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Diffstat (limited to 'theories7/Reals/Ranalysis4.v')
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diff --git a/theories7/Reals/Ranalysis4.v b/theories7/Reals/Ranalysis4.v deleted file mode 100644 index 061854dc..00000000 --- a/theories7/Reals/Ranalysis4.v +++ /dev/null @@ -1,313 +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: Ranalysis4.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) - -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis1. -Require Ranalysis3. -Require Exp_prop. -V7only [Import R_scope.]. Open Local Scope R_scope. - -(**********) -Lemma derivable_pt_inv : (f:R->R;x:R) ``(f x)<>0`` -> (derivable_pt f x) -> (derivable_pt (inv_fct f) x). -Intros; Cut (derivable_pt (div_fct (fct_cte R1) f) x) -> (derivable_pt (inv_fct f) x). -Intro; Apply X0. -Apply derivable_pt_div. -Apply derivable_pt_const. -Assumption. -Assumption. -Unfold div_fct inv_fct fct_cte; Intro; Elim X0; Intros; Unfold derivable_pt; Apply Specif.existT with x0; Unfold derivable_pt_abs; Unfold derivable_pt_lim; Unfold derivable_pt_abs in p; Unfold derivable_pt_lim in p; Intros; Elim (p eps H0); Intros; Exists x1; Intros; Unfold Rdiv in H1; Unfold Rdiv; Rewrite <- (Rmult_1l ``/(f x)``); Rewrite <- (Rmult_1l ``/(f (x+h))``). -Apply H1; Assumption. -Qed. - -(**********) -Lemma pr_nu_var : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) f==g -> (derive_pt f x pr1) == (derive_pt g x pr2). -Unfold derivable_pt derive_pt; Intros. -Elim pr1; Intros. -Elim pr2; Intros. -Simpl. -Rewrite H in p. -Apply unicite_limite with g x; Assumption. -Qed. - -(**********) -Lemma pr_nu_var2 : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) ((h:R)(f h)==(g h)) -> (derive_pt f x pr1) == (derive_pt g x pr2). -Unfold derivable_pt derive_pt; Intros. -Elim pr1; Intros. -Elim pr2; Intros. -Simpl. -Assert H0 := (unicite_step2 ? ? ? p). -Assert H1 := (unicite_step2 ? ? ? p0). -Cut (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h <> 0`` x1 ``0``). -Intro; Assert H3 := (unicite_step1 ? ? ? ? H0 H2). -Assumption. -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Unfold limit1_in in H1; Unfold limit_in in H1; Unfold dist in H1; Simpl in H1; Unfold R_dist in H1. -Intros; Elim (H1 eps H2); Intros. -Elim H3; Intros. -Exists x2. -Split. -Assumption. -Intros; Do 2 Rewrite H; Apply H5; Assumption. -Qed. - -(**********) -Lemma derivable_inv : (f:R->R) ((x:R)``(f x)<>0``)->(derivable f)->(derivable (inv_fct f)). -Intros. -Unfold derivable; Intro. -Apply derivable_pt_inv. -Apply (H x). -Apply (X x). -Qed. - -Lemma derive_pt_inv : (f:R->R;x:R;pr:(derivable_pt f x);na:``(f x)<>0``) (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) == ``-(derive_pt f x pr)/(Rsqr (f x))``. -Intros; Replace (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) with (derive_pt (div_fct (fct_cte R1) f) x (derivable_pt_div (fct_cte R1) f x (derivable_pt_const R1 x) pr na)). -Rewrite derive_pt_div; Rewrite derive_pt_const; Unfold fct_cte; Rewrite Rmult_Ol; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_Ol; Reflexivity. -Apply pr_nu_var2. -Intro; Unfold div_fct fct_cte inv_fct. -Unfold Rdiv; Ring. -Qed. - -(* Rabsolu *) -Lemma Rabsolu_derive_1 : (x:R) ``0<x`` -> (derivable_pt_lim Rabsolu x ``1``). -Intros. -Unfold derivable_pt_lim; Intros. -Exists (mkposreal x H); Intros. -Rewrite (Rabsolu_right x). -Rewrite (Rabsolu_right ``x+h``). -Rewrite Rplus_sym. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r. -Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. -Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H0. -Apply H1. -Apply Rle_sym1. -Case (case_Rabsolu h); Intro. -Rewrite (Rabsolu_left h r) in H2. -Left; Rewrite Rplus_sym; Apply Rlt_anti_compatibility with ``-h``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H2. -Apply ge0_plus_ge0_is_ge0. -Left; Apply H. -Apply Rle_sym2; Apply r. -Left; Apply H. -Qed. - -Lemma Rabsolu_derive_2 : (x:R) ``x<0`` -> (derivable_pt_lim Rabsolu x ``-1``). -Intros. -Unfold derivable_pt_lim; Intros. -Cut ``0< -x``. -Intro; Exists (mkposreal ``-x`` H1); Intros. -Rewrite (Rabsolu_left x). -Rewrite (Rabsolu_left ``x+h``). -Rewrite Rplus_sym. -Rewrite Ropp_distr1. -Unfold Rminus; Rewrite Ropp_Ropp; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l. -Rewrite Rplus_Or; Unfold Rdiv. -Rewrite Ropp_mul1. -Rewrite <- Rinv_r_sym. -Rewrite Ropp_Ropp; Rewrite Rplus_Ropp_l; Rewrite Rabsolu_R0; Apply H0. -Apply H2. -Case (case_Rabsolu h); Intro. -Apply Ropp_Rlt. -Rewrite Ropp_O; Rewrite Ropp_distr1; Apply gt0_plus_gt0_is_gt0. -Apply H1. -Apply Rgt_RO_Ropp; Apply r. -Rewrite (Rabsolu_right h r) in H3. -Apply Rlt_anti_compatibility with ``-x``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H3. -Apply H. -Apply Rgt_RO_Ropp; Apply H. -Qed. - -(* Rabsolu is derivable for all x <> 0 *) -Lemma derivable_pt_Rabsolu : (x:R) ``x<>0`` -> (derivable_pt Rabsolu x). -Intros. -Case (total_order_T x R0); Intro. -Elim s; Intro. -Unfold derivable_pt; Apply Specif.existT with ``-1``. -Apply (Rabsolu_derive_2 x a). -Elim H; Exact b. -Unfold derivable_pt; Apply Specif.existT with ``1``. -Apply (Rabsolu_derive_1 x r). -Qed. - -(* Rabsolu is continuous for all x *) -Lemma continuity_Rabsolu : (continuity Rabsolu). -Unfold continuity; Intro. -Case (Req_EM x R0); Intro. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists eps; Split. -Apply H0. -Intros; Rewrite H; Rewrite Rabsolu_R0; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Elim H1; Intros; Rewrite H in H3; Unfold Rminus in H3; Rewrite Ropp_O in H3; Rewrite Rplus_Or in H3; Apply H3. -Apply derivable_continuous_pt; Apply (derivable_pt_Rabsolu x H). -Qed. - -(* Finite sums : Sum a_k x^k *) -Lemma continuity_finite_sum : (An:nat->R;N:nat) (continuity [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). -Intros; Unfold continuity; Intro. -Induction N. -Simpl. -Apply continuity_pt_const. -Unfold constant; Intros; Reflexivity. -Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). -Apply continuity_pt_plus. -Apply HrecN. -Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). -Apply continuity_pt_scal. -Apply derivable_continuous_pt. -Apply derivable_pt_pow. -Reflexivity. -Reflexivity. -Qed. - -Lemma derivable_pt_lim_fs : (An:nat->R;x:R;N:nat) (lt O N) -> (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N))). -Intros; Induction N. -Elim (lt_n_n ? H). -Cut N=O\/(lt O N). -Intro; Elim H0; Intro. -Rewrite H1. -Simpl. -Replace [y:R]``(An O)*1+(An (S O))*(y*1)`` with (plus_fct (fct_cte ``(An O)*1``) (mult_real_fct ``(An (S O))`` (mult_fct id (fct_cte R1)))). -Replace ``1*(An (S O))*1`` with ``0+(An (S O))*(1*(fct_cte R1 x)+(id x)*0)``. -Apply derivable_pt_lim_plus. -Apply derivable_pt_lim_const. -Apply derivable_pt_lim_scal. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte id; Ring. -Reflexivity. -Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). -Replace (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))) with (Rplus (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) ``(An (S N))*((INR (S (pred (S N))))*(pow x (pred (S N))))``). -Apply derivable_pt_lim_plus. -Apply HrecN. -Assumption. -Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). -Apply derivable_pt_lim_scal. -Replace (pred (S N)) with N; [Idtac | Reflexivity]. -Pattern 3 N; Replace N with (pred (S N)). -Apply derivable_pt_lim_pow. -Reflexivity. -Reflexivity. -Cut (pred (S N)) = (S (pred N)). -Intro; Rewrite H2. -Rewrite tech5. -Apply