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Diffstat (limited to 'theories/Reals/Rpower.v')
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diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v new file mode 100644 index 00000000..7575d929 --- /dev/null +++ b/theories/Reals/Rpower.v @@ -0,0 +1,661 @@ +(************************************************************************) +(* 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: Rpower.v,v 1.17.2.1 2004/07/16 19:31:13 herbelin Exp $ i*) +(*i Due to L.Thery i*) + +(************************************************************) +(* Definitions of log and Rpower : R->R->R; main properties *) +(************************************************************) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import Exp_prop. +Require Import Rsqrt_def. +Require Import R_sqrt. +Require Import MVT. +Require Import Ranalysis4. Open Local Scope R_scope. + +Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y). +intros P x y H1 H2; unfold Rmin in |- *; case (Rle_dec x y); intro; + assumption. +Qed. + +Lemma exp_le_3 : exp 1 <= 3. +assert (exp_1 : exp 1 <> 0). +assert (H0 := exp_pos 1); red in |- *; intro; rewrite H in H0; + elim (Rlt_irrefl _ H0). +apply Rmult_le_reg_l with (/ exp 1). +apply Rinv_0_lt_compat; apply exp_pos. +rewrite <- Rinv_l_sym. +apply Rmult_le_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)). +unfold exp in |- *; case (exist_exp (-1)); intros; simpl in |- *; + unfold exp_in in e; + assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1). +cut + (sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (S (2 * 1)) <= x <= + sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (2 * 1)). +intro; elim H0; clear H0; intros H0 _; simpl in H0; unfold tg_alt in H0; + simpl in H0. +replace (/ 3) with + (1 * / 1 + -1 * 1 * / 1 + -1 * (-1 * 1) * / 2 + + -1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1)). +apply H0. +repeat rewrite Rinv_1; repeat rewrite Rmult_1_r; + rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; + rewrite Ropp_involutive; rewrite Rplus_opp_r; rewrite Rmult_1_r; + rewrite Rplus_0_l; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 6. +rewrite Rmult_plus_distr_l; replace (2 + 1 + 1 + 1 + 1) with 6. +rewrite <- (Rmult_comm (/ 6)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; replace 6 with 6. +do 2 rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm 3); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +ring. +discrR. +ring. +discrR. +apply H. +unfold Un_decreasing in |- *; intros; + apply Rmult_le_reg_l with (INR (fact n)). +apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S n))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply le_INR; apply fact_le; apply le_n_Sn. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +assert (H0 := cv_speed_pow_fact 1); unfold Un_cv in |- *; unfold Un_cv in H0; + intros; elim (H0 _ H1); intros; exists x0; intros; + unfold R_dist in H2; unfold R_dist in |- *; + replace (/ INR (fact n)) with (1 ^ n / INR (fact n)). +apply (H2 _ H3). +unfold Rdiv in |- *; rewrite pow1; rewrite Rmult_1_l; reflexivity. +unfold infinit_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0); + intros; exists x0; intros; + replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with + (sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n). +apply (H1 _ H2). +apply sum_eq; intros; apply Rmult_comm. +apply Rmult_eq_reg_l with (exp 1). +rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0; + rewrite <- Rinv_r_sym. +reflexivity. +assumption. +assumption. +discrR. +assumption. +Qed. + +(******************************************************************) +(* Properties of Exp *) +(******************************************************************) + +Theorem exp_increasing : forall x y:R, x < y -> exp x < exp y. +intros x y H. +assert (H0 : derivable exp). +apply derivable_exp. +assert (H1 := positive_derivative _ H0). +unfold strict_increasing in H1. +apply H1. +intro. +replace (derive_pt exp x0 (H0 x0)) with (exp x0). +apply exp_pos. +symmetry in |- *; apply derive_pt_eq_0. +apply (derivable_pt_lim_exp x0). +apply H. +Qed. + +Theorem exp_lt_inv : forall x y:R, exp x < exp y -> x < y. +intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]. +assumption. +rewrite H1 in H; elim (Rlt_irrefl _ H). +assert (H2 := exp_increasing _ _ H1). +elim (Rlt_irrefl _ (Rlt_trans _ _ _ H H2)). +Qed. + +Lemma exp_ineq1 : forall x:R, 0 < x -> 1 + x < exp x. +intros; apply Rplus_lt_reg_r with (- exp 0); rewrite <- (Rplus_comm (exp x)); + assert (H0 := MVT_cor1 exp 0 x derivable_exp H); elim H0; + intros; elim H1; intros; unfold Rminus in H2; rewrite H2; + rewrite Ropp_0; rewrite Rplus_0_r; + replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0). +rewrite exp_0; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + pattern x at 1 in |- *; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0)); + apply Rmult_lt_compat_l. +apply H. +rewrite <- exp_0; apply exp_increasing; elim H3; intros; assumption. +symmetry in |- *; apply derive_pt_eq_0; apply derivable_pt_lim_exp. +Qed. + +Lemma ln_exists1 : forall y:R, 0 < y -> 1 <= y -> sigT (fun z:R => y = exp z). +intros; set (f := fun x:R => exp x - y); cut (f 0 <= 0). +intro; cut (continuity f). +intro; cut (0 <= f y). +intro; cut (f 0 * f y <= 0). +intro; assert (X := IVT_cor f 0 y H2 (Rlt_le _ _ H) H4); elim X; intros t H5; + apply existT with t; elim H5; intros; unfold f in H7; + apply Rminus_diag_uniq_sym; exact H7. +pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)); + rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l; + assumption. +unfold f in |- *; apply Rplus_le_reg_l with y; left; + apply Rlt_trans with (1 + y). +rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1. +replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H) | ring ]. +unfold f in |- *; change (continuity (exp - fct_cte y)) in |- *; + apply continuity_minus; + [ apply derivable_continuous; apply derivable_exp + | apply derivable_continuous; apply derivable_const ]. +unfold f in |- *; rewrite exp_0; apply Rplus_le_reg_l with y; + rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H0 | ring ]. +Qed. + +(**********) +Lemma ln_exists : forall y:R, 0 < y -> sigT (fun z:R => y = exp z). +intros; case (Rle_dec 1 y); intro. +apply (ln_exists1 _ H r). +assert (H0 : 1 <= / y). +apply Rmult_le_reg_l with y. +apply H. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n). +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +assert (H1 : 0 < / y). +apply Rinv_0_lt_compat; apply H. +assert (H2 := ln_exists1 _ H1 H0); elim H2; intros; apply existT with (- x); + apply Rmult_eq_reg_l with (exp x / y). +unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc; + rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0; + rewrite Rmult_1_r; symmetry in |- *; apply p. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +unfold Rdiv in |- *; apply prod_neq_R0. +assert (H3 := exp_pos x); red in |- *; intro; rewrite H4 in H3; + elim (Rlt_irrefl _ H3). +apply Rinv_neq_0_compat; red in |- *; intro; rewrite H3 in H; + elim (Rlt_irrefl _ H). +Qed. + +(* Definition of log R+* -> R *) +Definition Rln (y:posreal) : R := + match ln_exists (pos y) (cond_pos y) with + | existT a b => a + end. + +(* Extension on R *) +Definition ln (x:R) : R := + match Rlt_dec 0 x with + | left a => Rln (mkposreal x a) + | right a => 0 + end. + +Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x. +intros; unfold ln in |- *; case (Rlt_dec 0 x); intro. +unfold Rln in |- *; + case (ln_exists (mkposreal x r) (cond_pos (mkposreal x r))); + intros. +simpl in e; symmetry in |- *; apply e. +elim n; apply H. +Qed. + +Theorem exp_inv : forall x y:R, exp x = exp y -> x = y. +intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]; auto; + assert (H2 := exp_increasing _ _ H1); rewrite H in H2; + elim (Rlt_irrefl _ H2). +Qed. + +Theorem exp_Ropp : forall x:R, exp (- x) = / exp x. +intros x; assert (H : exp x <> 0). +assert (H := exp_pos x); red in |- *; intro; rewrite H0 in H; + elim (Rlt_irrefl _ H). +apply Rmult_eq_reg_l with (r := exp x). +rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0. +apply Rinv_r_sym. +apply H. +apply H. +Qed. + +(******************************************************************) +(* Properties of Ln *) +(******************************************************************) + +Theorem ln_increasing : forall x y:R, 0 < x -> x < y -> ln x < ln y. +intros x y H H0; apply exp_lt_inv. +repeat rewrite exp_ln. +apply H0. +apply Rlt_trans with x; assumption. +apply H. +Qed. + +Theorem ln_exp : forall x:R, ln (exp x) = x. +intros x; apply exp_inv. +apply exp_ln. +apply exp_pos. +Qed. + +Theorem ln_1 : ln 1 = 0. +rewrite <- exp_0; rewrite ln_exp; reflexivity. +Qed. + +Theorem ln_lt_inv : forall x y:R, 0 < x -> 0 < y -> ln x < ln y -> x < y. +intros x y H H0 H1; rewrite <- (exp_ln x); try rewrite <- (exp_ln y). +apply exp_increasing; apply H1. +assumption. +assumption. +Qed. + +Theorem ln_inv : forall x y:R, 0 < x -> 0 < y -> ln x = ln y -> x = y. +intros x y H H0 H'0; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]; + auto. +assert (H2 := ln_increasing _ _ H H1); rewrite H'0 in H2; + elim (Rlt_irrefl _ H2). +assert (H2 := ln_increasing _ _ H0 H1); rewrite H'0 in H2; + elim (Rlt_irrefl _ H2). +Qed. + +Theorem ln_mult : forall x y:R, 0 < x -> 0 < y -> ln (x * y) = ln x + ln y. +intros x y H H0; apply exp_inv. +rewrite exp_plus. +repeat rewrite exp_ln. +reflexivity. +assumption. +assumption. +apply Rmult_lt_0_compat; assumption. +Qed. + +Theorem ln_Rinv : forall x:R, 0 < x -> ln (/ x) = - ln x. +intros x H; apply exp_inv; repeat rewrite exp_ln || rewrite exp_Ropp. +reflexivity. +assumption. +apply Rinv_0_lt_compat; assumption. +Qed. + +Theorem ln_continue : + forall y:R, 0 < y -> continue_in ln (fun x:R => 0 < x) y. +intros y H. +unfold continue_in, limit1_in, limit_in in |- *; intros eps Heps. +cut (1 < exp eps); [ intros H1 | idtac ]. +cut (exp (- eps) < 1); [ intros H2 | idtac ]. +exists (Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))); split. +red in |- *; apply P_Rmin. +apply Rmult_lt_0_compat. +assumption. +apply Rplus_lt_reg_r with 1. +rewrite Rplus_0_r; replace (1 + (exp eps - 1)) with (exp eps); + [ apply H1 | ring ]. +apply Rmult_lt_0_compat. +assumption. +apply Rplus_lt_reg_r with (exp (- eps)). +rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1; + [ apply H2 | ring ]. +unfold dist, R_met, R_dist in |- *; simpl in |- *. +intros x [[H3 H4] H5]. +cut (y * (x * / y) = x). +intro Hxyy. +replace (ln x - ln y) with (ln (x * / y)). +case (Rtotal_order x y); [ intros Hxy | intros [Hxy| Hxy] ]. +rewrite Rabs_left. +apply Ropp_lt_cancel; rewrite Ropp_involutive. +apply exp_lt_inv. +rewrite exp_ln. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy. +apply Ropp_lt_cancel. +apply Rplus_lt_reg_r with (r := y). +replace (y + - (y * exp (- eps))) with (y * (1 - exp (- eps))); + [ idtac | ring ]. +replace (y + - x) with (Rabs (x - y)); [ idtac | ring ]. +apply Rlt_le_trans with (1 := H5); apply Rmin_r. +rewrite Rabs_left; [ ring | idtac ]. +apply (Rlt_minus _ _ Hxy). +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +rewrite <- ln_1. +apply ln_increasing. +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy; rewrite Rmult_1_r; apply Hxy. +rewrite Hxy; rewrite Rinv_r. +rewrite ln_1; rewrite Rabs_R0; apply Heps. +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +rewrite Rabs_right. +apply exp_lt_inv. +rewrite exp_ln. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy. +apply Rplus_lt_reg_r with (r := - y). +replace (- y + y * exp eps) with (y * (exp eps - 1)); [ idtac | ring ]. +replace (- y + x) with (Rabs (x - y)); [ idtac | ring ]. +apply Rlt_le_trans with (1 := H5); apply Rmin_l. +rewrite Rabs_right; [ ring | idtac ]. +left; apply (Rgt_minus _ _ Hxy). +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +rewrite <- ln_1. +apply Rgt_ge; red in |- *; apply ln_increasing. +apply Rlt_0_1. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy; rewrite Rmult_1_r; apply Hxy. +rewrite ln_mult. +rewrite ln_Rinv. +ring. +assumption. +assumption. +apply Rinv_0_lt_compat; assumption. +rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +ring. +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +apply Rmult_lt_reg_l with (exp eps). +apply exp_pos. +rewrite <- exp_plus; rewrite Rmult_1_r; rewrite Rplus_opp_r; rewrite exp_0; + apply H1. +rewrite <- exp_0. +apply exp_increasing; apply Heps. +Qed. + +(******************************************************************) +(* Definition of Rpower *) +(******************************************************************) + +Definition Rpower (x y:R) := exp (y * ln x). + +Infix Local "^R" := Rpower (at level 30, right associativity) : R_scope. + +(******************************************************************) +(* Properties of Rpower *) +(******************************************************************) + +Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y. +intros x y z; unfold Rpower in |- *. +rewrite Rmult_plus_distr_r; rewrite exp_plus; auto. +Qed. + +Theorem Rpower_mult : forall x y z:R, (x ^R y) ^R z = x ^R (y * z). +intros x y z; unfold Rpower in |- *. +rewrite ln_exp. +replace (z * (y * ln x)) with (y * z * ln x). +reflexivity. +ring. +Qed. + +Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1. +intros x H; unfold Rpower in |- *. +rewrite Rmult_0_l; apply exp_0. +Qed. + +Theorem Rpower_1 : forall x:R, 0 < x -> x ^R 1 = x. +intros x H; unfold Rpower in |- *. +rewrite Rmult_1_l; apply exp_ln; apply H. +Qed. + +Theorem Rpower_pow : forall (n:nat) (x:R), 0 < x -> x ^R INR n = x ^ n. +intros n; elim n; simpl in |- *; auto; fold INR in |- *. +intros x H; apply Rpower_O; auto. +intros n1; case n1. +intros H x H0; simpl in |- *; rewrite Rmult_1_r; apply Rpower_1; auto. +intros n0 H x H0; rewrite Rpower_plus; rewrite H; try rewrite Rpower_1; + try apply Rmult_comm || assumption. +Qed. + +Theorem Rpower_lt : + forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z. +intros x y z H H0 H1. +unfold Rpower in |- *. +apply exp_increasing. +apply Rmult_lt_compat_r. +rewrite <- ln_1; apply ln_increasing. +apply Rlt_0_1. +apply H. +apply H1. +Qed. + +Theorem Rpower_sqrt : forall x:R, 0 < x -> x ^R (/ 2) = sqrt x. +intros x H. +apply ln_inv. +unfold Rpower in |- *; apply exp_pos. +apply sqrt_lt_R0; apply H. +apply Rmult_eq_reg_l with (INR 2). +apply exp_inv. +fold Rpower in |- *. +cut ((x ^R (/ 2)) ^R INR 2 = sqrt x ^R INR 2). +unfold Rpower in |- *; auto. +rewrite Rpower_mult. +rewrite Rinv_l. +replace 1 with (INR 1); auto. +repeat rewrite Rpower_pow; simpl in |- *. +pattern x at 1 in |- *; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)). +ring. +apply sqrt_lt_R0; apply H. +apply H. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +Qed. + +Theorem Rpower_Ropp : forall x y:R, x ^R (- y) = / x ^R y. +unfold Rpower in |- *. +intros x y; rewrite Ropp_mult_distr_l_reverse. +apply exp_Ropp. +Qed. + +Theorem Rle_Rpower : + forall e n m:R, 1 < e -> 0 <= n -> n <= m -> e ^R n <= e ^R m. +intros e n m H H0 H1; case H1. +intros H2; left; apply Rpower_lt; assumption. +intros H2; rewrite H2; right; reflexivity. +Qed. + +Theorem ln_lt_2 : / 2 < ln 2. +apply Rmult_lt_reg_l with (r := 2). +prove_sup0. +rewrite Rinv_r. +apply exp_lt_inv. +apply Rle_lt_trans with (1 := exp_le_3). +change (3 < 2 ^R 2) in |- *. +repeat rewrite Rpower_plus; repeat rewrite Rpower_1. +repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; + repeat rewrite Rmult_1_l. +pattern 3 at 1 in |- *; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1); + [ apply Rplus_lt_compat_l; apply Rlt_0_1 | ring ]. +prove_sup0. +discrR. +Qed. + +(**************************************) +(* Differentiability of Ln and Rpower *) +(**************************************) + +Theorem limit1_ext : + forall (f g:R -> R) (D:R -> Prop) (l x:R), + (forall x:R, D x -> f x = g x) -> limit1_in f D l x -> limit1_in g D l x. +intros f g D l x H; unfold limit1_in, limit_in in |- *. +intros H0 eps H1; case (H0 eps); auto. +intros x0 [H2 H3]; exists x0; split; auto. +intros x1 [H4 H5]; rewrite <- H; auto. +Qed. + +Theorem limit1_imp : + forall (f:R -> R) (D D1:R -> Prop) (l x:R), + (forall x:R, D1 x -> D x) -> limit1_in f D l x -> limit1_in f D1 l x. +intros f D D1 l x H; unfold limit1_in, limit_in in |- *. +intros H0 eps H1; case (H0 eps H1); auto. +intros alpha [H2 H3]; exists alpha; split; auto. +intros d [H4 H5]; apply H3; split; auto. +Qed. + +Theorem Rinv_Rdiv : forall x y:R, x <> 0 -> y <> 0 -> / (x / y) = y / x. +intros x y H1 H2; unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +assumption. +assumption. +apply Rinv_neq_0_compat; assumption. +Qed. + +Theorem Dln : forall y:R, 0 < y -> D_in ln Rinv (fun x:R => 0 < x) y. +intros y Hy; unfold D_in in |- *. +apply limit1_ext with + (f := fun x:R => / ((exp (ln x) - exp (ln y)) / (ln x - ln y))). +intros x [HD1 HD2]; repeat rewrite exp_ln. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +apply Rminus_eq_contra. +red in |- *; intros H2; case HD2. +symmetry in |- *; apply (ln_inv _ _ HD1 Hy H2). +apply Rminus_eq_contra; apply (sym_not_eq HD2). +apply Rinv_neq_0_compat; apply Rminus_eq_contra; red in |- *; intros H2; + case HD2; apply ln_inv; auto. +assumption. +assumption. +apply limit_inv with + (f := fun x:R => (exp (ln x) - exp (ln y)) / (ln x - ln y)). +apply limit1_imp with + (f := fun x:R => (fun x:R => (exp x - exp (ln y)) / (x - ln y)) (ln x)) + (D := Dgf (D_x (fun x:R => 0 < x) y) (D_x (fun x:R => True) (ln y)) ln). +intros x [H1 H2]; split. +split; auto. +split; auto. +red in |- *; intros H3; case H2; apply ln_inv; auto. +apply limit_comp with + (l := ln y) (g := fun x:R => (exp x - exp (ln y)) / (x - ln y)) (f := ln). +apply ln_continue; auto. +assert (H0 := derivable_pt_lim_exp (ln y)); unfold derivable_pt_lim in H0; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; elim (H0 _ H); + intros; exists (pos x); split. +apply (cond_pos x). +intros; pattern y at 3 in |- *; rewrite <- exp_ln. +pattern x0 at 1 in |- *; replace x0 with (ln y + (x0 - ln y)); + [ idtac | ring ]. +apply H1. +elim H2; intros H3 _; unfold D_x in H3; elim H3; clear H3; intros _ H3; + apply Rminus_eq_contra; apply (sym_not_eq (A:=R)); + apply H3. +elim H2; clear H2; intros _ H2; apply H2. +assumption. +red in |- *; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy). +Qed. + +Lemma derivable_pt_lim_ln : forall x:R, 0 < x -> derivable_pt_lim ln x (/ x). +intros; assert (H0 := Dln x H); unfold D_in in H0; unfold limit1_in in H0; + unfold limit_in in H0; simpl in H0; unfold R_dist in H0; + unfold derivable_pt_lim in |- *; intros; elim (H0 _ H1); + intros; elim H2; clear H2; intros; set (alp := Rmin x0 (x / 2)); + assert (H4 : 0 < alp). +unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec x0 (x / 2)); intro. +apply H2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +exists (mkposreal _ H4); intros; pattern h at 2 in |- *; + replace h with (x + h - x); [ idtac | ring ]. +apply H3; split. +unfold D_x in |- *; split. +case (Rcase_abs h); intro. +assert (H7 : Rabs h < x / 2). +apply Rlt_le_trans with alp. +apply H6. +unfold alp in |- *; apply Rmin_r. +apply Rlt_trans with (x / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +rewrite Rabs_left in H7. +apply Rplus_lt_reg_r with (- h - x / 2). +replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ]. +pattern x at 2 in |- *; rewrite double_var. +replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ]. +apply r. +apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply r ]. +apply (sym_not_eq (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h; + [ apply H5 | ring ]. +replace (x + h - x) with h; + [ apply Rlt_le_trans with alp; + [ apply H6 | unfold alp in |- *; apply Rmin_l ] + | ring ]. +Qed. + +Theorem D_in_imp : + forall (f g:R -> R) (D D1:R -> Prop) (x:R), + (forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x. +intros f g D D1 x H; unfold D_in in |- *. +intros H0; apply limit1_imp with (D := D_x D x); auto. +intros x1 [H1 H2]; split; auto. +Qed. + +Theorem D_in_ext : + forall (f g h:R -> R) (D:R -> Prop) (x:R), + f x = g x -> D_in h f D x -> D_in h g D x. +intros f g h D x H; unfold D_in in |- *. +rewrite H; auto. +Qed. + +Theorem Dpower : + forall y z:R, + 0 < y -> + D_in (fun x:R => x ^R z) (fun x:R => z * x ^R (z - 1)) ( + fun x:R => 0 < x) y. +intros y z H; + apply D_in_imp with (D := Dgf (fun x:R => 0 < x) (fun x:R => True) ln). +intros x H0; repeat split. +assumption. +apply D_in_ext with (f := fun x:R => / x * (z * exp (z * ln x))). +unfold Rminus in |- *; rewrite Rpower_plus; rewrite Rpower_Ropp; + rewrite (Rpower_1 _ H); ring. +apply Dcomp with + (f := ln) + (g := fun x:R => exp (z * x)) + (df := Rinv) + (dg := fun x:R => z * exp (z * x)). +apply (Dln _ H). +apply D_in_imp with + (D := Dgf (fun x:R => True) (fun x:R => True) (fun x:R => z * x)). +intros x H1; repeat split; auto. +apply + (Dcomp (fun _:R => True) (fun _:R => True) (fun x => z) exp + (fun x:R => z * x) exp); simpl in |- *. +apply D_in_ext with (f := fun x:R => z * 1). +apply Rmult_1_r. +apply (Dmult_const (fun x => True) (fun x => x) (fun x => 1)); apply Dx. +assert (H0 := derivable_pt_lim_D_in exp exp (z * ln y)); elim H0; clear H0; + intros _ H0; apply H0; apply derivable_pt_lim_exp. +Qed. + +Theorem derivable_pt_lim_power : + forall x y:R, + 0 < x -> derivable_pt_lim (fun x => x ^R y) x (y * x ^R (y - 1)). +intros x y H. +unfold Rminus in |- *; rewrite Rpower_plus. +rewrite Rpower_Ropp. +rewrite Rpower_1; auto. +rewrite <- Rmult_assoc. +unfold Rpower in |- *. +apply derivable_pt_lim_comp with (f1 := ln) (f2 := fun x => exp (y * x)). +apply derivable_pt_lim_ln; assumption. +rewrite (Rmult_comm y). +apply derivable_pt_lim_comp with (f1 := fun x => y * x) (f2 := exp). +pattern y at 2 in |- *; replace y with (0 * ln x + y * 1). +apply derivable_pt_lim_mult with (f1 := fun x:R => y) (f2 := fun x:R => x). +apply derivable_pt_lim_const with (a := y). +apply derivable_pt_lim_id. +ring. +apply derivable_pt_lim_exp. +Qed.
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