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Diffstat (limited to 'theories/Reals/Ranalysis3.v')
| -rw-r--r-- | theories/Reals/Ranalysis3.v | 793 |
1 files changed, 793 insertions, 0 deletions
diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v new file mode 100644 index 00000000..9f85b00a --- /dev/null +++ b/theories/Reals/Ranalysis3.v @@ -0,0 +1,793 @@ +(************************************************************************) +(* 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: Ranalysis3.v,v 1.10.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import Ranalysis2. Open Local Scope R_scope. + +(* Division *) +Theorem derivable_pt_lim_div : + forall (f1 f2:R -> R) (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> + f2 x <> 0 -> + derivable_pt_lim (f1 / f2) x ((l1 * f2 x - l2 * f1 x) / Rsqr (f2 x)). +intros. +cut (derivable_pt f2 x); + [ intro | unfold derivable_pt in |- *; apply existT with l2; exact H0 ]. +assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1). +elim H2; clear H2; intros eps_f2 H2. +unfold div_fct in |- *. +assert (H3 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3; + unfold limit_in in H3; unfold dist in H3. +simpl in H3; unfold R_dist in H3. +elim (H3 (Rabs (f2 x) / 2)); + [ idtac + | unfold Rdiv in |- *; change (0 < Rabs (f2 x) * / 2) in |- *; + apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +clear H3; intros alp_f2 H3. +cut + (forall x0:R, + Rabs (x0 - x) < alp_f2 -> Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2). +intro H4. +cut (forall a:R, Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)). +intro H5. +cut + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)). +intro Maj. +unfold derivable_pt_lim in |- *; intros. +elim (H (Rabs (eps * f2 x / 8))); + [ idtac + | unfold Rdiv in |- *; change (0 < Rabs (eps * f2 x * / 8)) in |- *; + apply Rabs_pos_lt; repeat apply prod_neq_R0; + [ red in |- *; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6) + | assumption + | apply Rinv_neq_0_compat; discrR ] ]. +intros alp_f1d H7. +case (Req_dec (f1 x) 0); intro. +case (Req_dec l1 0); intro. +(***********************************) +(* Cas n° 1 *) +(* (f1 x)=0 l1 =0 *) +(***********************************) +cut (0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)); + [ intro + | repeat apply Rmin_pos; + [ apply (cond_pos eps_f2) + | elim H3; intros; assumption + | apply (cond_pos alp_f1d) ] ]. +exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10). +simpl in |- *; intros. +assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)). +assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)). +assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)). +assert (H16 := Rlt_le_trans _ _ _ H13 (Rmin_l _ _)). +assert (H17 := H7 _ H11 H15). +rewrite formule; [ idtac | assumption | assumption | apply H2; apply H14 ]. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); + try assumption || apply H2. +apply H14. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +(***********************************) +(* Cas n° 2 *) +(* (f1 x)=0 l1<>0 *) +(***********************************) +assert (H10 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H10. +unfold continue_in in H10. +unfold limit1_in in H10. +unfold limit_in in H10. +unfold dist in H10. +simpl in H10. +unfold R_dist in H10. +elim (H10 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +clear H10; intros alp_f2t2 H10. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +intro H11. +cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)). +intro. +exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12). +simpl in |- *. +intros. +assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). +assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +clear H14 H15 H16. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +apply (cond_pos alp_f1d). +elim H3; intros; assumption. +elim H10; intros; assumption. +intros. +elim H10; intros. +case (Req_dec a 0); intro. +rewrite H14; rewrite Rplus_0_r. +unfold Rminus in |- *; rewrite Rplus_opp_r. +rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption. +apply H13. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. +apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + repeat apply prod_neq_R0. +red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6). +assumption. +assumption. +apply Rinv_neq_0_compat; repeat apply prod_neq_R0; + [ discrR | discrR | discrR | assumption ]. +(***********************************) +(* Cas n° 3 *) +(* (f1 x)<>0 l1=0 l2=0 *) +(***********************************) +case (Req_dec l1 0); intro. +case (Req_dec l2 0); intro. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); + [ idtac + | apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + repeat apply prod_neq_R0; + [ assumption + | assumption + | red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6) + | apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption ] ]. +intros alp_f2d H12. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)). +intro. +exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11). +simpl in |- *. +intros. +assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). +assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +clear H15 H16. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H10. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +(***********************************) +(* Cas n° 4 *) +(* (f1 x)<>0 l1=0 l2<>0 *) +(***********************************) +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); + [ idtac + | apply Rabs_pos_lt; unfold Rsqr, Rdiv in |- *; + repeat rewrite Rinv_mult_distr; repeat apply prod_neq_R0; + try assumption || discrR ]. +intros alp_f2d H11. +assert (H12 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H12. +unfold continue_in in H12. +unfold limit1_in in H12. +unfold limit_in in H12. +unfold dist in H12. +simpl in H12. +unfold R_dist in H12. +elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))). +intros alp_f2c H13. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))). +intro. +exists + (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))) + H14). +simpl in |- *; intros. +assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H24 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +clear H16 H17 H18 H19. +cut + (forall a:R, + Rabs a < alp_f2c -> + Rabs (f2 (x + a) - f2 x) < + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +intro. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite <- Rabs_mult. +apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H17; rewrite Rplus_0_r. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *. +repeat rewrite Rinv_mult_distr; try assumption. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. +apply Rinv_neq_0_compat; assumption. +discrR. +discrR. +discrR. +discrR. +discrR. +apply prod_neq_R0; [ discrR | assumption ]. +elim H13; intros. +apply H19. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H13; intros; assumption. +change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *. +apply Rabs_pos_lt. +unfold Rsqr, Rdiv in |- *. +repeat rewrite Rinv_mult_distr; try assumption || discrR. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. +apply Rinv_neq_0_compat; assumption. +apply prod_neq_R0; [ discrR | assumption ]. +red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. +(***********************************) +(* Cas n° 5 *) +(* (f1 x)<>0 l1<>0 l2=0 *) +(***********************************) +case (Req_dec l2 0); intro. +assert (H11 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H11. +unfold continue_in in H11. +unfold limit1_in in H11. +unfold limit_in in H11. +unfold dist in H11. +simpl in H11. +unfold R_dist in H11. +elim (H11 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +clear H11; intros alp_f2t2 H11. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))). +intros alp_f2d H12. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))). +intro. +exists + (mkposreal + (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13). +simpl in |- *. +intros. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +intro. +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)). +assert (H24 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)). +clear H15 H17 H18 H21. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H10. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H17; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +unfold Rsqr in |- *. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)). +elim H11; intros. +apply H19. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H11; intros; assumption. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). +(***********************************) +(* Cas n° 6 *) +(* (f1 x)<>0 l1<>0 l2<>0 *) +(***********************************) +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))). +intros alp_f2d H11. +assert (H12 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H12. +unfold continue_in in H12. +unfold limit1_in in H12. +unfold limit_in in H12. +unfold dist in H12. +simpl in H12. +unfold R_dist in H12. +elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))). +intros alp_f2c H13. +elim (H12 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +intros alp_f2t2 H14. +cut + (0 < + Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) + (Rmin alp_f2c alp_f2t2)). +intro. +exists + (mkposreal + (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) + (Rmin alp_f2c alp_f2t2)) H15). +simpl in |- *. +intros. +assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)). +assert (H24 := Rlt_le_trans _ _ _ H20 (Rmin_l _ _)). +assert (H25 := Rlt_le_trans _ _ _ H20 (Rmin_r _ _)). +assert (H26 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)). +assert (H27 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)). +clear H17 H18 H19 H20 H21. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +cut + (forall a:R, + Rabs a < alp_f2c -> + Rabs (f2 (x + a) - f2 x) < + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +intros. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite <- Rabs_mult. +apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). +apply prod_neq_R0; [ discrR | assumption ]. +apply prod_neq_R0; [ discrR | assumption ]. +assumption. +elim H13; intros. +apply H20. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +intros. +case (Req_dec a 0); intro. +rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). +discrR. +assumption. +elim H14; intros. +apply H20. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply Rminus_not_eq_right. +replace (x + a - x) with a; [ assumption | ring ]. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H13; intros; assumption. +elim H14; intros; assumption. +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)). +change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *; + apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)). +apply prod_neq_R0; [ discrR | assumption ]. +apply prod_neq_R0; [ discrR | assumption ]. +assumption. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; + [ idtac | discrR | assumption ]. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)). +intros. +unfold Rdiv in |- *. +apply Rmult_lt_reg_l with (Rabs (f2 (x + a))). +apply Rabs_pos_lt; apply H2. +apply Rlt_le_trans with (Rmin eps_f2 alp_f2). +assumption. +apply Rmin_l. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (Rabs (f2 x)). +apply Rabs_pos_lt; assumption. +rewrite Rmult_1_r. +rewrite (Rmult_comm (Rabs (f2 x))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply Rmult_lt_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +repeat rewrite (Rmult_comm (/ 2)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +unfold Rdiv in H5; apply H5. +replace (x + a - x) with a. +assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_r _ _)); assumption. +ring. +discrR. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; apply H2. +assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_l _ _)); assumption. +intros. +assert (H6 := H4 a H5). +rewrite <- (Rabs_Ropp (f2 a - f2 x)) in H6. +rewrite Ropp_minus_distr in H6. +assert (H7 := Rle_lt_trans _ _ _ (Rabs_triang_inv _ _) H6). +apply Rplus_lt_reg_r with (- Rabs (f2 a) + Rabs (f2 x) / 2). +rewrite Rplus_assoc. +rewrite <- double_var. +do 2 rewrite (Rplus_comm (- Rabs (f2 a))). +rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +unfold Rminus in H7; assumption. +intros. +case (Req_dec x x0); intro. +rewrite <- H5; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim H3; intros. +apply H7. +split. +unfold D_x, no_cond in |- *; split. +trivial. +assumption. +assumption. +Qed. + +Lemma derivable_pt_div : + forall (f1 f2:R -> R) (x:R), + derivable_pt f1 x -> + derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x. +unfold derivable_pt in |- *. +intros. +elim X; intros. +elim X0; intros. +apply existT with ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)). +apply derivable_pt_lim_div; assumption. +Qed. + +Lemma derivable_div : + forall f1 f2:R -> R, + derivable f1 -> + derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2). +unfold derivable in |- *; intros. +apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)). +Qed. + +Lemma derive_pt_div : + forall (f1 f2:R -> R) (x:R) (pr1:derivable_pt f1 x) + (pr2:derivable_pt f2 x) (na:f2 x <> 0), + derive_pt (f1 / f2) x (derivable_pt_div _ _ _ pr1 pr2 na) = + (derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) / Rsqr (f2 x). +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_div _ _ _ pr1 pr2 na)). +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_div; assumption. +Qed.
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