diff options
author | Guillaume Melquiond <guillaume.melquiond@inria.fr> | 2017-03-05 19:58:44 +0100 |
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committer | Maxime Dénès <mail@maximedenes.fr> | 2017-03-22 16:42:00 +0100 |
commit | 46d79ebd74876f34242c8c5d9ab3dcedbadba7cc (patch) | |
tree | 262da87e5fca237320e75be293244dd19628ca3a /theories/Reals/R_sqrt.v | |
parent | d2061bd8cd2d809d6e5a849dd150e9e0d74331fc (diff) |
Simplify some proofs using ring and field.
Diffstat (limited to 'theories/Reals/R_sqrt.v')
-rw-r--r-- | theories/Reals/R_sqrt.v | 117 |
1 files changed, 16 insertions, 101 deletions
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index a6b1a26e0..5c975c3f5 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -359,107 +359,22 @@ Lemma Rsqr_sol_eq_0_1 : x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0. Proof. intros; elim H0; intro. - unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv; - repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg; - rewrite Rsqr_sqrt. - rewrite Rsqr_inv. - unfold Rsqr; 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; 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; 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; - repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg; - rewrite Rsqr_sqrt. - rewrite Rsqr_inv. - unfold Rsqr; 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; rewrite Rmult_plus_distr_r. - rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; - pattern 2 at 2; 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. + rewrite H1. + unfold sol_x1, Delta, Rsqr. + field_simplify. + rewrite <- (Rsqr_pow2 (sqrt _)), Rsqr_sqrt. + field. + apply a. + apply H. + apply a. + rewrite H1. + unfold sol_x2, Delta, Rsqr. + field_simplify. + rewrite <- (Rsqr_pow2 (sqrt _)), Rsqr_sqrt. + field. + apply a. + apply H. + apply a. Qed. Lemma Rsqr_sol_eq_0_0 : |