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Diffstat (limited to 'theories/Reals/R_sqrt.v')
| -rw-r--r-- | theories/Reals/R_sqrt.v | 399 |
1 files changed, 399 insertions, 0 deletions
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v new file mode 100644 index 00000000..660b0527 --- /dev/null +++ b/theories/Reals/R_sqrt.v @@ -0,0 +1,399 @@ +(************************************************************************) +(* 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: R_sqrt.v,v 1.10.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Rsqrt_def. Open Local Scope R_scope. + +(* Here is a continuous extension of Rsqrt on R *) +Definition sqrt (x:R) : R := + match Rcase_abs x with + | left _ => 0 + | right a => Rsqrt (mknonnegreal x (Rge_le _ _ a)) + end. + +Lemma sqrt_positivity : forall x:R, 0 <= x -> 0 <= sqrt x. +intros. +unfold sqrt in |- *. +case (Rcase_abs x); intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). +apply Rsqrt_positivity. +Qed. + +Lemma sqrt_sqrt : forall x:R, 0 <= x -> sqrt x * sqrt x = x. +intros. +unfold sqrt in |- *. +case (Rcase_abs x); intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). +rewrite Rsqrt_Rsqrt; reflexivity. +Qed. + +Lemma sqrt_0 : sqrt 0 = 0. +apply Rsqr_eq_0; unfold Rsqr in |- *; apply sqrt_sqrt; right; reflexivity. +Qed. + +Lemma sqrt_1 : sqrt 1 = 1. +apply (Rsqr_inj (sqrt 1) 1); + [ apply sqrt_positivity; left + | left + | unfold Rsqr in |- *; rewrite sqrt_sqrt; [ ring | left ] ]; + apply Rlt_0_1. +Qed. + +Lemma sqrt_eq_0 : forall x:R, 0 <= x -> sqrt x = 0 -> x = 0. +intros; cut (Rsqr (sqrt x) = 0). +intro; unfold Rsqr in H1; rewrite sqrt_sqrt in H1; assumption. +rewrite H0; apply Rsqr_0. +Qed. + +Lemma sqrt_lem_0 : forall x y:R, 0 <= x -> 0 <= y -> sqrt x = y -> y * y = x. +intros; rewrite <- H1; apply (sqrt_sqrt x H). +Qed. + +Lemma sqtr_lem_1 : forall x y:R, 0 <= x -> 0 <= y -> y * y = x -> sqrt x = y. +intros; apply Rsqr_inj; + [ apply (sqrt_positivity x H) + | assumption + | unfold Rsqr in |- *; rewrite H1; apply (sqrt_sqrt x H) ]. +Qed. + +Lemma sqrt_def : forall x:R, 0 <= x -> sqrt x * sqrt x = x. +intros; apply (sqrt_sqrt x H). +Qed. + +Lemma sqrt_square : forall x:R, 0 <= x -> sqrt (x * x) = x. +intros; + apply + (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (Rle_0_sqr x)) H); + unfold Rsqr in |- *; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)). +Qed. + +Lemma sqrt_Rsqr : forall x:R, 0 <= x -> sqrt (Rsqr x) = x. +intros; unfold Rsqr in |- *; apply sqrt_square; assumption. +Qed. + +Lemma sqrt_Rsqr_abs : forall x:R, sqrt (Rsqr x) = Rabs x. +intro x; rewrite Rsqr_abs; apply sqrt_Rsqr; apply Rabs_pos. +Qed. + +Lemma Rsqr_sqrt : forall x:R, 0 <= x -> Rsqr (sqrt x) = x. +intros x H1; unfold Rsqr in |- *; apply (sqrt_sqrt x H1). +Qed. + +Lemma sqrt_mult : + forall x y:R, 0 <= x -> 0 <= y -> sqrt (x * y) = sqrt x * sqrt y. +intros x y H1 H2; + apply + (Rsqr_inj (sqrt (x * y)) (sqrt x * sqrt y) + (sqrt_positivity (x * y) (Rmult_le_pos x y H1 H2)) + (Rmult_le_pos (sqrt x) (sqrt y) (sqrt_positivity x H1) + (sqrt_positivity y H2))); rewrite Rsqr_mult; + repeat rewrite Rsqr_sqrt; + [ ring | assumption | assumption | apply (Rmult_le_pos x y H1 H2) ]. +Qed. + +Lemma sqrt_lt_R0 : forall x:R, 0 < x -> 0 < sqrt x. +intros x H1; apply Rsqr_incrst_0; + [ rewrite Rsqr_0; rewrite Rsqr_sqrt; [ assumption | left; assumption ] + | right; reflexivity + | apply (sqrt_positivity x (Rlt_le 0 x H1)) ]. +Qed. + +Lemma sqrt_div : + forall x y:R, 0 <= x -> 0 < y -> sqrt (x / y) = sqrt x / sqrt y. +intros x y H1 H2; apply Rsqr_inj; + [ apply sqrt_positivity; apply (Rmult_le_pos x (/ y)); + [ assumption + | generalize (Rinv_0_lt_compat y H2); clear H2; intro H2; left; + assumption ] + | apply (Rmult_le_pos (sqrt x) (/ sqrt y)); + [ apply (sqrt_positivity x H1) + | generalize (sqrt_lt_R0 y H2); clear H2; intro H2; + generalize (Rinv_0_lt_compat (sqrt y) H2); clear H2; + intro H2; left; assumption ] + | rewrite Rsqr_div; repeat rewrite Rsqr_sqrt; + [ reflexivity + | left; assumption + | assumption + | generalize (Rinv_0_lt_compat y H2); intro H3; + generalize (Rlt_le 0 (/ y) H3); intro H4; + apply (Rmult_le_pos x (/ y) H1 H4) + | red in |- *; intro H3; generalize (Rlt_le 0 y H2); intro H4; + generalize (sqrt_eq_0 y H4 H3); intro H5; rewrite H5 in H2; + elim (Rlt_irrefl 0 H2) ] ]. +Qed. + +Lemma sqrt_lt_0 : forall x y:R, 0 <= x -> 0 <= y -> sqrt x < sqrt y -> x < y. +intros x y H1 H2 H3; + generalize + (Rsqr_incrst_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) + (sqrt_positivity y H2)); intro H4; rewrite (Rsqr_sqrt x H1) in H4; + rewrite (Rsqr_sqrt y H2) in H4; assumption. +Qed. + +Lemma sqrt_lt_1 : forall x y:R, 0 <= x -> 0 <= y -> x < y -> sqrt x < sqrt y. +intros x y H1 H2 H3; apply Rsqr_incrst_0; + [ rewrite (Rsqr_sqrt x H1); rewrite (Rsqr_sqrt y H2); assumption + | apply (sqrt_positivity x H1) + | apply (sqrt_positivity y H2) ]. +Qed. + +Lemma sqrt_le_0 : + forall x y:R, 0 <= x -> 0 <= y -> sqrt x <= sqrt y -> x <= y. +intros x y H1 H2 H3; + generalize + (Rsqr_incr_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) + (sqrt_positivity y H2)); intro H4; rewrite (Rsqr_sqrt x H1) in H4; + rewrite (Rsqr_sqrt y H2) in H4; assumption. +Qed. + +Lemma sqrt_le_1 : + forall x y:R, 0 <= x -> 0 <= y -> x <= y -> sqrt x <= sqrt y. +intros x y H1 H2 H3; apply Rsqr_incr_0; + [ rewrite (Rsqr_sqrt x H1); rewrite (Rsqr_sqrt y H2); assumption + | apply (sqrt_positivity x H1) + | apply (sqrt_positivity y H2) ]. +Qed. + +Lemma sqrt_inj : forall x y:R, 0 <= x -> 0 <= y -> sqrt x = sqrt y -> x = y. +intros; cut (Rsqr (sqrt x) = Rsqr (sqrt y)). +intro; rewrite (Rsqr_sqrt x H) in H2; rewrite (Rsqr_sqrt y H0) in H2; + assumption. +rewrite H1; reflexivity. +Qed. + +Lemma sqrt_less : forall x:R, 0 <= x -> 1 < x -> sqrt x < x. +intros x H1 H2; generalize (sqrt_lt_1 1 x (Rlt_le 0 1 Rlt_0_1) H1 H2); + intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x)); + intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 2 in |- *; + rewrite <- (sqrt_def x H1); + apply + (Rmult_lt_compat_l (sqrt x) 1 (sqrt x) + (sqrt_lt_R0 x (Rlt_trans 0 1 x Rlt_0_1 H2)) H3). +Qed. + +Lemma sqrt_more : forall x:R, 0 < x -> x < 1 -> x < sqrt x. +intros x H1 H2; + generalize (sqrt_lt_1 x 1 (Rlt_le 0 x H1) (Rlt_le 0 1 Rlt_0_1) H2); + intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x)); + intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1 