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Diffstat (limited to 'theories7/Reals/Rtrigo_alt.v')
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diff --git a/theories7/Reals/Rtrigo_alt.v b/theories7/Reals/Rtrigo_alt.v new file mode 100644 index 00000000..db0e2fea --- /dev/null +++ b/theories7/Reals/Rtrigo_alt.v @@ -0,0 +1,294 @@ +(************************************************************************) +(* 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_alt.v,v 1.1.2.1 2004/07/16 19:31:36 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. + +(*****************************************************************) +(* Using series definitions of cos and sin *) +(*****************************************************************) + +Definition sin_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``. + +Definition cos_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (mult (S (S O)) i))/(INR (fact (mult (S (S O)) i)))``. + +Definition sin_approx [a:R;n:nat] : R := (sum_f_R0 (sin_term a) n). + +Definition cos_approx [a:R;n:nat] : R := (sum_f_R0 (cos_term a) n). + +(**********) +Lemma PI_4 : ``PI<=4``. +Assert H0 := (PI_ineq O). +Elim H0; Clear H0; Intros _ H0. +Unfold tg_alt PI_tg in H0; Simpl in H0. +Rewrite Rinv_R1 in H0; Rewrite Rmult_1r in H0; Unfold Rdiv in H0. +Apply Rle_monotony_contra with ``/4``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rinv_l_sym; [Rewrite Rmult_sym; Assumption | DiscrR]. +Qed. + +(**********) +Theorem sin_bound : (a:R; n:nat) ``0 <= a``->``a <= PI``->``(sin_approx a (plus (mult (S (S O)) n) (S O))) <= (sin a)<= (sin_approx a (mult (S (S O)) (plus n (S O))))``. +Intros; Case (Req_EM a R0); Intro Hyp_a. +Rewrite Hyp_a; Rewrite sin_0; Split; Right; Unfold sin_approx; Apply sum_eq_R0 Orelse (Symmetry; Apply sum_eq_R0); Intros; Unfold sin_term; Rewrite pow_add; Simpl; Unfold Rdiv; Rewrite Rmult_Ol; Ring. +Unfold sin_approx; Cut ``0<a``. +Intro Hyp_a_pos. +Rewrite (decomp_sum (sin_term a) (plus (mult (S (S O)) n) (S O))). +Rewrite (decomp_sum (sin_term a) (mult (S (S O)) (plus n (S O)))). +Replace (sin_term a O) with a. +Cut (Rle (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O)))) ``(sin a)-a``)/\(Rle ``(sin a)-a`` (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O)))))) -> (Rle (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O))))) (sin a))/\(Rle (sin a) (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O))))))). +Intro; Apply H1. +Pose Un := [n:nat]``(pow a (plus (mult (S (S O)) (S n)) (S O)))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))``. +Replace (pred (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) n). +Replace (pred (mult (S (S O)) (plus n (S O)))) with (S (mult (S (S O)) n)). +Replace (sum_f_R0 [i:nat](sin_term a (S i)) (mult (S (S O)) n)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``. +Replace (sum_f_R0 [i:nat](sin_term a (S i)) (S (mult (S (S O)) n))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. +Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))<=a-(sin a)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n)) <= (sin a)-a <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. +Intro; Apply H2. +Apply alternated_series_ineq. +Unfold Un_decreasing Un; Intro; Cut (plus (mult (S (S O)) (S (S n0))) (S O))=(S (S (plus (mult (S (S O)) (S n0)) (S O)))). +Intro; Rewrite H3. +Replace ``(pow a (S (S (plus (mult (S (S O)) (S n0)) (S O)))))`` with ``(pow a (plus (mult (S (S O)) (S n0)) (S O)))*(a*a)``. +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply pow_lt; Assumption. +Apply Rle_monotony_contra with ``(INR (fact (S (S (plus (mult (S (S O)) (S n0)) (S O))))))``. +Rewrite <- H3; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H5 := (sym_eq ? ? ? H4); Elim (fact_neq_0 ? H5). +Rewrite <- H3; Rewrite (Rmult_sym ``(INR (fact (plus (mult (S (S O)) (S (S n0))) (S O))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite H3; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n0)+1)+(0+1)+1+1)*((0+1+1)*((INR n0)+1)+(0+1)+1)`` with ``4*(INR n0)*(INR n0)+18*(INR n0)+20``; [Idtac | Ring]. +Apply Rle_trans with ``20``. +Apply Rle_trans with ``16``. +Replace ``16`` with ``(Rsqr 4)``; [Idtac | SqRing]. +Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]. +Apply Rsqr_incr_1. +Apply Rle_trans with PI; [Assumption | Apply PI_4]. +Assumption. +Left; Sup0. +Rewrite <- (Rplus_Or ``16``); Replace ``20`` with ``16+4``; [Apply Rle_compatibility; Left; Sup0 | Ring]. +Rewrite <- (Rplus_sym ``20``); Pattern 1 ``20``; Rewrite <- Rplus_Or; Apply Rle_compatibility. +Apply ge0_plus_ge0_is_ge0. +Repeat Apply Rmult_le_pos. +Left; Sup0. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply Rmult_le_pos. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assert H3 := (cv_speed_pow_fact a); Unfold Un; Unfold Un_cv in H3; Unfold R_dist in H3; Unfold Un_cv; Unfold R_dist; Intros; Elim (H3 eps H4); Intros N H5. +Exists N; Intros; Apply H5. +Replace (plus (mult (2) (S n0)) (1)) with (S (mult (2) (S n0))). +Unfold ge; Apply le_trans with (mult (2) (S n0)). +Apply le_trans with (mult (2) (S N)). +Apply le_trans with (mult (2) N). +Apply le_n_2n. +Apply mult_le; Apply le_n_Sn. +Apply mult_le; Apply le_n_S; Assumption. +Apply le_n_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Reflexivity. +Assert X := (exist_sin (Rsqr a)); Elim X; Intros. +Cut ``x==(sin a)/a``. +Intro; Rewrite H3 in p; Unfold sin_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/(Rabsolu a)``. +Intro; Elim (p ? H5); Intros N H6. +Exists N; Intros. +Replace (sum_f_R0 (tg_alt Un) n0) with (Rmult a (Rminus R1 (sum_f_R0 [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)))). +Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym a); Rewrite (Rplus_sym ``-a``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rlt_monotony_contra with ``/(Rabsolu a)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Pattern 1 ``/(Rabsolu a)``; Rewrite <- (Rabsolu_Rinv a Hyp_a). +Rewrite <- Rabsolu_mult; Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1l | Assumption]; Rewrite (Rmult_sym ``/a``); Rewrite (Rmult_sym ``/(Rabsolu a)``); Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus Rdiv in H6; Apply H6; Unfold ge; Apply le_trans with n0; [Exact H7 | Apply le_n_Sn]. +Rewrite (decomp_sum [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)). +Replace (sin_n O) with R1. +Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Rewrite scal_sum; Apply sum_eq. +Intros; Unfold sin_n Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. +Replace ``(pow a (plus (mult (S (S O)) (S i)) (S O)))`` with ``(Rsqr a)*(pow (Rsqr a) i)*a``. +Unfold Rdiv; Ring. +Rewrite pow_add; Rewrite pow_Rsqr; Simpl; Ring. +Simpl; Ring. +Unfold sin_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_O_Sn. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Unfold sin; Case (exist_sin (Rsqr a)). +Intros; Cut x==x0. +Intro; Rewrite H3; Unfold Rdiv. +Symmetry; Apply Rinv_r_simpl_m; Assumption. +Unfold sin_in in p; Unfold sin_in in s; EApply unicity_sum. +Apply p. +Apply s. +Intros; Elim H2; Intros. +Replace ``(sin a)-a`` with ``-(a-(sin a))``; [Idtac | Ring]. +Split; Apply Rle_Ropp1; Assumption. +Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros; Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros. +Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). +Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intro; Elim H1; Intros. +Split. +Apply Rle_anti_compatibility with ``-a``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H2. +Apply Rle_anti_compatibility with ``-a``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H3. +Unfold sin_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. +Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). +Apply lt_O_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). +Apply lt_O_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Inversion H; [Assumption | Elim Hyp_a; Symmetry; Assumption]. +Qed. + +(**********) +Lemma cos_bound : (a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``. +Cut ((a:R; n:nat) ``0 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``) -> ((a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``). +Intros H a n; Apply H. +Intros; Unfold cos_approx. +Rewrite (decomp_sum (cos_term a0) (plus (mult (S (S O)) n0) (S O))). +Rewrite (decomp_sum (cos_term a0) (mult (S (S O)) (plus n0 (S O)))). +Replace (cos_term a0 O) with R1. +Cut (Rle (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O)))) ``(cos a0)-1``)/\(Rle ``(cos a0)-1`` (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O)))))) -> (Rle (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O))))) (cos a0))/\(Rle (cos a0) (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O))))))). +Intro; Apply H2. +Pose Un := [n:nat]``(pow a0 (mult (S (S O)) (S n)))/(INR (fact (mult (S (S O)) (S n))))``. +Replace (pred (plus (mult (S (S O)) n0) (S O))) with (mult (S (S O)) n0). +Replace (pred (mult (S (S O)) (plus n0 (S O)))) with (S (mult (S (S O)) n0)). +Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (mult (S (S O)) n0)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``. +Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (S (mult (S (S O)) n0))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. +Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))<=1-(cos a0)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0)) <= (cos a0)-1 <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. +Intro; Apply H3. +Apply alternated_series_ineq. +Unfold Un_decreasing; Intro; Unfold