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(* * 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 *)
(* * (see LICENSE file for the text of the license) *)
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(** * Basic lemmas for the classical real numbers *)
(*********************************************************)
Require Export Raxioms.
Require Import Rpow_def.
Require Import Zpower.
Require Export ZArithRing.
Require Import Omega.
Require Export RealField.
Local Open Scope Z_scope.
Local Open Scope R_scope.
Implicit Type r : R.
(*********************************************************)
(** ** Relation between orders and equality *)
(*********************************************************)
(** Reflexivity of the large order *)
Lemma Rle_refl : forall r, r <= r.
Proof.
intro; right; reflexivity.
Qed.
Hint Immediate Rle_refl: rorders.
Lemma Rge_refl : forall r, r <= r.
Proof. exact Rle_refl. Qed.
Hint Immediate Rge_refl: rorders.
(** Irreflexivity of the strict order *)
Lemma Rlt_irrefl : forall r, ~ r < r.
Proof.
intros r H; eapply Rlt_asym; eauto.
Qed.
Hint Resolve Rlt_irrefl: real.
Lemma Rgt_irrefl : forall r, ~ r > r.
Proof. exact Rlt_irrefl. Qed.
Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2.
Proof.
red; intros r1 r2 H H0; apply (Rlt_irrefl r1).
pattern r1 at 2; rewrite H0; trivial.
Qed.
Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2.
Proof.
intros; apply not_eq_sym; apply Rlt_not_eq; auto with real.
Qed.
(**********)
Lemma Rlt_dichotomy_converse : forall r1 r2, r1 < r2 \/ r1 > r2 -> r1 <> r2.
Proof.
intuition.
- apply Rlt_not_eq in H1. eauto.
- apply Rgt_not_eq in H1. eauto.
Qed.
Hint Resolve Rlt_dichotomy_converse: real.
(** Reasoning by case on equality and order *)
(**********)
Lemma Req_dec : forall r1 r2, r1 = r2 \/ r1 <> r2.
Proof.
intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse;
unfold not; intuition eauto 3.
Qed.
Hint Resolve Req_dec: real.
(**********)
Lemma Rtotal_order : forall r1 r2, r1 < r2 \/ r1 = r2 \/ r1 > r2.
Proof.
intros; generalize (total_order_T r1 r2); tauto.
Qed.
(**********)
Lemma Rdichotomy : forall r1 r2, r1 <> r2 -> r1 < r2 \/ r1 > r2.
Proof.
intros; generalize (total_order_T r1 r2); tauto.
Qed.
(*********************************************************)
(** ** Relating [<], [>], [<=] and [>=] *)
(*********************************************************)
(*********************************************************)
(** ** Order *)
(*********************************************************)
(** *** Relating strict and large orders *)
Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2.
Proof.
intros; red; tauto.
Qed.
Hint Resolve Rlt_le: real.
Lemma Rgt_ge : forall r1 r2, r1 > r2 -> r1 >= r2.
Proof.
intros; red; tauto.
Qed.
(**********)
Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1.
Proof.
destruct 1; red; auto with real.
Qed.
Hint Immediate Rle_ge: real.
Hint Resolve Rle_ge: rorders.
Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1.
Proof.
destruct 1; red; auto with real.
Qed.
Hint Resolve Rge_le: real.
Hint Immediate Rge_le: rorders.
(**********)
Lemma Rlt_gt : forall r1 r2, r1 < r2 -> r2 > r1.
Proof.
trivial.
Qed.
Hint Resolve Rlt_gt: rorders.
Lemma Rgt_lt : forall r1 r2, r1 > r2 -> r2 < r1.
Proof.
trivial.
Qed.
Hint Immediate Rgt_lt: rorders.
(**********)
Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1.
Proof.
intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle; tauto.
Qed.
Hint Immediate Rnot_le_lt: real.
Lemma Rnot_ge_gt : forall r1 r2, ~ r1 >= r2 -> r2 > r1.