Rplus_plus_r. -Rewrite <- H2. -Replace (pred (S N)) with N; [Idtac | Reflexivity]. -Ring. -Simpl. -Apply S_pred with O; Assumption. -Unfold plus_fct. -Simpl; Reflexivity. -Inversion H. -Left; Reflexivity. -Right; Apply lt_le_trans with (1); [Apply lt_O_Sn | Assumption]. -Qed. - -Lemma derivable_pt_lim_finite_sum : (An:(nat->R); x:R; N:nat) (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (Cases N of O => R0 | _ => (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) end)). -Intros. -Induction N. -Simpl. -Rewrite Rmult_1r. -Replace [_:R]``(An O)`` with (fct_cte (An O)); [Apply derivable_pt_lim_const | Reflexivity]. -Apply derivable_pt_lim_fs; Apply lt_O_Sn. -Qed. - -Lemma derivable_pt_finite_sum : (An:nat->R;N:nat;x:R) (derivable_pt [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x). -Intros. -Unfold derivable_pt. -Assert H := (derivable_pt_lim_finite_sum An x N). -Induction N. -Apply Specif.existT with R0; Apply H. -Apply Specif.existT with (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))); Apply H. -Qed. - -Lemma derivable_finite_sum : (An:nat->R;N:nat) (derivable [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). -Intros; Unfold derivable; Intro; Apply derivable_pt_finite_sum. -Qed. - -(* Regularity of hyperbolic functions *) -Lemma derivable_pt_lim_cosh : (x:R) (derivable_pt_lim cosh x ``(sinh x)``). -Intro. -Unfold cosh sinh; Unfold Rdiv. -Replace [x0:R]``((exp x0)+(exp ( -x0)))*/2`` with (mult_fct (plus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. -Replace ``((exp x)-(exp ( -x)))*/2`` with ``((exp x)+((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((plus_fct exp (comp exp (opp_fct id))) x)*0``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_plus. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_comp. -Apply derivable_pt_lim_opp. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_const. -Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. -Qed. - -Lemma derivable_pt_lim_sinh : (x:R) (derivable_pt_lim sinh x ``(cosh x)``). -Intro. -Unfold cosh sinh; Unfold Rdiv. -Replace [x0:R]``((exp x0)-(exp ( -x0)))*/2`` with (mult_fct (minus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. -Replace ``((exp x)+(exp ( -x)))*/2`` with ``((exp x)-((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((minus_fct exp (comp exp (opp_fct id))) x)*0``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_comp. -Apply derivable_pt_lim_opp. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_const. -Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. -Qed. - -Lemma derivable_pt_exp : (x:R) (derivable_pt exp x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (exp x). -Apply derivable_pt_lim_exp. -Qed. - -Lemma derivable_pt_cosh : (x:R) (derivable_pt cosh x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (sinh x). -Apply derivable_pt_lim_cosh. -Qed. - -Lemma derivable_pt_sinh : (x:R) (derivable_pt sinh x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (cosh x). -Apply derivable_pt_lim_sinh. -Qed. - -Lemma derivable_exp : (derivable exp). -Unfold derivable; Apply derivable_pt_exp. -Qed. - -Lemma derivable_cosh : (derivable cosh). -Unfold derivable; Apply derivable_pt_cosh. -Qed. - -Lemma derivable_sinh : (derivable sinh). -Unfold derivable; Apply derivable_pt_sinh. -Qed. - -Lemma derive_pt_exp : (x:R) (derive_pt exp x (derivable_pt_exp x))==(exp x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_exp. -Qed. - -Lemma derive_pt_cosh : (x:R) (derive_pt cosh x (derivable_pt_cosh x))==(sinh x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_cosh. -Qed. - -Lemma derive_pt_sinh : (x:R) (derive_pt sinh x (derivable_pt_sinh x))==(cosh x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_sinh. -Qed. |