apply Rabsolu_no_R0. +Apply prod_neq_R0. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Apply pow_nonzero; Assumption. +Assert H0 := Alembert_cos. +Unfold cos_n in H0; Unfold Un_cv in H0; Unfold Un_cv; Intros. +Cut ``0<eps/(Rsqr r)``. +Intro; Elim (H0 ? H2); Intros N0 H3. +Exists N0; Intros. +Unfold R_dist; Assert H5 := (H3 ? H4). +Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (mult (S (S O)) (S n))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (mult (S (S O)) (S n))))/((pow ( -1) n)/(INR (fact (mult (S (S O)) n))))))``. +Apply Rlt_monotony_contra with ``/(Rsqr r)``. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Pattern 1 ``/(Rsqr r)``; Replace ``/(Rsqr r)`` with ``(Rabsolu (/(Rsqr r)))``. +Rewrite <- Rabsolu_mult; Rewrite Rminus_distr; Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Apply H5. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_right. +Reflexivity. +Apply Rle_sym1; Apply pos_Rsqr. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Rewrite (Rmult_sym (Rsqr r)); Unfold Rdiv; Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs; Rewrite Rmult_1l; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_mult; Rewrite (pow_1_abs n); Rewrite Rmult_1l; Rewrite <- Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite Rinv_Rmult. +Rewrite Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite (Rmult_sym ``(Rabsolu (Rabsolu (pow r (mult (S (S O)) (S n)))))``); Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Do 2 Rewrite Rabsolu_Rabsolu; Repeat Rewrite Rabsolu_right. +Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. +Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Unfold Rsqr; Ring. +Apply pow_nonzero; Assumption. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply Rabsolu_no_R0; Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply INR_fact_neq_0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply H1. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Qed. + +(**********) +Lemma continuity_cos : (continuity cos). +Pose fn := [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``. +Cut (CVN_R fn). +Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). +Intro cv; Cut ((n:nat)(continuity (fn n))). +Intro; Cut (x:R)(cos x)==(SFL fn cv x). +Intro; Cut (continuity (SFL fn cv))->(continuity cos). +Intro; Apply H1. +Apply SFL_continuity; Assumption. +Unfold continuity; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Elim (H1 x ? H2); Intros. +Exists x0; Intros. +Elim H3; Intros. +Split. +Apply H4. +Intros; Rewrite (H0 x); Rewrite (H0 x1); Apply H5; Apply H6. +Intro; Unfold cos SFL. +Case (cv x); Case (exist_cos (Rsqr x)); Intros. +Symmetry; EApply UL_sequence. +Apply u. +Unfold cos_in in c; Unfold infinit_sum in c; Unfold Un_cv; Intros. +Elim (c ? H0); Intros N0 H1. +Exists N0; Intros. +Unfold R_dist in H1; Unfold R_dist SP. +Replace (sum_f_R0 [k:nat](fn k x) n) with (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr x) i)`` n). +Apply H1; Assumption. +Apply sum_eq; Intros. +Unfold cos_n fn; Apply Rmult_mult_r. +Unfold Rsqr; Rewrite pow_sqr; Reflexivity. +Intro; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. +Apply continuity_mult. +Apply derivable_continuous; Apply derivable_const. +Apply derivable_continuous; Apply (derivable_pow (mult (2) n)). +Apply CVN_R_CVS; Apply X. +Apply CVN_R_cos; Unfold fn; Reflexivity. +Qed. + +(**********) +Lemma continuity_sin : (continuity