Proof. intros; red; apply Rnot_le_lt. auto with real. Qed.
Lemma Rnot_le_gt : forall r1 r2, ~ r1 <= r2 -> r1 > r2.
Proof. intros; red; apply Rnot_le_lt. auto with real. Qed.
Lemma Rnot_ge_lt : forall r1 r2, ~ r1 >= r2 -> r1 < r2.
Proof. intros; apply Rnot_le_lt. auto with real. Qed.
Lemma Rnot_lt_le : forall r1 r2, ~ r1 < r2 -> r2 <= r1.
Proof.
intros r1 r2 H; destruct (Rtotal_order r1 r2) as [ | [ H0 | H0 ] ].
contradiction. subst; auto with rorders. auto with real.
Qed.
Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2.
Proof. auto using Rnot_lt_le with real. Qed.
Lemma Rnot_gt_ge : forall r1 r2, ~ r1 > r2 -> r2 >= r1.
Proof. intros; eauto using Rnot_lt_le with rorders. Qed.
Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2.
Proof. eauto using Rnot_gt_ge with rorders. Qed.
(**********)
Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2.
Proof.
generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle.
unfold not; intuition eauto 3.
Qed.
Hint Immediate Rlt_not_le: real.
Lemma Rgt_not_le : forall r1 r2, r1 > r2 -> ~ r1 <= r2.
Proof. exact Rlt_not_le. Qed.
Lemma Rlt_not_ge : forall r1 r2, r1 < r2 -> ~ r1 >= r2.
Proof. red; intros; eapply Rlt_not_le; eauto with real. Qed.
Hint Immediate Rlt_not_ge: real.
Lemma Rgt_not_ge : forall r1 r2, r2 > r1 -> ~ r1 >= r2.
Proof. exact Rlt_not_ge. Qed.
Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2.
Proof.
intros r1 r2. generalize (Rlt_asym r1 r2) (Rlt_dichotomy_converse r1 r2).
unfold Rle; intuition.
Qed.
Lemma Rge_not_lt : forall r1 r2, r1 >= r2 -> ~ r1 < r2.
Proof. intros; apply Rle_not_lt; auto with real. Qed.
Lemma Rle_not_gt : forall r1 r2, r1 <= r2 -> ~ r1 > r2.
Proof. do 2 intro; apply Rle_not_lt. Qed.
Lemma Rge_not_gt : forall r1 r2, r2 >= r1 -> ~ r1 > r2.
Proof. do 2 intro; apply Rge_not_lt. Qed.
(**********)
Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2.
Proof.
unfold Rle; tauto.
Qed.
Hint Immediate Req_le: real.
Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2.
Proof.
unfold Rge; tauto.
Qed.
Hint Immediate Req_ge: real.
Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2.
Proof.
unfold Rle; auto.
Qed.
Hint Immediate Req_le_sym: real.
Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2.
Proof.
unfold Rge; auto.
Qed.
Hint Immediate Req_ge_sym: real.
(** *** Asymmetry *)
(** Remark: [Rlt_asym] is an axiom *)
Lemma Rgt_asym : forall r1 r2:R, r1 > r2 -> ~ r2 > r1.
Proof. do 2 intro; apply Rlt_asym. Qed.
(** *** Antisymmetry *)
Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2.
Proof.
intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle; intuition.
Qed.
Hint Resolve Rle_antisym: real.
Lemma Rge_antisym : forall r1 r2, r1 >= r2 -> r2 >= r1 -> r1 = r2.
Proof. auto with real. Qed.
(**********)
Lemma Rle_le_eq : forall r1 r2, r1 <= r2 /\ r2 <= r1 <-> r1 = r2.
Proof.
intuition.
Qed.
Lemma Rge_ge_eq : forall r1 r2, r1 >= r2 /\ r2 >= r1 <-> r1 = r2.
Proof.
intuition.
Qed.
(** *** Compatibility with equality *)
Lemma Rlt_eq_compat :
forall r1 r2 r3 r4, r1 = r2 -> r2 < r4 -> r4 = r3 -> r1 < r3.