in |- *; + rewrite <- (sqrt_def x (Rlt_le 0 x H1)); + apply (Rmult_lt_compat_l (sqrt x) (sqrt x) 1 (sqrt_lt_R0 x H1) H3). +Qed. + +Lemma sqrt_cauchy : + forall a b c d:R, + a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d). +intros a b c d; apply Rsqr_incr_0_var; + [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr in |- *; + [ replace ((a * c + b * d) * (a * c + b * d)) with + (a * a * c * c + b * b * d * d + 2 * a * b * c * d); + [ replace ((a * a + b * b) * (c * c + d * d)) with + (a * a * c * c + b * b * d * d + (a * a * d * d + b * b * c * c)); + [ apply Rplus_le_compat_l; + replace (a * a * d * d + b * b * c * c) with + (2 * a * b * c * d + + (a * a * d * d + b * b * c * c - 2 * a * b * c * d)); + [ pattern (2 * a * b * c * d) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; + replace (a * a * d * d + b * b * c * c - 2 * a * b * c * d) + with (Rsqr (a * d - b * c)); + [ apply Rle_0_sqr | unfold Rsqr in |- *; ring ] + | ring ] + | ring ] + | ring ] + | apply + (Rplus_le_le_0_compat (Rsqr c) (Rsqr d) (Rle_0_sqr c) (Rle_0_sqr d)) + | apply + (Rplus_le_le_0_compat (Rsqr a) (Rsqr b) (Rle_0_sqr a) (Rle_0_sqr b)) ] + | apply Rmult_le_pos; apply sqrt_positivity; apply Rplus_le_le_0_compat; + apply Rle_0_sqr ]. +Qed. + +(************************************************************) +(* Resolution of [a*X^2+b*X+c=0] *) +(************************************************************) + +Definition Delta (a:nonzeroreal) (b c:R) : R := Rsqr b - 4 * a * c. + +Definition Delta_is_pos (a:nonzeroreal) (b c:R) : Prop := 0 <= Delta a b c. + +Definition sol_x1 (a:nonzeroreal) (b c:R) : R := + (- b + sqrt (Delta a b c)) / (2 * a). + +Definition sol_x2 (a:nonzeroreal) (b c:R) : R := + (- b - sqrt (Delta a b c)) / (2 * a). + +Lemma Rsqr_sol_eq_0_1 : + forall (a:nonzeroreal) (b c x:R), + Delta_is_pos a b c -> + x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0. +intros; elim H0; intro. +unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg; + rewrite Rsqr_sqrt. +rewrite Rsqr_inv. +unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc; rewrite (Rmult_comm a). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite + (Rmult_plus_distr_r (- b) (sqrt (b * b - 2 * (2 * (a * c)))) (/ 2 * / a)) + . +rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. +replace + (- b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + + (b * (- b * (/ 2 * / a)) + + (b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + c))) with + (b * (- b * (/ 2 * / a)) + c). +unfold Rminus in |- *; repeat rewrite <- Rplus_assoc. +replace (b * b + b * b) with (2 * (b * b)). +rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm a); rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- Rmult_opp_opp. +ring. +apply (cond_nonzero a). +discrR. +discrR. +discrR. +ring. +ring. +discrR. +apply (cond_nonzero a). +discrR. +apply (cond_nonzero a). +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +assumption. +unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg; + rewrite Rsqr_sqrt. +rewrite Rsqr_inv. +unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr; + repeat rewrite Rmult_assoc. +rewrite (Rmult_comm a); repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; unfold Rminus in |- *; rewrite Rmult_plus_distr_r. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; + pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; + rewrite + (Rmult_plus_distr_r (- b) (- sqrt (b * b + - (2 * (2 * (a * c))))) + (/ 2 * / a)). +rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. +rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_involutive. +replace + (b * (sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + + (b * (- b * (/ 2 * / a)) + + (b * (- sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + c))) with + (b * (- b * (/ 2 * / a)) + c). +repeat rewrite <- Rplus_assoc; replace (b * b + b * b) with (2 * (b * b)). +rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; repeat rewrite Rmult_assoc; rewrite (Rmult_comm a); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- Rmult_opp_opp; ring. +apply (cond_nonzero a). +discrR. +discrR. +discrR. +ring. +ring. +discrR. +apply (cond_nonzero a). +discrR. +discrR. +apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +assumption. +Qed. + +Lemma Rsqr_sol_eq_0_0 : + forall (a:nonzeroreal) (b c x:R), + Delta_is_pos a b c -> + a * Rsqr x + b * x + c = 0 -> x = sol_x1 a b c \/ x = sol_x2 a b c. +intros; rewrite (canonical_Rsqr a b c x) in H0; rewrite Rplus_comm in H0; + generalize + (Rplus_opp_r_uniq ((4 * a * c - Rsqr b) / (4 * a)) + (a * Rsqr (x + b / (2 * a))) H0); cut (Rsqr b - 4 * a * c = Delta a b c). +intro; + replace (- ((4 * a * c - Rsqr b) / (4 * a))) with + ((Rsqr b - 4 * a * c) / (4 * a)). +rewrite H1; intro; + generalize + (Rmult_eq_compat_l (/ a) (a * Rsqr (x + b / (2 * a))) + (Delta a b c / (4 * a)) H2); + replace (/ a * (a * Rsqr (x + b / (2 * a)))) with (Rsqr (x + b / (2 * a))). +replace (/ a * (Delta a b c / (4 * a))) with + (Rsqr (sqrt (Delta a b c) / (2 * a))). +intro; + generalize (Rsqr_eq (x + b / (2 * a)) (sqrt (Delta a b c) / (2 * a)) H3); + intro; elim H4; intro. +left; unfold sol_x1 in |- *; + generalize + (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) + (sqrt (Delta a b c) / (2 * a)) H5); + replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. +intro; rewrite H6; unfold Rdiv in |- *; ring. +ring. +right; unfold sol_x2 in |- *; + generalize + (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) + (- (sqrt (Delta a b c) / (2 * a))) H5); + replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. +intro; rewrite H6; unfold Rdiv in |- *; ring. +ring. +rewrite Rsqr_div. +rewrite Rsqr_sqrt. +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (/ a)). +rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace (2 * (2 * a) * a) with (Rsqr (2 * a)). +reflexivity. +ring_Rsqr. +rewrite <- Rmult_assoc; apply prod_neq_R0; + [ discrR | apply (cond_nonzero a) ]. +apply (cond_nonzero a). +assumption. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +symmetry in |- *; apply Rmult_1_l. +apply (cond_nonzero a). +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. +rewrite Ropp_minus_distr. +reflexivity. +reflexivity. +Qed.
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