sin). +Unfold continuity; Intro. +Assert H0 := (continuity_cos ``PI/2-x``). +Unfold continuity_pt in H0; Unfold continue_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0; Simpl in H0; Unfold R_dist in H0; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Elim (H0 ? H); Intros. +Exists x0; Intros. +Elim H1; Intros. +Split. +Assumption. +Intros; Rewrite <- (cos_shift x); Rewrite <- (cos_shift x1); Apply H3. +Elim H4; Intros. +Split. +Unfold D_x no_cond; Split. +Trivial. +Red; Intro; Unfold D_x no_cond in H5; Elim H5; Intros _ H8; Elim H8; Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp x1); Apply eq_Ropp; Apply r_Rplus_plus with ``PI/2``; Apply H7. +Replace ``PI/2-x1-(PI/2-x)`` with ``x-x1``; [Idtac | Ring]; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply H6. +Qed. + +Lemma CVN_R_sin : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). +Unfold CVN_R; Unfold CVN_r; Intros fn H r. +Apply Specif.existT with [n:nat]``/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))``. +Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow r (mult (S (S O)) k))``) n) l)). +Intro; Elim X; Intros. +Apply existTT with x. +Split. +Apply p. +Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult; Rewrite pow_1_abs; Rewrite Rmult_1l. +Cut ``0</(INR (fact (plus (mult (S (S O)) n) (S O))))``. +Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). +Apply Rle_monotony. +Left; Apply H1. +Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. +Rewrite Rabsolu_Rabsolu; Unfold Boule in H0; Rewrite minus_R0 in H0; Left; Apply H0. +Apply Rlt_Rinv; Apply INR_fact_lt_0. +Cut (r::R)<>``0``. +Intro; Apply Alembert_C2. +Intro; Apply Rabsolu_no_R0. +Apply prod_neq_R0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply pow_nonzero; Assumption. +Assert H1 := Alembert_sin. +Unfold sin_n in H1; Unfold Un_cv in H1; Unfold Un_cv; Intros. +Cut ``0<eps/(Rsqr r)``. +Intro; Elim (H1 ? H3); Intros N0 H4. +Exists N0; Intros. +Unfold R_dist; Assert H6 := (H4 ? H5). +Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))/((pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O)))))))``. +Apply Rlt_monotony_contra with ``/(Rsqr r)``. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Pattern 1 ``/(Rsqr r)``; Rewrite <- (Rabsolu_right ``/(Rsqr r)``). +Rewrite <- Rabsolu_mult. +Rewrite Rminus_distr. +Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps). +Apply H6. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Unfold Rdiv; Rewrite (Rmult_sym (Rsqr r)); Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs. +Rewrite Rmult_1l. +Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rinv. +Rewrite pow_1_abs; Rewrite Rinv_R1; Rewrite Rmult_1l. +Rewrite Rinv_Rmult. +Rewrite <- Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite Rabsolu_mult. +Do 2 Rewrite Rabsolu_Rabsolu. +Rewrite (Rmult_sym ``(Rabsolu (pow r (mult (S (S O)) (S n))))``). +Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_Rabsolu. +Repeat Rewrite Rabsolu_right. +Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. +Do 2 Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Unfold Rsqr; Ring. +Apply pow_nonzero; Assumption. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply INR_fact_neq_0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption]. +Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Qed. + +(* (sin h)/h -> 1 when h -> 0 *) +Lemma derivable_pt_lim_sin_0 : (derivable_pt_lim sin R0 R1). +Unfold derivable_pt_lim; Intros. +Pose fn := [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``. +Cut (CVN_R fn). +Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). +Intro cv. +Pose r := (mkposreal ? Rlt_R0_R1). +Cut (CVN_r fn r). +Intro; Cut ((n:nat; y:R)(Boule ``0`` r y)->(continuity_pt (fn n) y)). +Intro; Cut (Boule R0 r R0). +Intro; Assert H2 := (SFL_continuity_pt ? cv ? X0 H0 ? H1). +Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. +Elim (H2 ? H); Intros alp H3. +Elim H3; Intros. +Exists (mkposreal ? H4). +Simpl; Intros. +Rewrite sin_0; Rewrite Rplus_Ol; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Cut ``(Rabsolu ((SFL fn cv h)-(SFL fn cv 0))) < eps``. +Intro; Cut (SFL fn cv R0)==R1. +Intro; Cut (SFL fn cv h)==``(sin h)/h``. +Intro; Rewrite H9 in H8; Rewrite H10 in H8. +Apply H8. +Unfold SFL sin. +Case (cv h); Intros. +Case (exist_sin (Rsqr h)); Intros. +Unfold Rdiv; Rewrite (Rinv_r_simpl_m h x0 H6). +EApply UL_sequence. +Apply u. +Unfold sin_in in s; Unfold sin_n infinit_sum in s; Unfold SP fn Un_cv; Intros. +Elim (s ? H10); Intros N0 H11. +Exists N0; Intros. +Unfold R_dist; Unfold R_dist in H11. +Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow h (mult (S (S O)) k))`` n) with (sum_f_R0 [i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (Rsqr h) i)`` n). +Apply H11; Assumption. +Apply sum_eq; Intros; Apply Rmult_mult_r; Unfold Rsqr; Rewrite pow_sqr; Reflexivity. +Unfold SFL sin. +Case (cv R0); Intros. +EApply UL_sequence. +Apply u. +Unfold SP fn; Unfold Un_cv; Intros; Exists (S O); Intros. +Unfold R_dist; Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (mult (S (S O)) k))`` n) with R1. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Rewrite decomp_sum. +Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite Rinv_R1; Rewrite Rmult_1r; Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rplus_plus_r. +Symmetry; Apply sum_eq_R0; Intros. +Rewrite Rmult_Ol; Rewrite Rmult_Or; Reflexivity. +Unfold ge in H10; Apply lt_le_trans with (1); [Apply lt_n_Sn | Apply H10]. +Apply H5. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply not_sym; Apply H6. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply H7. +Unfold Boule; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_R0; Apply (cond_pos r). +Intros; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. +Apply continuity_pt_mult. +Apply derivable_continuous_pt. +Apply derivable_pt_const. +Apply derivable_continuous_pt. +Apply (derivable_pt_pow (mult (2) n) y). +Apply (X r). +Apply (CVN_R_CVS ? X). +Apply CVN_R_sin; Unfold fn; Reflexivity. +Qed. + +(* ((cos h)-1)/h -> 0 when h -> 0 *) +Lemma derivable_pt_lim_cos_0 : (derivable_pt_lim cos ``0`` ``0``). +Unfold derivable_pt_lim; Intros. +Assert H0 := derivable_pt_lim_sin_0. +Unfold derivable_pt_lim in H0. +Cut ``0<eps/2``. +Intro; Elim (H0 ? H1); Intros del H2. +Cut (continuity_pt sin ``0``). +Intro; Unfold continuity_pt in H3; Unfold continue_in in H3; Unfold limit1_in in H3; Unfold limit_in in H3; Simpl in H3; Unfold R_dist in H3. +Cut ``0<eps/2``; [Intro | Assumption]. +Elim (H3 ? H4); Intros del_c H5. +Cut ``0<(Rmin del del_c)``. +Intro; Pose delta := (mkposreal ? H6). +Exists delta; Intros. +Rewrite Rplus_Ol; Replace ``((cos h)-(cos 0))`` with ``-2*(Rsqr (sin (h/2)))``. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Unfold Rdiv; Do 2 Rewrite Ropp_mul1. +Rewrite Rabsolu_Ropp. +Replace ``2*(Rsqr (sin (h*/2)))*/h`` with ``(sin (h/2))*((sin (h/2))/(h/2)-1)+(sin (h/2))``. +Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))*((sin (h/2))/(h/2)-1)))+(Rabsolu ((sin (h/2))))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))/(h/2)-1))``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin (h/2))/(h/2)-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H9 := (SIN_bound ``h/2``). +Unfold Rabsolu; Case (case_Rabsolu ``(sin (h/2))``); Intro. +Pattern 3 R1; Rewrite <- (Ropp_Ropp ``1``). +Apply Rle_Ropp1. +Elim H9; Intros; Assumption. +Elim H9; Intros; Assumption. +Cut ``(Rabsolu (h/2))<del``. +Intro; Cut ``h/2<>0``. +Intro; Assert H11 := (H2 ? H10 H9). +Rewrite Rplus_Ol in H11; Rewrite sin_0 in H11. +Rewrite minus_R0 in H11; Apply H11. +Unfold Rdiv; Apply prod_neq_R0. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Apply Rlt_trans with ``del/2``. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right ``/2``). +Do 2 Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_le_trans with (pos delta). +Apply H8. +Unfold delta; Simpl; Apply Rmin_l. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. +Rewrite <- (Rplus_Or ``del/2``); Pattern 1 del; Rewrite (double_var del); Apply Rlt_compatibility; Unfold Rdiv; Apply Rmult_lt_pos. +Apply (cond_pos del). +Apply Rlt_Rinv; Sup0. +Elim H5; Intros; Assert H11 := (H10 ``h/2``). +Rewrite sin_0 in H11; Do 2 Rewrite minus_R0 in H11. +Apply H11. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply not_sym; Unfold Rdiv; Apply prod_neq_R0. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Apply Rlt_trans with ``del_c/2``. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right ``/2``). +Do 2 Rewrite <- (Rmult_sym ``/2``). +Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_le_trans with (pos delta). +Apply H8. +Unfold delta; Simpl; Apply Rmin_r. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. +Rewrite <- (Rplus_Or ``del_c/2``); Pattern 2 del_c; Rewrite (double_var del_c); Apply Rlt_compatibility. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply H9. +Apply Rlt_Rinv; Sup0. +Rewrite Rminus_distr; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite (Rmult_sym ``2``); Unfold Rdiv Rsqr. +Repeat Rewrite Rmult_assoc. +Repeat Apply Rmult_mult_r. +Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Apply Rmult_sym. +DiscrR. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Pattern 2 h; Replace h with ``2*(h/2)``. +Rewrite (cos_2a_sin ``h/2``). +Rewrite cos_0; Unfold Rsqr; Ring. +Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +DiscrR. +Unfold Rmin; Case (total_order_Rle del del_c); Intro. +Apply (cond_pos del). +Elim H5; Intros; Assumption. +Apply continuity_sin. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Qed. + +(**********) +Theorem derivable_pt_lim_sin : (x:R)(derivable_pt_lim sin x (cos x)). +Intro; Assert H0 := derivable_pt_lim_sin_0. +Assert H := derivable_pt_lim_cos_0. +Unfold derivable_pt_lim in H0 H. +Unfold derivable_pt_lim; Intros. +Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Apply H1 | Apply Rlt_Rinv; Sup0]]. +Elim (H0 ? H2); Intros alp1 H3. +Elim (H ? H2); Intros alp2 H4. +Pose alp := (Rmin alp1 alp2). +Cut ``0<alp``. +Intro; Exists (mkposreal ? H5); Intros. +Replace ``((sin (x+h))-(sin x))/h-(cos x)`` with ``(sin x)*((cos h)-1)/h+(cos x)*((sin h)/h-1)``. +Apply Rle_lt_trans with ``(Rabsolu ((sin x)*((cos h)-1)/h))+(Rabsolu ((cos