Proof.
intros x x' y y'; intros; replace x with x'; replace y with y'; assumption.
Qed.
Lemma Rgt_eq_compat :
forall r1 r2 r3 r4, r1 = r2 -> r2 > r4 -> r4 = r3 -> r1 > r3.
Proof. intros; red; apply Rlt_eq_compat with (r2:=r4) (r4:=r2); auto. Qed.
(** *** Transitivity *)
(** Remark: [Rlt_trans] is an axiom *)
Lemma Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3.
Proof.
generalize eq_trans Rlt_trans Rlt_eq_compat.
unfold Rle.
intuition eauto 2.
Qed.
Lemma Rge_trans : forall r1 r2 r3, r1 >= r2 -> r2 >= r3 -> r1 >= r3.
Proof. eauto using Rle_trans with rorders. Qed.
Lemma Rgt_trans : forall r1 r2 r3, r1 > r2 -> r2 > r3 -> r1 > r3.
Proof. eauto using Rlt_trans with rorders. Qed.
(**********)
Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3.
Proof.
generalize Rlt_trans Rlt_eq_compat.
unfold Rle.
intuition eauto 2.
Qed.
Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3.
Proof.
generalize Rlt_trans Rlt_eq_compat; unfold Rle; intuition eauto 2.
Qed.
Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3.
Proof. eauto using Rlt_le_trans with rorders. Qed.
Lemma Rgt_ge_trans : forall r1 r2 r3, r1 > r2 -> r2 >= r3 -> r1 > r3.
Proof. eauto using Rle_lt_trans with rorders. Qed.
(** *** (Classical) decidability *)
Lemma Rlt_dec : forall r1 r2, {r1 < r2} + {~ r1 < r2}.
Proof.
intros; generalize (total_order_T r1 r2) (Rlt_dichotomy_converse r1 r2);
intuition.
Qed.
Lemma Rle_dec : forall r1 r2, {r1 <= r2} + {~ r1 <= r2}.
Proof.
intros r1 r2.
generalize (total_order_T r1 r2) (Rlt_dichotomy_converse r1 r2).
intuition eauto 4 with real.
Qed.
Lemma Rgt_dec : forall r1 r2, {r1 > r2} + {~ r1 > r2}.
Proof. do 2 intro; apply Rlt_dec. Qed.
Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}.
Proof. intros; edestruct Rle_dec; [left|right]; eauto with rorders. Qed.
Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}.
Proof.
intros; generalize (total_order_T r1 r2); intuition.
Qed.
Lemma Rgt_ge_dec : forall r1 r2, {r1 > r2} + {r2 >= r1}.
Proof. intros; edestruct Rlt_le_dec; [left|right]; eauto with rorders. Qed.
Lemma Rle_lt_dec : forall r1 r2, {r1 <= r2} + {r2 < r1}.
Proof.
intros; generalize (total_order_T r1 r2); intuition.
Qed.
Lemma Rge_gt_dec : forall r1 r2, {r1 >= r2} + {r2 > r1}.
Proof. intros; edestruct Rle_lt_dec; [left|right]; eauto with rorders. Qed.
Lemma Rlt_or_le : forall r1 r2, r1 < r2 \/ r2 <= r1.
Proof.
intros n m; elim (Rle_lt_dec m n); auto with real.
Qed.
Lemma Rgt_or_ge : forall r1 r2, r1 > r2 \/ r2 >= r1.
Proof. intros; edestruct Rlt_or_le; [left|right]; eauto with rorders. Qed.
Lemma Rle_or_lt : forall r1 r2, r1 <= r2 \/ r2 < r1.
Proof.
intros n m; elim (Rlt_le_dec m n); auto with real.
Qed.
Lemma Rge_or_gt : forall r1 r2, r1 >= r2 \/ r2 > r1.
Proof. intros; edestruct Rle_or_lt; [left|right]; eauto with rorders. Qed.
