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diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v new file mode 100644 index 00000000..918ebfc0 --- /dev/null +++ b/theories/Reals/Ranalysis1.v @@ -0,0 +1,1479 @@ +(************************************************************************) +(* 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: Ranalysis1.v,v 1.21.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Export Rlimit. +Require Export Rderiv. Open Local Scope R_scope. +Implicit Type f : R -> R. + +(****************************************************) +(** Basic operations on functions *) +(****************************************************) +Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x. +Definition opp_fct f (x:R) : R := - f x. +Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x. +Definition mult_real_fct (a:R) f (x:R) : R := a * f x. +Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x. +Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x. +Definition div_real_fct (a:R) f (x:R) : R := a / f x. +Definition comp f1 f2 (x:R) : R := f1 (f2 x). +Definition inv_fct f (x:R) : R := / f x. + +Infix "+" := plus_fct : Rfun_scope. +Notation "- x" := (opp_fct x) : Rfun_scope. +Infix "*" := mult_fct : Rfun_scope. +Infix "-" := minus_fct : Rfun_scope. +Infix "/" := div_fct : Rfun_scope. +Notation Local "f1 'o' f2" := (comp f1 f2) + (at level 20, right associativity) : Rfun_scope. +Notation "/ x" := (inv_fct x) : Rfun_scope. + +Delimit Scope Rfun_scope with F. + +Definition fct_cte (a x:R) : R := a. +Definition id (x:R) := x. + +(****************************************************) +(** Variations of functions *) +(****************************************************) +Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y. +Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x. +Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y. +Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x. +Definition constant f : Prop := forall x y:R, f x = f y. + +(**********) +Definition no_cond (x:R) : Prop := True. + +(**********) +Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop := + forall x:R, D x -> f x = c. + +(***************************************************) +(** Definition of continuity as a limit *) +(***************************************************) + +(**********) +Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0. +Definition continuity f : Prop := forall x:R, continuity_pt f x. + +Arguments Scope continuity_pt [Rfun_scope R_scope]. +Arguments Scope continuity [Rfun_scope]. + +(**********) +Lemma continuity_pt_plus : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0. +unfold continuity_pt, plus_fct in |- *; unfold continue_in in |- *; intros; + apply limit_plus; assumption. +Qed. + +Lemma continuity_pt_opp : + forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0. +unfold continuity_pt, opp_fct in |- *; unfold continue_in in |- *; intros; + apply limit_Ropp; assumption. +Qed. + +Lemma continuity_pt_minus : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0. +unfold continuity_pt, minus_fct in |- *; unfold continue_in in |- *; intros; + apply limit_minus; assumption. +Qed. + +Lemma continuity_pt_mult : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0. +unfold continuity_pt, mult_fct in |- *; unfold continue_in in |- *; intros; + apply limit_mul; assumption. +Qed. + +Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0. +unfold constant, continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + intros; exists 1; split; + [ apply Rlt_0_1 + | intros; generalize (H x x0); intro; rewrite H2; simpl in |- *; + rewrite R_dist_eq; assumption ]. +Qed. + +Lemma continuity_pt_scal : + forall f (a x0:R), + continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0. +unfold continuity_pt, mult_real_fct in |- *; unfold continue_in in |- *; + intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0). +unfold limit1_in in |- *; unfold limit_in in |- *; intros; exists 1; split. +apply Rlt_0_1. +intros; rewrite R_dist_eq; assumption. +assumption. +Qed. + +Lemma continuity_pt_inv : + forall f (x0:R), continuity_pt f x0 -> f x0 <> 0 -> continuity_pt (/ f) x0. +intros. +replace (/ f)%F with (fun x:R => / f x). +unfold continuity_pt in |- *; unfold continue_in in |- *; intros; + apply limit_inv; assumption. +unfold inv_fct in |- *; reflexivity. +Qed. + +Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F. +intros; reflexivity. +Qed. + +Lemma continuity_pt_div : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> + continuity_pt f2 x0 -> f2 x0 <> 0 -> continuity_pt (f1 / f2) x0. +intros; rewrite (div_eq_inv f1 f2); apply continuity_pt_mult; + [ assumption | apply continuity_pt_inv; assumption ]. +Qed. + +Lemma continuity_pt_comp : + forall f1 f2 (x:R), + continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x. +unfold continuity_pt in |- *; unfold continue_in in |- *; intros; + unfold comp in |- *. +cut + (limit1_in (fun x0:R => f2 (f1 x0)) + (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) ( + f2 (f1 x)) x -> + limit1_in (fun x0:R => f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x). +intro; apply H1. +eapply limit_comp. +apply H. +apply H0. +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +assert (H3 := H1 eps H2). +elim H3; intros. +exists x0. +split. +elim H4; intros; assumption. +intros; case (Req_dec (f1 x) (f1 x1)); intro. +rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + assumption. +elim H4; intros; apply H8. +split. +unfold Dgf, D_x, no_cond in |- *. +split. +split. +trivial. +elim H5; unfold D_x, no_cond in |- *; intros. +elim H9; intros; assumption. +split. +trivial. +assumption. +elim H5; intros; assumption. +Qed. + +(**********) +Lemma continuity_plus : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2). +unfold continuity in |- *; intros; + apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_opp : forall f, continuity f -> continuity (- f). +unfold continuity in |- *; intros; apply (continuity_pt_opp f x (H x)). +Qed. + +Lemma continuity_minus : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2). +unfold continuity in |- *; intros; + apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_mult : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2). +unfold continuity in |- *; intros; + apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_const : forall f, constant f -> continuity f. +unfold continuity in |- *; intros; apply (continuity_pt_const f x H). +Qed. + +Lemma continuity_scal : + forall f (a:R), continuity f -> continuity (mult_real_fct a f). +unfold continuity in |- *; intros; apply (continuity_pt_scal f a x (H x)). +Qed. + +Lemma continuity_inv : + forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f). +unfold continuity in |- *; intros; apply (continuity_pt_inv f x (H x) (H0 x)). +Qed. + +Lemma continuity_div : + forall f1 f2, + continuity f1 -> + continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2). +unfold continuity in |- *; intros; + apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). +Qed. + +Lemma continuity_comp : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1). +unfold continuity in |- *; intros. +apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). +Qed. + + +(*****************************************************) +(** Derivative's definition using Landau's kernel *) +(*****************************************************) + +Definition derivable_pt_lim f (x l:R) : Prop := + forall eps:R, + 0 < eps -> + exists delta : posreal, + (forall h:R, + h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps). + +Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l. + +Definition derivable_pt f (x:R) := sigT (derivable_pt_abs f x). +Definition derivable f := forall x:R, derivable_pt f x. + +Definition derive_pt f (x:R) (pr:derivable_pt f x) := projT1 pr. +Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x). + +Arguments Scope derivable_pt_lim [Rfun_scope R_scope]. +Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope]. +Arguments Scope derivable_pt [Rfun_scope R_scope]. +Arguments Scope derivable [Rfun_scope]. +Arguments Scope derive_pt [Rfun_scope R_scope _]. +Arguments Scope derive [Rfun_scope _]. + +Definition antiderivative f (g:R -> R) (a b:R) : Prop := + (forall x:R, + a <= x <= b -> exists pr : derivable_pt g x, f x = derive_pt g x pr) /\ + a <= b. +(************************************) +(** Class of differential functions *) +(************************************) +Record Differential : Type := mkDifferential + {d1 :> R -> R; cond_diff : derivable d1}. + +Record Differential_D2 : Type := mkDifferential_D2 + {d2 :> R -> R; + cond_D1 : derivable d2; + cond_D2 : derivable (derive d2 cond_D1)}. + +(**********) +Lemma uniqueness_step1 : + forall f (x l1 l2:R), + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l1 0 -> + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l2 0 -> + l1 = l2. +intros; + apply + (single_limit (fun h:R => (f (x + h) - f x) / h) ( + fun h:R => h <> 0) l1 l2 0); try assumption. +unfold adhDa in |- *; intros; exists (alp / 2). +split. +unfold Rdiv in |- *; apply prod_neq_R0. +red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). +apply Rinv_neq_0_compat; discrR. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite Rabs_mult. +replace (Rabs (/ 2)) with (/ 2). +replace (Rabs alp) with alp. +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double; + pattern alp at 1 in |- *; replace alp with (alp + 0); + [ idtac | ring ]; apply Rplus_lt_compat_l; assumption. +symmetry in |- *; apply Rabs_right; left; assumption. +symmetry in |- *; apply Rabs_right; left; change (0 < / 2) in |- *; + apply Rinv_0_lt_compat; prove_sup0. +Qed. + +Lemma uniqueness_step2 : + forall f (x l:R), + derivable_pt_lim f x l -> + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0. +unfold derivable_pt_lim in |- *; intros; unfold limit1_in in |- *; + unfold limit_in in |- *; intros. +assert (H1 := H eps H0). +elim H1; intros. +exists (pos x0). +split. +apply (cond_pos x0). +simpl in |- *; unfold R_dist in |- *; intros. +elim H3; intros. +apply H2; + [ assumption + | unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5; + assumption ]. +Qed. + +Lemma uniqueness_step3 : + forall f (x l:R), + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 -> + derivable_pt_lim f x l. +unfold limit1_in, derivable_pt_lim in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; intros. +elim (H eps H0). +intros; elim H1; intros. +exists (mkposreal x0 H2). +simpl in |- *; intros; unfold R_dist in H3; apply (H3 h). +split; + [ assumption + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; assumption ]. +Qed. + +Lemma uniqueness_limite : + forall f (x l1 l2:R), + derivable_pt_lim f x l1 -> derivable_pt_lim f x l2 -> l1 = l2. +intros. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +assert (H3 := uniqueness_step1 _ _ _ _ H1 H2). +assumption. +Qed. + +Lemma derive_pt_eq : + forall f (x l:R) (pr:derivable_pt f x), + derive_pt f x pr = l <-> derivable_pt_lim f x l. +intros; split. +intro; assert (H1 := projT2 pr); unfold derive_pt in H; rewrite H in H1; + assumption. +intro; assert (H1 := projT2 pr); unfold derivable_pt_abs in H1. +assert (H2 := uniqueness_limite _ _ _ _ H H1). +unfold derive_pt in |- *; unfold derivable_pt_abs in |- *. +symmetry in |- *; assumption. +Qed. + +(**********) +Lemma derive_pt_eq_0 : + forall f (x l:R) (pr:derivable_pt f x), + derivable_pt_lim f x l -> derive_pt f x pr = l. +intros; elim (derive_pt_eq f x l pr); intros. +apply (H1 H). +Qed. + +(**********) +Lemma derive_pt_eq_1 : + forall f (x l:R) (pr:derivable_pt f x), + derive_pt f x pr = l -> derivable_pt_lim f x l. +intros; elim (derive_pt_eq f x l pr); intros. +apply (H0 H). +Qed. + + +(********************************************************************) +(** Equivalence of this definition with the one using limit concept *) +(********************************************************************) +Lemma derive_pt_D_in : + forall f (df:R -> R) (x:R) (pr:derivable_pt f x), + D_in f df no_cond x <-> derive_pt f x pr = df x. +intros; split. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +apply derive_pt_eq_0. +unfold derivable_pt_lim in |- *. +intros; elim (H eps H0); intros alpha H1; elim H1; intros; + exists (mkposreal alpha H2); intros; generalize (H3 (x + h)); + intro; cut (x + h - x = h); + [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha); + [ intro; generalize (H6 H8); rewrite H7; intro; assumption + | split; + [ unfold D_x in |- *; split; + [ unfold no_cond in |- *; trivial + | apply Rminus_not_eq_right; rewrite H7; assumption ] + | rewrite H7; assumption ] ] + | ring ]. +intro. +assert (H0 := derive_pt_eq_1 f x (df x) pr H). +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +elim (H0 eps H1); intros alpha H2; exists (pos alpha); split. +apply (cond_pos alpha). +intros; elim H3; intros; unfold D_x in H4; elim H4; intros; cut (x0 - x <> 0). +intro; generalize (H2 (x0 - x) H8 H5); replace (x + (x0 - x)) with x0. +intro; assumption. +ring. +auto with real. +Qed. + +Lemma derivable_pt_lim_D_in : + forall f (df:R -> R) (x:R), + D_in f df no_cond x <-> derivable_pt_lim f x (df x). +intros; split. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +unfold derivable_pt_lim in |- *. +intros; elim (H eps H0); intros alpha H1; elim H1; intros; + exists (mkposreal alpha H2); intros; generalize (H3 (x + h)); + intro; cut (x + h - x = h); + [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha); + [ intro; generalize (H6 H8); rewrite H7; intro; assumption + | split; + [ unfold D_x in |- *; split; + [ unfold no_cond in |- *; trivial + | apply Rminus_not_eq_right; rewrite H7; assumption ] + | rewrite H7; assumption ] ] + | ring ]. +intro. +unfold derivable_pt_lim in H. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +elim (H eps H0); intros alpha H2; exists (pos alpha); split. +apply (cond_pos alpha). +intros. +elim H1; intros; unfold D_x in H3; elim H3; intros; cut (x0 - x <> 0). +intro; generalize (H2 (x0 - x) H7 H4); replace (x + (x0 - x)) with x0. +intro; assumption. +ring. +auto with real. +Qed. + + +(***********************************) +(** derivability -> continuity *) +(***********************************) +(**********) +Lemma derivable_derive : + forall f (x:R) (pr:derivable_pt f x), exists l : R, derive_pt f x pr = l. +intros; exists (projT1 pr). +unfold derive_pt in |- *; reflexivity. +Qed. + +Theorem derivable_continuous_pt : + forall f (x:R), derivable_pt f x -> continuity_pt f x. +intros. +generalize (derivable_derive f x X); intro. +elim H; intros l H1. +cut (l = fct_cte l x). +intro. +rewrite H0 in H1. +generalize (derive_pt_D_in f (fct_cte l) x); intro. +elim (H2 X); intros. +generalize (H4 H1); intro. +unfold continuity_pt in |- *. +apply (cont_deriv f (fct_cte l) no_cond x H5). +unfold fct_cte in |- *; reflexivity. +Qed. + +Theorem derivable_continuous : forall f, derivable f -> continuity f. +unfold derivable, continuity in |- *; intros. +apply (derivable_continuous_pt f x (X x)). +Qed. + +(****************************************************************) +(** Main rules *) +(****************************************************************) + +Lemma derivable_pt_lim_plus : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 + f2) x (l1 + l2). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +unfold plus_fct in |- *. +cut + (forall h:R, + (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h = + (f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h). +intro. +generalize + (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h') + (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H4 eps H5); intros. +exists x0. +elim H6; intros. +split. +assumption. +intros; rewrite H3; apply H8; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_opp : + forall f (x l:R), derivable_pt_lim f x l -> derivable_pt_lim (- f) x (- l). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +unfold opp_fct in |- *. +cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)). +intro. +generalize + (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H2 eps H3); intros. +exists x0. +elim H4; intros. +split. +assumption. +intros; rewrite H0; apply H6; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_minus : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 - f2) x (l1 - l2). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +unfold minus_fct in |- *. +cut + (forall h:R, + (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h = + (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h). +intro. +generalize + (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h') + (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H4 eps H5); intros. +exists x0. +elim H6; intros. +split. +assumption. +intros; rewrite <- H3; apply H8; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_mult : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> + derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2). +intros. +assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x). +elim H1; intros. +assert (H4 := H3 H). +assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) x). +elim H5; intros. +assert (H8 := H7 H0). +clear H1 H2 H3 H5 H6 H7. +assert + (H1 := + derivable_pt_lim_D_in (f1 * f2)%F (fun y:R => l1 * f2 x + f1 x * l2) x). +elim H1; intros. +clear H1 H3. +apply H2. +unfold mult_fct in |- *. +apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption. +Qed. + +Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0. +intros; unfold fct_cte, derivable_pt_lim in |- *. +intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus in |- *; + rewrite Rplus_opp_r; unfold Rdiv in |- *; rewrite Rmult_0_l; + rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +Qed. + +Lemma derivable_pt_lim_scal : + forall f (a x l:R), + derivable_pt_lim f x l -> derivable_pt_lim (mult_real_fct a f) x (a * l). +intros. +assert (H0 := derivable_pt_lim_const a x). +replace (mult_real_fct a f) with (fct_cte a * f)%F. +replace (a * l) with (0 * f x + a * l); [ idtac | ring ]. +apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption. +unfold mult_real_fct, mult_fct, fct_cte in |- *; reflexivity. +Qed. + +Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1. +intro; unfold derivable_pt_lim in |- *. +intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2; + unfold id in |- *; replace ((x + h - x) / h - 1) with 0. +rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h). +apply Rabs_pos. +assumption. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite (Rplus_comm x); + rewrite Rplus_assoc. +rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite <- Rinv_r_sym. +symmetry in |- *; apply Rplus_opp_r. +assumption. +Qed. + +Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x). +intro; unfold derivable_pt_lim in |- *. +unfold Rsqr in |- *; intros eps Heps; exists (mkposreal eps Heps); + intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h. +assumption. +replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h); + [ idtac | ring ]. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +repeat rewrite <- Rinv_r_sym; [ idtac | assumption ]. +ring. +Qed. + +Lemma derivable_pt_lim_comp : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 (f1 x) l2 -> derivable_pt_lim (f2 o f1) x (l2 * l1). +intros; assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x). +elim H1; intros. +assert (H4 := H3 H). +assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) (f1 x)). +elim H5; intros. +assert (H8 := H7 H0). +clear H1 H2 H3 H5 H6 H7. +assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x). +elim H1; intros. +clear H1 H3; apply H2. +unfold comp in |- *; + cut + (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) + (Dgf no_cond no_cond f1) x -> + D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) no_cond x). +intro; apply H1. +rewrite Rmult_comm; + apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); + assumption. +unfold Dgf, D_in, no_cond in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; unfold dist in |- *; simpl in |- *; + unfold R_dist in |- *; intros. +elim (H1 eps H3); intros. +exists x0; intros; split. +elim H5; intros; assumption. +intros; elim H5; intros; apply H9; split. +unfold D_x in |- *; split. +split; trivial. +elim H6; intros; unfold D_x in H10; elim H10; intros; assumption. +elim H6; intros; assumption. +Qed. + +Lemma derivable_pt_plus : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 + x1). +apply derivable_pt_lim_plus; assumption. +Qed. + +Lemma derivable_pt_opp : + forall f (x:R), derivable_pt f x -> derivable_pt (- f) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +apply existT with (- x0). +apply derivable_pt_lim_opp; assumption. +Qed. + +Lemma derivable_pt_minus : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 - x1). +apply derivable_pt_lim_minus; assumption. +Qed. + +Lemma derivable_pt_mult : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 * f2 x + f1 x * x1). +apply