x)*((sin h)/h-1)))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply Rle_lt_trans with ``(Rabsolu ((cos h)-1)/h)``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu (((cos h)-1)/h))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H8 := (SIN_bound x); Elim H8; Intros. +Unfold Rabsolu; Case (case_Rabsolu (sin x)); Intro. +Rewrite <- (Ropp_Ropp R1). +Apply Rle_Ropp1; Assumption. +Assumption. +Cut ``(Rabsolu h)<alp2``. +Intro; Assert H9 := (H4 ? H6 H8). +Rewrite cos_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. +Apply Rlt_le_trans with alp. +Apply H7. +Unfold alp; Apply Rmin_r. +Apply Rle_lt_trans with ``(Rabsolu ((sin h)/h-1))``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin h)/h-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H8 := (COS_bound x); Elim H8; Intros. +Unfold Rabsolu; Case (case_Rabsolu (cos x)); Intro. +Rewrite <- (Ropp_Ropp R1); Apply Rle_Ropp1; Assumption. +Assumption. +Cut ``(Rabsolu h)<alp1``. +Intro; Assert H9 := (H3 ? H6 H8). +Rewrite sin_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. +Apply Rlt_le_trans with alp. +Apply H7. +Unfold alp; Apply Rmin_l. +Rewrite sin_plus; Unfold Rminus Rdiv; Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; Repeat Rewrite Rmult_assoc; Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Rewrite (Rplus_sym ``(sin x)*( -1*/h)``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Rewrite Ropp_mul3; Rewrite Ropp_mul1; Rewrite Rmult_1r; Rewrite Rmult_1l; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Apply Rplus_sym. +Unfold alp; Unfold Rmin; Case (total_order_Rle alp1 alp2); Intro. +Apply (cond_pos alp1). +Apply (cond_pos alp2). +Qed. + +Lemma derivable_pt_lim_cos : (x:R) (derivable_pt_lim cos x ``-(sin x)``). +Intro; Cut (h:R)``(sin (h+PI/2))``==(cos h). +Intro; Replace ``-(sin x)`` with (Rmult (cos ``x+PI/2``) (Rplus R1 R0)). +Generalize (derivable_pt_lim_comp (plus_fct id (fct_cte ``PI/2``)) sin); Intros. +Cut (derivable_pt_lim (plus_fct id (fct_cte ``PI/2``)) x ``1+0``). +Cut (derivable_pt_lim sin (plus_fct id (fct_cte ``PI/2``) x) ``(cos (x+PI/2))``). +Intros; Generalize (H0 ? ? ? H2 H1); Replace (comp sin (plus_fct id (fct_cte ``PI/2``))) with [x:R]``(sin (x+PI/2))``; [Idtac | Reflexivity]. +Unfold derivable_pt_lim; Intros. +Elim (H3 eps H4); Intros. +Exists x0. +Intros; Rewrite <- (H ``x+h``); Rewrite <- (H x); Apply H5; Assumption. +Apply derivable_pt_lim_sin. +Apply derivable_pt_lim_plus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Rewrite sin_cos; Rewrite <- (Rplus_sym x); Ring. +Intro; Rewrite cos_sin; Rewrite Rplus_sym; Reflexivity. +Qed. + +Lemma derivable_pt_sin : (x:R) (derivable_pt sin x). +Unfold derivable_pt; Intro. +Apply Specif.existT with (cos x). +Apply derivable_pt_lim_sin. +Qed. + +Lemma derivable_pt_cos : (x:R) (derivable_pt cos x). +Unfold derivable_pt; Intro. +Apply Specif.existT with ``-(sin x)``. +Apply derivable_pt_lim_cos. +Qed. + +Lemma derivable_sin : (derivable sin). +Unfold derivable; Intro; Apply derivable_pt_sin. +Qed. + +Lemma derivable_cos : (derivable cos). +Unfold derivable; Intro; Apply derivable_pt_cos. +Qed. + +Lemma derive_pt_sin : (x:R) ``(derive_pt sin x (derivable_pt_sin ?))==(cos x)``. +Intros; Apply derive_pt_eq_0. +Apply derivable_pt_lim_sin. +Qed. + +Lemma derive_pt_cos : (x:R) ``(derive_pt cos x (derivable_pt_cos ?))==-(sin x)``. +Intros; Apply derive_pt_eq_0. +Apply derivable_pt_lim_cos. +Qed. |