Lemma Rle_lt_or_eq_dec : forall r1 r2, r1 <= r2 -> {r1 < r2} + {r1 = r2}.
Proof.
intros r1 r2 H; generalize (total_order_T r1 r2); intuition.
Qed.
(**********)
Lemma inser_trans_R :
forall r1 r2 r3 r4, r1 <= r2 < r3 -> {r1 <= r2 < r4} + {r4 <= r2 < r3}.
Proof.
intros n m p q; intros; generalize (Rlt_le_dec m q); intuition.
Qed.
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(** ** Addition *)
(*********************************************************)
(** Remark: [Rplus_0_l] is an axiom *)
Lemma Rplus_0_r : forall r, r + 0 = r.
Proof.
intro; ring.
Qed.
Hint Resolve Rplus_0_r: real.
Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r.
Proof.
split; ring.
Qed.
Hint Resolve Rplus_ne: real.
(**********)
(** Remark: [Rplus_opp_r] is an axiom *)
Lemma Rplus_opp_l : forall r, - r + r = 0.
Proof.
intro; ring.
Qed.
Hint Resolve Rplus_opp_l: real.
(**********)
Lemma Rplus_opp_r_uniq : forall r1 r2, r1 + r2 = 0 -> r2 = - r1.
Proof.
intros x y H;
replace y with (- x + x + y) by ring.
rewrite Rplus_assoc; rewrite H; ring.
Qed.
Definition f_equal_R := (f_equal (A:=R)).
Hint Resolve f_equal_R : real.
Lemma Rplus_eq_compat_l : forall r r1 r2, r1 = r2 -> r + r1 = r + r2.
Proof.
intros r r1 r2.
apply f_equal.
Qed.
Lemma Rplus_eq_compat_r : forall r r1 r2, r1 = r2 -> r1 + r = r2 + r.
Proof.
intros r r1 r2.
apply (f_equal (fun v => v + r)).
Qed.
(**********)
Lemma Rplus_eq_reg_l : forall r r1 r2, r + r1 = r + r2 -> r1 = r2.
Proof.
intros; transitivity (- r + r + r1).
ring.
transitivity (- r + r + r2).
repeat rewrite Rplus_assoc; rewrite <- H; reflexivity.
ring.
Qed.
Hint Resolve Rplus_eq_reg_l: real.
Lemma Rplus_eq_reg_r : forall r r1 r2, r1 + r = r2 + r -> r1 = r2.
Proof.
intros r r1 r2 H.
apply Rplus_eq_reg_l with r.
now rewrite 2!(Rplus_comm r).
Qed.
(**********)
Lemma Rplus_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0.
Proof.
intros r b; pattern r at 2; replace r with (r + 0); eauto with real.
Qed.
(***********)
Lemma Rplus_eq_0_l :
forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0.
Proof.
intros a b H [H0| H0] H1; auto with real.
absurd (0 < a + b).
rewrite H1; auto with real.
apply Rle_lt_trans with (a + 0).
rewrite Rplus_0_r; assumption.
auto using Rplus_lt_compat_l with real.
rewrite <- H0, Rplus_0_r in H1; assumption.
Qed.
Lemma Rplus_eq_R0 :
forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0 /\ r2 = 0.
Proof.
intros a b; split.
apply Rplus_eq_0_l with b; auto with real.
apply Rplus_eq_0_l with a; auto with real.
rewrite Rplus_comm; auto with real.
Qed.
(*********************************************************)
(** ** Multiplication *)
(*********************************************************)
(**********)
Lemma Rinv_r : forall r, r <> 0 -> r * / r = 1.
Proof.
intros; field; trivial.
Qed.
Hint Resolve Rinv_r: real.
Lemma Rinv_l_sym : forall r, r <> 0 -> 1 = / r * r.
Proof.
intros; field; trivial.
Qed.
Hint Resolve Rinv_l_sym: real.