derivable_pt_lim_mult; assumption. +Qed. + +Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x. +intros; unfold derivable_pt in |- *. +apply existT with 0. +apply derivable_pt_lim_const. +Qed. + +Lemma derivable_pt_scal : + forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +apply existT with (a * x0). +apply derivable_pt_lim_scal; assumption. +Qed. + +Lemma derivable_pt_id : forall x:R, derivable_pt id x. +unfold derivable_pt in |- *; intro. +exists 1. +apply derivable_pt_lim_id. +Qed. + +Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x. +unfold derivable_pt in |- *; intro; apply existT with (2 * x). +apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derivable_pt_comp : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x1 * x0). +apply derivable_pt_lim_comp; assumption. +Qed. + +Lemma derivable_plus : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2). +unfold derivable in |- *; intros. +apply (derivable_pt_plus _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_opp : forall f, derivable f -> derivable (- f). +unfold derivable in |- *; intros. +apply (derivable_pt_opp _ x (X _)). +Qed. + +Lemma derivable_minus : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2). +unfold derivable in |- *; intros. +apply (derivable_pt_minus _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_mult : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2). +unfold derivable in |- *; intros. +apply (derivable_pt_mult _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_const : forall a:R, derivable (fct_cte a). +unfold derivable in |- *; intros. +apply derivable_pt_const. +Qed. + +Lemma derivable_scal : + forall f (a:R), derivable f -> derivable (mult_real_fct a f). +unfold derivable in |- *; intros. +apply (derivable_pt_scal _ a x (X _)). +Qed. + +Lemma derivable_id : derivable id. +unfold derivable in |- *; intro; apply derivable_pt_id. +Qed. + +Lemma derivable_Rsqr : derivable Rsqr. +unfold derivable in |- *; intro; apply derivable_pt_Rsqr. +Qed. + +Lemma derivable_comp : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1). +unfold derivable in |- *; intros. +apply (derivable_pt_comp _ _ x (X _) (X0 _)). +Qed. + +Lemma derive_pt_plus : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 + f2) x (derivable_pt_plus _ _ _ pr1 pr2) = + derive_pt f1 x pr1 + derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 + f2)%F x (derivable_pt_plus _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_plus; assumption. +Qed. + +Lemma derive_pt_opp : + forall f (x:R) (pr1:derivable_pt f x), + derive_pt (- f) x (derivable_pt_opp _ _ pr1) = - derive_pt f x pr1. +intros. +assert (H := derivable_derive f x pr1). +assert (H0 := derivable_derive (- f)%F x (derivable_pt_opp _ _ pr1)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +rewrite H; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +apply derivable_pt_lim_opp; assumption. +Qed. + +Lemma derive_pt_minus : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 - f2) x (derivable_pt_minus _ _ _ pr1 pr2) = + derive_pt f1 x pr1 - derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 - f2)%F x (derivable_pt_minus _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_minus; assumption. +Qed. + +Lemma derive_pt_mult : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 * f2) x (derivable_pt_mult _ _ _ pr1 pr2) = + derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 * f2)%F x (derivable_pt_mult _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_mult; assumption. +Qed. + +Lemma derive_pt_const : + forall a x:R, derive_pt (fct_cte a) x (derivable_pt_const a x) = 0. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_const. +Qed. + +Lemma derive_pt_scal : + forall f (a x:R) (pr:derivable_pt f x), + derive_pt (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr) = + a * derive_pt f x pr. +intros. +assert (H := derivable_derive f x pr). +assert + (H0 := derivable_derive (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +rewrite H; apply derive_pt_eq_0. +assert (H3 := projT2 pr). +unfold derive_pt in H; rewrite H in H3. +apply derivable_pt_lim_scal; assumption. +Qed. + +Lemma derive_pt_id : forall x:R, derive_pt id x (derivable_pt_id _) = 1. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_id. +Qed. + +Lemma derive_pt_Rsqr : + forall x:R, derive_pt Rsqr x (derivable_pt_Rsqr _) = 2 * x. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derive_pt_comp : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 (f1 x)), + derive_pt (f2 o f1) x (derivable_pt_comp _ _ _ pr1 pr2) = + derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 (f1 x) pr2). +assert + (H1 := derivable_derive (f2 o f1)%F x (derivable_pt_comp _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_comp; assumption. +Qed. + +(* Pow *) +Definition pow_fct (n:nat) (y:R) : R := y ^ n. + +Lemma derivable_pt_lim_pow_pos : + forall (x:R) (n:nat), + (0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n). +intros. +induction n as [| n Hrecn]. +elim (lt_irrefl _ H). +cut (n = 0%nat \/ (0 < n)%nat). +intro; elim H0; intro. +rewrite H1; simpl in |- *. +replace (fun y:R => y * 1) with (id * fct_cte 1)%F. +replace (1 * 1) with (1 * fct_cte 1 x + id x * 0). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte, id in |- *; ring. +reflexivity. +replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n). +replace (pred (S n)) with n; [ idtac | reflexivity ]. +replace (fun y:R => y * y ^ n) with (id * (fun y:R => y ^ n))%F. +set (f := fun y:R => y ^ n). +replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_id. +unfold f in |- *; apply Hrecn; assumption. +unfold f in |- *. +pattern n at 1 5 in |- *; replace n with (S (pred n)). +unfold id in |- *; rewrite S_INR; simpl in |- *. +ring. +symmetry in |- *; apply S_pred with 0%nat; assumption. +unfold mult_fct, id in |- *; reflexivity. +reflexivity. +inversion H. +left; reflexivity. +right. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +assumption. +Qed. + +Lemma derivable_pt_lim_pow : + forall (x:R) (n:nat), + derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n). +intros. +induction n as [| n Hrecn]. +simpl in |- *. +rewrite Rmult_0_l. +replace (fun _:R => 1) with (fct_cte 1); + [ apply derivable_pt_lim_const | reflexivity ]. +apply derivable_pt_lim_pow_pos. +apply lt_O_Sn. +Qed. + +Lemma derivable_pt_pow : + forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x. +intros; unfold derivable_pt in |- *. +apply existT with (INR n * x ^ pred n). +apply derivable_pt_lim_pow. +Qed. + +Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n). +intro; unfold derivable in |- *; intro; apply derivable_pt_pow. +Qed. + +Lemma derive_pt_pow : + forall (n:nat) (x:R), + derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_pow. +Qed. + +Lemma pr_nu : + forall f (x:R) (pr1 pr2:derivable_pt f x), + derive_pt f x pr1 = derive_pt f x pr2. +intros. +unfold derivable_pt in pr1. +unfold derivable_pt in pr2. +elim pr1; intros. +elim pr2; intros. +unfold derivable_pt_abs in p. +unfold derivable_pt_abs in p0. +simpl in |- *. +apply (uniqueness_limite f x x0 x1 p p0). +Qed. + + +(************************************************************) +(** Local extremum's condition *) +(************************************************************) + +Theorem deriv_maximum : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> + (forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0. +intros; case (Rtotal_order 0 (derive_pt f c pr)); intro. +assert (H3 := derivable_derive f c pr). +elim H3; intros l H4; rewrite H4 in H2. +assert (H5 := derive_pt_eq_1 f c l pr H4). +cut (0 < l / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H5 (l / 2) H6); intros delta H7. +cut (0 < (b - c) / 2). +intro; cut (Rmin (delta / 2) ((b - c) / 2) <> 0). +intro; cut (Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta). +intro. +assert (H11 := H7 (Rmin (delta / 2) ((b - c) / 2)) H9 H10). +cut (0 < Rmin (delta / 2) ((b - c) / 2)). +intro; cut (a < c + Rmin (delta / 2) ((b - c) / 2)). +intro; cut (c + Rmin (delta / 2) ((b - c) / 2) < b). +intro; assert (H15 := H1 (c + Rmin (delta / 2) ((b - c) / 2)) H13 H14). +cut + ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) / + Rmin (delta / 2) ((b - c) / 2) <= 0). +intro; cut (- l < 0). +intro; unfold Rminus in H11. +cut + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l < 0). +intro; + cut + (Rabs + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2). +unfold Rabs in |- *; + case + (Rcase_abs + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l)); intro. +replace + (- + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l)) with + (l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2))). +intro; + generalize + (Rplus_lt_compat_l (- l) + (l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2))) (l / 2) H19); + repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; replace (- l + l / 2) with (- (l / 2)). +intro; + generalize + (Ropp_lt_gt_contravar + (- + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2))) (- (l / 2)) H20); + repeat rewrite Ropp_involutive; intro; + generalize + (Rlt_trans 0 (l / 2) + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2)) H6 H21); intro; + elim + (Rlt_irrefl 0 + (Rlt_le_trans 0 + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)). +pattern l at 2 in |- *; rewrite double_var. +ring. +ring. +intro. +assert + (H20 := + Rge_le + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l) 0 r). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H20 H18)). +assumption. +rewrite <- Ropp_0; + replace + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l) with + (- + (l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) / + Rmin (delta / 2) ((b + - c) / 2)))). +apply Ropp_gt_lt_contravar; + change + (0 < + l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) / + Rmin (delta / 2) ((b + - c) / 2))) in |- *; apply Rplus_lt_le_0_compat; + [ assumption + | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ]. +ring. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +replace + ((f (c + Rmin (delta / 2) ((b - c) / 2)) - f c) / + Rmin (delta / 2) ((b - c) / 2)) with + (- + ((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) / + Rmin (delta / 2) ((b - c) / 2))). +rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; + unfold Rdiv in |- *; apply Rmult_le_pos; + [ generalize + (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2))) + (f (c + Rmin (delta * / 2) ((b - c) * / 2))) ( + f c) H15); rewrite Rplus_opp_r; intro; assumption + | left; apply Rinv_0_lt_compat; assumption ]. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))). +apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)). +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +repeat rewrite Rmult_1_l. +ring. +red in |- *; intro. +unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). +red in |- *; intro. +unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). +assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)). +assert + (H15 := + Rplus_le_compat_l c (Rmin (delta / 2) ((b - c) / 2)) ((b - c) / 2) H14). +apply Rle_lt_trans with (c + (b - c) / 2). +assumption. +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (c + (b - c) / 2)) with (c + b). +replace (2 * b) with (b + b). +apply Rplus_lt_compat_r; assumption. +ring. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_l. +repeat rewrite (Rmult_comm 2). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +ring. +discrR. +apply Rlt_trans with c. +assumption. +pattern c at 1 in |- *; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l; + assumption. +cut (0 < delta / 2). +intro; + apply + (Rmin_stable_in_posreal (mkposreal (delta / 2) H12) + (mkposreal ((b - c) / 2) H8)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rabs in |- *; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))). +intro. +cut (0 < delta / 2). +intro. +generalize + (Rmin_stable_in_posreal (mkposreal (delta / 2) H10) + (mkposreal ((b - c) / 2) H8)); simpl in |- *; intro; + elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 r)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; apply Rle_lt_trans with (delta / 2). +apply Rmin_l. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +replace (2 * delta) with (delta + delta). +pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + apply Rplus_lt_compat_l. +rewrite Rplus_0_r; apply (cond_pos delta). +symmetry in |- *; apply double. +discrR. +cut (0 < delta / 2). +intro; + generalize + (Rmin_stable_in_posreal (mkposreal (delta / 2) H9) + (mkposreal ((b - c) / 2) H8)); simpl in |- *; + intro; red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro; + assumption. +apply Rinv_0_lt_compat; prove_sup0. +elim H2; intro. +symmetry in |- *; assumption. +generalize (derivable_derive f c pr); intro; elim H4; intros l H5. +rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro; + cut (0 < - (l / 2)). +intro; elim (H6 (- (l / 2)) H7); intros delta H9. +cut (0 < (c - a) / 2). +intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) < 0). +intro; cut (Rmax (- (delta / 2)) ((a - c) / 2) <> 0). +intro; cut (Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta). +intro; generalize (H9 (Rmax (- (delta / 2)) ((a - c) / 2)) H11 H12); intro; + cut (a < c + Rmax (- (delta / 2)) ((a - c) / 2)). +cut (c + Rmax (- (delta / 2)) ((a - c) / 2) < b). +intros; generalize (H1 (c + Rmax (- (delta / 2)) ((a - c) / 2)) H15 H14); + intro; + cut + (0 <= + (f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) / + Rmax (- (delta / 2)) ((a - c) / 2)). +intro; cut (0 < - l). +intro; unfold Rminus in H13; + cut + (0 < + (f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l). +intro; + cut + (Rabs + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) < + - (l / 2)). +unfold Rabs in |- *; + case + (Rcase_abs + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l)). +intro; + elim + (Rlt_irrefl 0 + (Rlt_trans 0 + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) 0 H19 r)). +intros; + generalize + (Rplus_lt_compat_r l + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) ( + - (l / 2)) H20); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; replace (- (l / 2) + l) with (l / 2). +cut (l / 2 < 0). +intros; + generalize + (Rlt_trans + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2)) (l / 2) 0 H22 H21); + intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 + ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) / + Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)). +rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0; + apply Ropp_lt_gt_contravar; assumption. +pattern l at 3 in |- *; rewrite double_var. +ring. +assumption. +apply Rplus_le_lt_0_compat; assumption. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +unfold Rdiv in |- *; + replace + ((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) * + / Rmax (- (delta * / 2)) ((a - c) * / 2)) with + (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) * + / - Rmax (- (delta * / 2)) ((a - c) * / 2)). +apply Rmult_le_pos. +generalize + (Rplus_le_compat_l (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) + (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) ( + f c) H16); rewrite Rplus_opp_l; + replace (- (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c)) with + (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) + f c). +intro; assumption. +ring. +left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; + assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_inv_permute. +rewrite Rmult_opp_opp. +reflexivity. +unfold Rdiv in H11; assumption. +generalize (Rplus_lt_compat_l c (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H10); + rewrite Rplus_0_r; intro; apply Rlt_trans with c; + assumption. +generalize (RmaxLess2 (- (delta / 2)) ((a - c) / 2)); intro; + generalize + (Rplus_le_compat_l c ((a - c) / 2) (Rmax (- (delta / 2)) ((a - c) / 2)) H14); + intro; apply Rlt_le_trans with (c + (a - c) / 2). +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (c + (a - c) / 2)) with (a + c). +rewrite double. +apply Rplus_lt_compat_l; assumption. +ring. +rewrite <- Rplus_assoc. +rewrite <- double_var. +ring. +assumption. +unfold Rabs in |- *; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))). +intro; generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro; + generalize + (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2)) + H12); rewrite Ropp_involutive; intro; + generalize (Rge_le (delta / 2) (- Rmax (- (delta / 2)) ((a - c) / 2)) H13); + intro; apply Rle_lt_trans with (delta / 2). +assumption. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double. +pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta). +discrR. +cut (- (delta / 2) < 0). +cut ((a - c) / 2 < 0). +intros; + generalize + (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13) + (mknegreal ((a - c) / 2) H12)); simpl in |- *; + intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 r); + intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H15 H14)). +rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); + apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). +assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite (Ropp_minus_distr a c). +reflexivity. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). +cut ((a - c) / 2 < 0). +intro; cut (- (delta / 2) < 0). +intro; + apply + (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11) + (mknegreal ((a - c) / 2) H10)). +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); + apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). +assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite (Ropp_minus_distr a c). +reflexivity. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro; + assumption + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +replace (- (l / 2)) with (- l / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ]. +unfold Rdiv in |- *; apply Ropp_mult_distr_l_reverse. +Qed. + +Theorem deriv_minimum : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> + (forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0. +intros. +rewrite <- (Ropp_involutive (derive_pt f c pr)). +apply Ropp_eq_0_compat. +rewrite <- (derive_pt_opp f c pr). +cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c). +intro. +apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2). +intros; unfold opp_fct in |- *; apply Ropp_ge_le_contravar; apply Rle_ge. +apply (H1 x H2 H3). +Qed. + +Theorem deriv_constant2 : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0. +intros. +eapply deriv_maximum with a b; try assumption. +intros; right; apply (H1 x H2 H3). +Qed. + +(**********) +Lemma nonneg_derivative_0 : + forall f (pr:derivable f), + increasing f -> forall x:R, 0 <= derive_pt f x (pr x). +intros; unfold increasing in H. +assert (H0 := derivable_derive f x (pr x)). +elim H0; intros l H1. +rewrite H1; case (Rtotal_order 0 l); intro. +left; assumption. +elim H2; intro. +right; assumption. +assert (H4 := derive_pt_eq_1 f x l (pr x) H1). +cut (0 < - (l / 2)). +intro; elim (H4 (- (l / 2)) H5); intros delta H6. +cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta). +intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11); + cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)). +intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l). +intro; unfold Rabs in |- *; + case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)). +intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 H12 r)). +intros; + generalize + (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l) + (- (l / 2)) H13); unfold Rminus in |- *; + replace (- (l / 2) + l) with (l / 2). +rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro; + generalize + (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) (l / 2) H9 H14); + intro; cut (l / 2 < 0). +intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (l / 2) 0 H15 H16)). +rewrite <- Ropp_0 in H5; + generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5); + repeat rewrite Ropp_involutive; intro; assumption. +pattern l at 3 in |- *; rewrite double_var. +ring. +unfold Rminus in |- *; apply Rplus_le_le_0_compat. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H x (x + delta * / 2) H12); intro; + generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H x (x + delta * / 2) H9); intro; + generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +split. +unfold Rdiv in |- *; apply prod_neq_R0. +generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H7; + elim (Rlt_irrefl 0 H7). +apply Rinv_neq_0_compat; discrR. +split. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +replace (Rabs (delta / 2)) with (delta / 2). +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +pattern (pos delta) at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l; apply (cond_pos delta). +symmetry in |- *; apply Rabs_right. +left; change (0 < delta / 2) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; + apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with l. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption. +apply Rinv_0_lt_compat; prove_sup0. +Qed. |