diff options
Diffstat (limited to 'theories/Reals/RIneq.v')
| -rw-r--r-- | theories/Reals/RIneq.v | 46 |
1 files changed, 19 insertions, 27 deletions
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index f02db3d4..70f4ff0d 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -1,14 +1,12 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: RIneq.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - (*********************************************************) (** * Basic lemmas for the classical real numbers *) (*********************************************************) @@ -1603,7 +1601,7 @@ Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (nat_of_P p). Proof. intro; apply lt_0_INR. simpl in |- *; auto with real. - apply lt_O_nat_of_P. + apply nat_of_P_pos. Qed. Hint Resolve pos_INR_nat_of_P: real. @@ -1712,38 +1710,32 @@ Qed. Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n). Proof. simple induction n; auto with real. - intros; simpl in |- *; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; + intros; simpl in |- *; rewrite nat_of_P_of_succ_nat; auto with real. Qed. Lemma plus_IZR_NEG_POS : forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q). Proof. - intros. - case (lt_eq_lt_dec (nat_of_P p) (nat_of_P q)). - intros [H| H]; simpl in |- *. - rewrite nat_of_P_lt_Lt_compare_complement_morphism; simpl in |- *; trivial. - rewrite (nat_of_P_minus_morphism q p). - rewrite minus_INR; auto with arith; ring. - apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial. - rewrite (nat_of_P_inj p q); trivial. - rewrite Pcompare_refl; simpl in |- *; auto with real. - intro H; simpl in |- *. - rewrite nat_of_P_gt_Gt_compare_complement_morphism; simpl in |- *; - auto with arith. - rewrite (nat_of_P_minus_morphism p q). - rewrite minus_INR; auto with arith; ring. - apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial. + intros p q; simpl. rewrite Z.pos_sub_spec. + case Pcompare_spec; intros H; simpl. + subst. ring. + rewrite Pminus_minus by trivial. + rewrite minus_INR by (now apply lt_le_weak, Plt_lt). + ring. + rewrite Pminus_minus by trivial. + rewrite minus_INR by (now apply lt_le_weak, Plt_lt). + ring. Qed. (**********) Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m. Proof. intro z; destruct z; intro t; destruct t; intros; auto with real. - simpl in |- *; intros; rewrite nat_of_P_plus_morphism; auto with real. + simpl; intros; rewrite Pplus_plus; auto with real. apply plus_IZR_NEG_POS. rewrite Zplus_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS. - simpl in |- *; intros; rewrite nat_of_P_plus_morphism; rewrite plus_INR; + simpl; intros; rewrite Pplus_plus; rewrite plus_INR; auto with real. Qed. @@ -1751,14 +1743,14 @@ Qed. Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m. Proof. intros z t; case z; case t; simpl in |- *; auto with real. - intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. - intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. + intros t1 z1; rewrite Pmult_mult; auto with real. + intros t1 z1; rewrite Pmult_mult; auto with real. rewrite Rmult_comm. rewrite Ropp_mult_distr_l_reverse; auto with real. apply Ropp_eq_compat; rewrite mult_comm; auto with real. - intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. + intros t1 z1; rewrite Pmult_mult; auto with real. rewrite Ropp_mult_distr_l_reverse; auto with real. - intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. + intros t1 z1; rewrite Pmult_mult; auto with real. rewrite Rmult_opp_opp; auto with real. Qed. @@ -1766,7 +1758,7 @@ Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Zpower z (Z_of_nat n)). Proof. intros z [|n];simpl;trivial. rewrite Zpower_pos_nat. - rewrite nat_of_P_o_P_of_succ_nat_eq_succ. unfold Zpower_nat;simpl. + rewrite nat_of_P_of_succ_nat. unfold Zpower_nat;simpl. rewrite mult_IZR. induction n;simpl;trivial. rewrite mult_IZR;ring[IHn]. |