Lemma Rinv_r_sym : forall r, r <> 0 -> 1 = r * / r.
Proof.
intros; field; trivial.
Qed.
Hint Resolve Rinv_r_sym: real.
(**********)
Lemma Rmult_0_r : forall r, r * 0 = 0.
Proof.
intro; ring.
Qed.
Hint Resolve Rmult_0_r: real.
(**********)
Lemma Rmult_0_l : forall r, 0 * r = 0.
Proof.
intro; ring.
Qed.
Hint Resolve Rmult_0_l: real.
(**********)
Lemma Rmult_ne : forall r, r * 1 = r /\ 1 * r = r.
Proof.
intro; split; ring.
Qed.
Hint Resolve Rmult_ne: real.
(**********)
Lemma Rmult_1_r : forall r, r * 1 = r.
Proof.
intro; ring.
Qed.
Hint Resolve Rmult_1_r: real.
(**********)
Lemma Rmult_eq_compat_l : forall r r1 r2, r1 = r2 -> r * r1 = r * r2.
Proof.
auto with real.
Qed.
Lemma Rmult_eq_compat_r : forall r r1 r2, r1 = r2 -> r1 * r = r2 * r.
Proof.
intros.
rewrite <- 2!(Rmult_comm r).
now apply Rmult_eq_compat_l.
Qed.
(**********)
Lemma Rmult_eq_reg_l : forall r r1 r2, r * r1 = r * r2 -> r <> 0 -> r1 = r2.
Proof.
intros; transitivity (/ r * r * r1).
field; trivial.
transitivity (/ r * r * r2).
repeat rewrite Rmult_assoc; rewrite H; trivial.
field; trivial.
Qed.
Lemma Rmult_eq_reg_r : forall r r1 r2, r1 * r = r2 * r -> r <> 0 -> r1 = r2.
Proof.
intros.
apply Rmult_eq_reg_l with (2 := H0).
now rewrite 2!(Rmult_comm r).
Qed.
(**********)
Lemma Rmult_integral : forall r1 r2, r1 * r2 = 0 -> r1 = 0 \/ r2 = 0.
Proof.
intros; case (Req_dec r1 0); [ intro Hz | intro Hnotz ].
auto.
right; apply Rmult_eq_reg_l with r1; trivial.
rewrite H; auto with real.
Qed.
(**********)
Lemma Rmult_eq_0_compat : forall r1 r2, r1 = 0 \/ r2 = 0 -> r1 * r2 = 0.
Proof.
intros r1 r2 [H| H]; rewrite H; auto with real.
Qed.
Hint Resolve Rmult_eq_0_compat: real.
(**********)
Lemma Rmult_eq_0_compat_r : forall r1 r2, r1 = 0 -> r1 * r2 = 0.
Proof.
auto with real.
Qed.
(**********)
Lemma Rmult_eq_0_compat_l : forall r1 r2, r2 = 0 -> r1 * r2 = 0.
Proof.
auto with real.
Qed.
(**********)
Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0.
Proof.
intros r1 r2 H; split; red; intro; apply H; auto with real.
Qed.
(**********)
Lemma Rmult_integral_contrapositive :
forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0.
Proof.
red; intros r1 r2 [H1 H2] H.
case (Rmult_integral r1 r2); auto with real.
Qed.
Hint Resolve Rmult_integral_contrapositive: real.
Lemma Rmult_integral_contrapositive_currified :
forall r1 r2, r1 <> 0 -> r2 <> 0 -> r1 * r2 <> 0.
Proof. auto using Rmult_integral_contrapositive. Qed.
(**********)
Lemma Rmult_plus_distr_r :
forall r1 r2 r3, (r1 + r2) * r3 = r1 * r3 + r2 * r3.
Proof.
intros; ring.
Qed.
(*********************************************************)
(** ** Square function *)
(*********************************************************)
(***********)
Definition Rsqr r : R := r * r.
Notation "r ²" := (Rsqr r) (at level 1, format "r ²") : R_scope.
