diff options
Diffstat (limited to 'theories/Reals/Sqrt_reg.v')
| -rw-r--r-- | theories/Reals/Sqrt_reg.v | 351 |
1 files changed, 351 insertions, 0 deletions
diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v new file mode 100644 index 00000000..b11e51f0 --- /dev/null +++ b/theories/Reals/Sqrt_reg.v @@ -0,0 +1,351 @@ +(************************************************************************) +(* 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: Sqrt_reg.v,v 1.9.2.1 2004/07/16 19:31:15 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import R_sqrt. Open Local Scope R_scope. + +(**********) +Lemma sqrt_var_maj : + forall h:R, Rabs h <= 1 -> Rabs (sqrt (1 + h) - 1) <= Rabs h. +intros; cut (0 <= 1 + h). +intro; apply Rle_trans with (Rabs (sqrt (Rsqr (1 + h)) - 1)). +case (total_order_T h 0); intro. +elim s; intro. +repeat rewrite Rabs_left. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)). +do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive; + apply Rplus_le_compat_l. +apply Ropp_le_contravar; apply sqrt_le_1. +apply Rle_0_sqr. +apply H0. +pattern (1 + h) at 2 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + apply Rmult_le_compat_l. +apply H0. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; + unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r. +pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +apply Rle_0_sqr. +left; apply Rlt_0_1. +pattern 1 at 2 in |- *; rewrite <- Rsqr_1; apply Rsqr_incrst_1. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +apply H0. +left; apply Rlt_0_1. +apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; + unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r. +pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +apply H0. +left; apply Rlt_0_1. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +rewrite b; rewrite Rplus_0_r; rewrite Rsqr_1; rewrite sqrt_1; right; + reflexivity. +repeat rewrite Rabs_right. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)); + apply Rplus_le_compat_l. +apply sqrt_le_1. +apply H0. +apply Rle_0_sqr. +pattern (1 + h) at 1 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + apply Rmult_le_compat_l. +apply H0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +apply Rle_ge; apply Rplus_le_reg_l with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_le_1. +left; apply Rlt_0_1. +apply Rle_0_sqr. +pattern 1 at 1 in |- *; rewrite <- Rsqr_1; apply Rsqr_incr_1. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +left; apply Rlt_0_1. +apply H0. +apply Rle_ge; left; apply Rplus_lt_reg_r with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +left; apply Rlt_0_1. +apply H0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +rewrite sqrt_Rsqr. +replace (1 + h - 1) with h; [ right; reflexivity | ring ]. +apply H0. +case (total_order_T h 0); intro. +elim s; intro. +rewrite (Rabs_left h a) in H. +apply Rplus_le_reg_l with (- h). +rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; exact H. +left; rewrite b; rewrite Rplus_0_r; apply Rlt_0_1. +left; apply Rplus_lt_0_compat. +apply Rlt_0_1. +apply r. +Qed. + +(* sqrt is continuous in 1 *) +Lemma sqrt_continuity_pt_R1 : continuity_pt sqrt 1. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +set (alpha := Rmin eps 1). +exists alpha; intros. +split. +unfold alpha in |- *; unfold Rmin in |- *; case (Rle_dec eps 1); intro. +assumption. +apply Rlt_0_1. +intros; elim H0; intros. +rewrite sqrt_1; replace x with (1 + (x - 1)); [ idtac | ring ]; + apply Rle_lt_trans with (Rabs (x - 1)). +apply sqrt_var_maj. +apply Rle_trans with alpha. +left; apply H2. +unfold alpha in |- *; apply Rmin_r. +apply Rlt_le_trans with alpha; + [ apply H2 | unfold alpha in |- *; apply Rmin_l ]. +Qed. + +(* sqrt is continuous forall x>0 *) +Lemma sqrt_continuity_pt : forall x:R, 0 < x -> continuity_pt sqrt x. +intros; generalize sqrt_continuity_pt_R1. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +cut (0 < eps / sqrt x). +intro; elim (H0 _ H2); intros alp_1 H3. +elim H3; intros. +set (alpha := alp_1 * x). +exists (Rmin alpha x); intros. +split. +change (0 < Rmin alpha x) in |- *; unfold Rmin in |- *; + case (Rle_dec alpha x); intro. +unfold alpha in |- *; apply Rmult_lt_0_compat; assumption. +apply H. +intros; replace x0 with (x + (x0 - x)); [ idtac | ring ]; + replace (sqrt (x + (x0 - x)) - sqrt x) with + (sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1)). +rewrite Rabs_mult; rewrite (Rabs_right (sqrt x)). +apply Rmult_lt_reg_l with (/ sqrt x). +apply Rinv_0_lt_compat; apply sqrt_lt_R0; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite Rmult_comm. +unfold Rdiv in H5. +case (Req_dec x x0); intro. +rewrite H7; unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; + rewrite Rmult_0_l; rewrite Rplus_0_r; rewrite Rplus_opp_r; + rewrite Rabs_R0. +apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; rewrite <- H7; apply sqrt_lt_R0; assumption. +apply H5. +split. +unfold D_x, no_cond in |- *. +split. +trivial. +red in |- *; intro. +cut ((x0 - x) * / x = 0). +intro. +elim (Rmult_integral _ _ H9); intro. +elim H7. +apply (Rminus_diag_uniq_sym _ _ H10). +assert (H11 := Rmult_eq_0_compat_r _ x H10). +rewrite <- Rinv_l_sym in H11. +elim R1_neq_R0; exact H11. +red in |- *; intro; rewrite H12 in H; elim (Rlt_irrefl _ H). +symmetry in |- *; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r; + unfold Rdiv in H8; exact H8. +unfold Rminus in |- *; rewrite Rplus_comm; rewrite <- Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_l; elim H6; intros. +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite Rabs_Rinv. +rewrite (Rabs_right x). +rewrite Rmult_comm; apply Rmult_lt_reg_l with x. +apply H. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha in |- *. +apply Rlt_le_trans with (Rmin alpha x). +apply H9. +apply Rmin_l. +red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). +apply Rle_ge; left; apply H. +red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). +assert (H7 := sqrt_lt_R0 x H). +red in |- *; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7). +apply Rle_ge; apply sqrt_positivity. +left; apply H. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l; + rewrite Ropp_mult_distr_r_reverse; repeat rewrite <- sqrt_mult. +rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + unfold Rdiv in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +red in |- *; intro; rewrite H7 in H; elim (Rlt_irrefl _ H). +left; apply H. +left; apply Rlt_0_1. +left; apply H. +elim H6; intros. +case (Rcase_abs (x0 - x)); intro. +rewrite (Rabs_left (x0 - x) r) in H8. +rewrite Rplus_comm. +apply Rplus_le_reg_l with (- ((x0 - x) / x)). +rewrite Rplus_0_r; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_le_reg_l with x. +apply H. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; left; apply Rlt_le_trans with (Rmin alpha x). +apply H8. +apply Rmin_r. +red in |- *; intro; rewrite H9 in H; elim (Rlt_irrefl _ H). +apply Rplus_le_le_0_compat. +left; apply Rlt_0_1. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply Rge_le; exact r. +left; apply Rinv_0_lt_compat; apply H. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply sqrt_lt_R0; apply H. +Qed. + +(* sqrt is derivable for all x>0 *) +Lemma derivable_pt_lim_sqrt : + forall x:R, 0 < x -> derivable_pt_lim sqrt x (/ (2 * sqrt x)). +intros; set (g := fun h:R => sqrt x + sqrt (x + h)). +cut (continuity_pt g 0). +intro; cut (g 0 <> 0). +intro; assert (H2 := continuity_pt_inv g 0 H0 H1). +unfold derivable_pt_lim in |- *; intros; unfold continuity_pt in H2; + unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 eps H3); intros alpha H4. +elim H4; intros. +set (alpha1 := Rmin alpha x). +cut (0 < alpha1). +intro; exists (mkposreal alpha1 H7); intros. +replace ((sqrt (x + h) - sqrt x) / h) with (/ (sqrt x + sqrt (x + h))). +unfold inv_fct, g in H6; replace (2 * sqrt x) with (sqrt x + sqrt (x + 0)). +apply H6. +split. +unfold D_x, no_cond in |- *. +split. +trivial. +apply (sym_not_eq (A:=R)); exact H8. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + apply Rlt_le_trans with alpha1. +exact H9. +unfold alpha1 in |- *; apply Rmin_l. +rewrite Rplus_0_r; ring. +cut (0 <= x + h). +intro; cut (0 < sqrt x + sqrt (x + h)). +intro; apply Rmult_eq_reg_l with (sqrt x + sqrt (x + h)). +rewrite <- Rinv_r_sym. +rewrite Rplus_comm; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rsqr_plus_minus; repeat rewrite Rsqr_sqrt. +rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym. +reflexivity. +apply H8. +left; apply H. +assumption. +red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). +red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). +apply Rplus_lt_le_0_compat. +apply sqrt_lt_R0; apply H. +apply sqrt_positivity; apply H10. +case (Rcase_abs h); intro. +rewrite (Rabs_left h r) in H9. +apply Rplus_le_reg_l with (- h). +rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; left; apply Rlt_le_trans with alpha1. +apply H9. +unfold alpha1 in |- *; apply Rmin_r. +apply Rplus_le_le_0_compat. +left; assumption. +apply Rge_le; apply r. +unfold alpha1 in |- *; unfold Rmin in |- *; case (Rle_dec alpha x); intro. +apply H5. +apply H. +unfold g in |- *; rewrite Rplus_0_r. +cut (0 < sqrt x + sqrt x). +intro; red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). +apply Rplus_lt_0_compat; apply sqrt_lt_R0; apply H. +replace g with (fct_cte (sqrt x) + comp sqrt (fct_cte x + id))%F; + [ idtac | reflexivity ]. +apply continuity_pt_plus. +apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + reflexivity. +apply continuity_pt_comp. +apply continuity_pt_plus. +apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + reflexivity. +apply derivable_continuous_pt; apply derivable_pt_id. +apply sqrt_continuity_pt. +unfold plus_fct, fct_cte, id in |- *; rewrite Rplus_0_r; apply H. +Qed. + +(**********) +Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x. +unfold derivable_pt in |- *; intros. +apply existT with (/ (2 * sqrt x)). +apply derivable_pt_lim_sqrt; assumption. +Qed. + +(**********) +Lemma derive_pt_sqrt : + forall (x:R) (pr:0 < x), + derive_pt sqrt x (derivable_pt_sqrt _ pr) = / (2 * sqrt x). +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_sqrt; assumption. +Qed. + +(* We show that sqrt is continuous for all x>=0 *) +(* Remark : by definition of sqrt (as extension of Rsqrt on |R), *) +(* we could also show that sqrt is continuous for all x *) +Lemma continuity_pt_sqrt : forall x:R, 0 <= x -> continuity_pt sqrt x. +intros; case (Rtotal_order 0 x); intro. +apply (sqrt_continuity_pt x H0). +elim H0; intro. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +exists (Rsqr eps); intros. +split. +change (0 < Rsqr eps) in |- *; apply Rsqr_pos_lt. +red in |- *; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2). +intros; elim H3; intros. +rewrite <- H1; rewrite sqrt_0; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5; + rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5. +case (Rcase_abs x0); intro. +unfold sqrt in |- *; case (Rcase_abs x0); intro. +rewrite Rabs_R0; apply H2. +assert (H6 := Rge_le _ _ r0); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 r)). +rewrite Rabs_right. +apply Rsqr_incrst_0. +rewrite Rsqr_sqrt. +rewrite (Rabs_right x0 r) in H5; apply H5. +apply Rge_le; exact r. +apply sqrt_positivity; apply Rge_le; exact r. +left; exact H2. +apply Rle_ge; apply sqrt_positivity; apply Rge_le; exact r. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H1 H)). +Qed.
\ No newline at end of file |