diff options
Diffstat (limited to 'theories/Reals/R_sqr.v')
| -rw-r--r-- | theories/Reals/R_sqr.v | 63 |
1 files changed, 9 insertions, 54 deletions
diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v index 445ffcb2..a60bb7cf 100644 --- a/theories/Reals/R_sqr.v +++ b/theories/Reals/R_sqr.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) Require Import Rbase. @@ -296,56 +298,9 @@ Lemma canonical_Rsqr : a * Rsqr (x + b / (2 * a)) + (4 * a * c - Rsqr b) / (4 * a). Proof. intros. - rewrite Rsqr_plus. - repeat rewrite Rmult_plus_distr_l. - repeat rewrite Rplus_assoc. - apply Rplus_eq_compat_l. - unfold Rdiv, Rminus. - replace (2 * 1 + 2 * 1) with 4; [ idtac | ring ]. - rewrite (Rmult_plus_distr_r (4 * a * c) (- Rsqr b) (/ (4 * a))). - rewrite Rsqr_mult. - repeat rewrite Rinv_mult_distr. - repeat rewrite (Rmult_comm a). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm (/ 2)). - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm a). - repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - rewrite (Rmult_comm 2). - repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - repeat rewrite Rplus_assoc. - rewrite (Rplus_comm (Rsqr b * (Rsqr (/ a * / 2) * a))). - repeat rewrite Rplus_assoc. - rewrite (Rmult_comm x). - apply Rplus_eq_compat_l. - rewrite (Rmult_comm (/ a)). - unfold Rsqr; repeat rewrite Rmult_assoc. - rewrite <- Rinv_l_sym. - rewrite Rmult_1_r. - ring. - apply (cond_nonzero a). - discrR. - apply (cond_nonzero a). - discrR. - discrR. - apply (cond_nonzero a). - discrR. - discrR. - discrR. - apply (cond_nonzero a). - discrR. - apply (cond_nonzero a). + unfold Rsqr. + field. + apply a. Qed. Lemma Rsqr_eq : forall x y:R, Rsqr x = Rsqr y -> x = y \/ x = - y. |