(***********)
Lemma Rsqr_0 : Rsqr 0 = 0.
unfold Rsqr; auto with real.
Qed.
(***********)
Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0.
unfold Rsqr; intros; elim (Rmult_integral r r H); trivial.
Qed.
(*********************************************************)
(** ** Opposite *)
(*********************************************************)
(**********)
Lemma Ropp_eq_compat : forall r1 r2, r1 = r2 -> - r1 = - r2.
Proof.
auto with real.
Qed.
Hint Resolve Ropp_eq_compat: real.
(**********)
Lemma Ropp_0 : -0 = 0.
Proof.
ring.
Qed.
Hint Resolve Ropp_0: real.
(**********)
Lemma Ropp_eq_0_compat : forall r, r = 0 -> - r = 0.
Proof.
intros; rewrite H; auto with real.
Qed.
Hint Resolve Ropp_eq_0_compat: real.
(**********)
Lemma Ropp_involutive : forall r, - - r = r.
Proof.
intro; ring.
Qed.
Hint Resolve Ropp_involutive: real.
(*********)
Lemma Ropp_neq_0_compat : forall r, r <> 0 -> - r <> 0.
Proof.
red; intros r H H0.
apply H.
transitivity (- - r); auto with real.
Qed.
Hint Resolve Ropp_neq_0_compat: real.
(**********)
Lemma Ropp_plus_distr : forall r1 r2, - (r1 + r2) = - r1 + - r2.
Proof.
intros; ring.
Qed.
Hint Resolve Ropp_plus_distr: real.
(*********************************************************)
(** ** Opposite and multiplication *)
(*********************************************************)
Lemma Ropp_mult_distr_l : forall r1 r2, - (r1 * r2) = - r1 * r2.
Proof.
intros; ring.
Qed.
Lemma Ropp_mult_distr_l_reverse : forall r1 r2, - r1 * r2 = - (r1 * r2).
Proof.
intros; ring.
Qed.
Hint Resolve Ropp_mult_distr_l_reverse: real.
(**********)
Lemma Rmult_opp_opp : forall r1 r2, - r1 * - r2 = r1 * r2.
Proof.
intros; ring.
Qed.
Hint Resolve Rmult_opp_opp: real.
Lemma Ropp_mult_distr_r : forall r1 r2, - (r1 * r2) = r1 * - r2.
Proof.
intros; ring.
Qed.
Lemma Ropp_mult_distr_r_reverse : forall r1 r2, r1 * - r2 = - (r1 * r2).
Proof.
intros; ring.
Qed.
(*********************************************************)
(** ** Subtraction *)
(*********************************************************)
Lemma Rminus_0_r : forall r, r - 0 = r.
Proof.
intro; ring.
Qed.
Hint Resolve Rminus_0_r: real.
Lemma Rminus_0_l : forall r, 0 - r = - r.
Proof.
intro; ring.
Qed.
Hint Resolve Rminus_0_l: real.
(**********)
Lemma Ropp_minus_distr : forall r1 r2, - (r1 - r2) = r2 - r1.
Proof.
intros; ring.
Qed.
Hint Resolve Ropp_minus_distr: real.
Lemma Ropp_minus_distr' : forall r1 r2, - (r2 - r1) = r1 - r2.
Proof.
intros; ring.
Qed.
(**********)
Lemma Rminus_diag_eq : forall r1 r2, r1 = r2 -> r1 - r2 = 0.
Proof.
intros; rewrite H; ring.
Qed.
Hint Resolve Rminus_diag_eq: real.
(**********)
Lemma Rminus_diag_uniq : forall r1 r2, r1 - r2 = 0 -> r1 = r2.
Proof.
intros r1 r2; unfold Rminus; rewrite Rplus_comm; intro.
rewrite <- (Ropp_involutive r2); apply (Rplus_opp_r_uniq (- r2) r1 H).
Qed.
Hint Immediate Rminus_diag_uniq: real.
