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Diffstat (limited to 'theories7/Reals/Rtrigo_reg.v')
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diff --git a/theories7/Reals/Rtrigo_reg.v b/theories7/Reals/Rtrigo_reg.v deleted file mode 100644 index 02e40caf..00000000 --- a/theories7/Reals/Rtrigo_reg.v +++ /dev/null @@ -1,497 +0,0 @@ -(************************************************************************) -(* 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. |