Un. +Cut (mult (S (S O)) (S (S n1)))=(S (S (mult (S (S O)) (S n1)))). +Intro; Rewrite H4; Replace ``(pow a0 (S (S (mult (S (S O)) (S n1)))))`` with ``(pow a0 (mult (S (S O)) (S n1)))*(a0*a0)``. +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le; Assumption. +Apply Rle_monotony_contra with ``(INR (fact (S (S (mult (S (S O)) (S n1))))))``. +Rewrite <- H4; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H6 := (sym_eq ? ? ? H5); Elim (fact_neq_0 ? H6). +Rewrite <- H4; Rewrite (Rmult_sym ``(INR (fact (mult (S (S O)) (S (S n1)))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite H4; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Do 2 Rewrite S_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n1)+1)+1+1)*((0+1+1)*((INR n1)+1)+1)`` with ``4*(INR n1)*(INR n1)+14*(INR n1)+12``; [Idtac | Ring]. +Apply Rle_trans with ``12``. +Apply Rle_trans with ``4``. +Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]. +Replace ``a0*a0`` with (Rsqr a0); [Idtac | Reflexivity]. +Apply Rsqr_incr_1. +Apply Rle_trans with ``PI/2``. +Assumption. +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. +Replace ``2*2`` with ``4``; [Apply PI_4 | Ring]. +DiscrR. +Assumption. +Left; Sup0. +Pattern 1 ``4``; Rewrite <- Rplus_Or; Replace ``12`` with ``4+8``; [Apply Rle_compatibility; Left; Sup0 | Ring]. +Rewrite <- (Rplus_sym ``12``); Pattern 1 ``12``; Rewrite <- Rplus_Or; Apply Rle_compatibility. +Apply ge0_plus_ge0_is_ge0. +Repeat Apply Rmult_le_pos. +Left; Sup0. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply Rmult_le_pos. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assert H4 := (cv_speed_pow_fact a0); Unfold Un; Unfold Un_cv in H4; Unfold R_dist in H4; Unfold Un_cv; Unfold R_dist; Intros; Elim (H4 eps H5); Intros N H6; Exists N; Intros. +Apply H6; Unfold ge; Apply le_trans with (mult (2) (S N)). +Apply le_trans with (mult (2) N). +Apply le_n_2n. +Apply mult_le; Apply le_n_Sn. +Apply mult_le; Apply le_n_S; Assumption. +Assert X := (exist_cos (Rsqr a0)); Elim X; Intros. +Cut ``x==(cos a0)``. +Intro; Rewrite H4 in p; Unfold cos_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. +Elim (p ? H5); Intros N H6. +Exists N; Intros. +Replace (sum_f_R0 (tg_alt Un) n1) with (Rminus R1 (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1))). +Unfold Rminus; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym R1); Rewrite (Rplus_sym ``-1``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus in H6; Apply H6. +Unfold ge; Apply le_trans with n1. +Exact H7. +Apply le_n_Sn. +Rewrite (decomp_sum [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1)). +Replace (cos_n O) with R1. +Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Replace (Ropp (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)) with (Rmult ``-1`` (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)); [Idtac | Ring]; Rewrite scal_sum; Apply sum_eq; Intros; Unfold cos_n Un tg_alt. +Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. +Replace ``(pow a0 (mult (S (S O)) (S i)))`` with ``(Rsqr a0)*(pow (Rsqr a0) i)``. +Unfold Rdiv; Ring. +Rewrite pow_Rsqr; Reflexivity. +Simpl; Ring. +Unfold cos_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_O_Sn. +Unfold cos; Case (exist_cos (Rsqr a0)); Intros; Unfold cos_in in p; Unfold cos_in in c; EApply unicity_sum. +Apply p. +Apply c. +Intros; Elim H3; Intros; Replace ``(cos a0)-1`` with ``-(1-(cos a0))``; [Idtac | Ring]. +Split; Apply Rle_Ropp1; Assumption. +Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``; [Rewrite scal_sum | Ring]; Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). +Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intro; Elim H2; Intros; Split. +Apply Rle_anti_compatibility with ``-1``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H3. +Apply Rle_anti_compatibility with ``-1``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H4. +Unfold cos_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. +Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). +Apply lt_O_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). +Apply lt_O_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intros; Case (total_order_T R0 a); Intro. +Elim s; Intro. +Apply H; [Left; Assumption | Assumption]. +Apply H; [Right; Assumption | Assumption]. +Cut ``0< -a``. +Intro; Cut (x:R;n:nat) (cos_approx x n)==(cos_approx ``-x`` n). +Intro; Rewrite H3; Rewrite (H3 a (mult (S (S O)) (plus n (S O)))); Rewrite cos_sym; Apply H. +Left; Assumption. +Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rle_Ropp1; Unfold Rdiv; Unfold Rdiv in H0; Rewrite <- Ropp_mul1; Exact H0. +Intros; Unfold cos_approx; Apply sum_eq; Intros; Unfold cos_term; Do 2 Rewrite pow_Rsqr; Rewrite Rsqr_neg; Unfold Rdiv; Reflexivity. +Apply Rgt_RO_Ropp; Assumption. +Qed. |