Lemma Rminus_diag_uniq_sym : forall r1 r2, r2 - r1 = 0 -> r1 = r2.
Proof.
intros; generalize (Rminus_diag_uniq r2 r1 H); clear H; intro H; rewrite H;
ring.
Qed.
Hint Immediate Rminus_diag_uniq_sym: real.
Lemma Rplus_minus : forall r1 r2, r1 + (r2 - r1) = r2.
Proof.
intros; ring.
Qed.
Hint Resolve Rplus_minus: real.
(**********)
Lemma Rminus_eq_contra : forall r1 r2, r1 <> r2 -> r1 - r2 <> 0.
Proof.
red; intros r1 r2 H H0.
apply H; auto with real.
Qed.
Hint Resolve Rminus_eq_contra: real.
Lemma Rminus_not_eq : forall r1 r2, r1 - r2 <> 0 -> r1 <> r2.
Proof.
red; intros; elim H; apply Rminus_diag_eq; auto.
Qed.
Hint Resolve Rminus_not_eq: real.
Lemma Rminus_not_eq_right : forall r1 r2, r2 - r1 <> 0 -> r1 <> r2.
Proof.
red; intros; elim H; rewrite H0; ring.
Qed.
Hint Resolve Rminus_not_eq_right: real.
(**********)
Lemma Rmult_minus_distr_l :
forall r1 r2 r3, r1 * (r2 - r3) = r1 * r2 - r1 * r3.
Proof.
intros; ring.
Qed.
(*********************************************************)
(** ** Inverse *)
(*********************************************************)
Lemma Rinv_1 : / 1 = 1.
Proof.
field.
Qed.
Hint Resolve Rinv_1: real.
(*********)
Lemma Rinv_neq_0_compat : forall r, r <> 0 -> / r <> 0.
Proof.
red; intros; apply R1_neq_R0.
replace 1 with (/ r * r); auto with real.
Qed.
Hint Resolve Rinv_neq_0_compat: real.
(*********)
Lemma Rinv_involutive : forall r, r <> 0 -> / / r = r.
Proof.
intros; field; trivial.
Qed.
Hint Resolve Rinv_involutive: real.
(*********)
Lemma Rinv_mult_distr :
forall r1 r2, r1 <> 0 -> r2 <> 0 -> / (r1 * r2) = / r1 * / r2.
Proof.
intros; field; auto.
Qed.
(*********)
Lemma Ropp_inv_permute : forall r, r <> 0 -> - / r = / - r.
Proof.
intros; field; trivial.
Qed.
Lemma Rinv_r_simpl_r : forall r1 r2, r1 <> 0 -> r1 * / r1 * r2 = r2.
Proof.
intros; transitivity (1 * r2); auto with real.
rewrite Rinv_r; auto with real.
Qed.
Lemma Rinv_r_simpl_l : forall r1 r2, r1 <> 0 -> r2 * r1 * / r1 = r2.
Proof.
intros; transitivity (r2 * 1); auto with real.
transitivity (r2 * (r1 * / r1)); auto with real.
Qed.
Lemma Rinv_r_simpl_m : forall r1 r2, r1 <> 0 -> r1 * r2 * / r1 = r2.
Proof.
intros; transitivity (r2 * 1); auto with real.
transitivity (r2 * (r1 * / r1)); auto with real.
ring.
Qed.
Hint Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m: real.
(*********)
Lemma Rinv_mult_simpl :
forall r1 r2 r3, r1 <> 0 -> r1 * / r2 * (r3 * / r1) = r3 * / r2.
Proof.
intros a b c; intros.
transitivity (a * / a * (c * / b)); auto with real.
ring.
Qed.
(*********************************************************)
(** ** Order and addition *)
(*********************************************************)
(** *** Compatibility *)
(** Remark: [Rplus_lt_compat_l] is an axiom *)
Lemma Rplus_gt_compat_l : forall r r1 r2, r1 > r2 -> r + r1 > r + r2.
