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-rw-r--r--theories/Reals/R_sqrt.v131
1 files changed, 24 insertions, 107 deletions
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v
index a6b1a26e..d4035fad 100644
--- a/theories/Reals/R_sqrt.v
+++ b/theories/Reals/R_sqrt.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.
@@ -359,107 +361,22 @@ 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 :
@@ -505,10 +422,10 @@ Proof.
rewrite (Rmult_comm (/ a)).
rewrite Rmult_assoc.
rewrite <- Rinv_mult_distr.
- replace (2 * (2 * a) * a) with (Rsqr (2 * a)).
+ replace (4 * a * a) with (Rsqr (2 * a)).
reflexivity.
ring_Rsqr.
- rewrite <- Rmult_assoc; apply prod_neq_R0;
+ apply prod_neq_R0;
[ discrR | apply (cond_nonzero a) ].
apply (cond_nonzero a).
assumption.
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-07 10:51:49 +0000