Proof. eauto using Rplus_lt_compat_l with rorders. Qed.
Hint Resolve Rplus_gt_compat_l: real.
(**********)
Lemma Rplus_lt_compat_r : forall r r1 r2, r1 < r2 -> r1 + r < r2 + r.
Proof.
intros.
rewrite (Rplus_comm r1 r); rewrite (Rplus_comm r2 r); auto with real.
Qed.
Hint Resolve Rplus_lt_compat_r: real.
Lemma Rplus_gt_compat_r : forall r r1 r2, r1 > r2 -> r1 + r > r2 + r.
Proof. do 3 intro; apply Rplus_lt_compat_r. Qed.
(**********)
Lemma Rplus_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2.
Proof.
unfold Rle; intros; elim H; intro.
left; apply (Rplus_lt_compat_l r r1 r2 H0).
right; rewrite <- H0; auto with zarith real.
Qed.
Lemma Rplus_ge_compat_l : forall r r1 r2, r1 >= r2 -> r + r1 >= r + r2.
Proof. auto using Rplus_le_compat_l with rorders. Qed.
Hint Resolve Rplus_ge_compat_l: real.
(**********)
Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r.
Proof.
unfold Rle; intros; elim H; intro.
left; apply (Rplus_lt_compat_r r r1 r2 H0).
right; rewrite <- H0; auto with real.
Qed.
Hint Resolve Rplus_le_compat_l Rplus_le_compat_r: real.
Lemma Rplus_ge_compat_r : forall r r1 r2, r1 >= r2 -> r1 + r >= r2 + r.
Proof. auto using Rplus_le_compat_r with rorders. Qed.
(*********)
Lemma Rplus_lt_compat :
forall r1 r2 r3 r4, r1 < r2 -> r3 < r4 -> r1 + r3 < r2 + r4.
Proof.
intros; apply Rlt_trans with (r2 + r3); auto with real.
Qed.
Hint Immediate Rplus_lt_compat: real.
Lemma Rplus_le_compat :
forall r1 r2 r3 r4, r1 <= r2 -> r3 <= r4 -> r1 + r3 <= r2 + r4.
Proof.
intros; apply Rle_trans with (r2 + r3); auto with real.
Qed.
Hint Immediate Rplus_le_compat: real.
Lemma Rplus_gt_compat :
forall r1 r2 r3 r4, r1 > r2 -> r3 > r4 -> r1 + r3 > r2 + r4.
Proof. auto using Rplus_lt_compat with rorders. Qed.
Lemma Rplus_ge_compat :
forall r1 r2 r3 r4, r1 >= r2 -> r3 >= r4 -> r1 + r3 >= r2 + r4.
Proof. auto using Rplus_le_compat with rorders. Qed.
(*********)
Lemma Rplus_lt_le_compat :
forall r1 r2 r3 r4, r1 < r2 -> r3 <= r4 -> r1 + r3 < r2 + r4.
Proof.
intros; apply Rlt_le_trans with (r2 + r3); auto with real.
Qed.
Lemma Rplus_le_lt_compat :
forall r1 r2 r3 r4, r1 <= r2 -> r3 < r4 -> r1 + r3 < r2 + r4.
Proof.
intros; apply Rle_lt_trans with (r2 + r3); auto with real.
Qed.
Hint Immediate Rplus_lt_le_compat Rplus_le_lt_compat: real.
Lemma Rplus_gt_ge_compat :
forall r1 r2 r3 r4, r1 > r2 -> r3 >= r4 -> r1 + r3 > r2 + r4.
Proof. auto using Rplus_lt_le_compat with rorders. Qed.
Lemma Rplus_ge_gt_compat :
forall r1 r2 r3 r4, r1 >= r2 -> r3 > r4 -> r1 + r3 > r2 + r4.
Proof. auto using Rplus_le_lt_compat with rorders. Qed.
(**********)
Lemma Rplus_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 + r2.
Proof.
intros x y; intros; apply Rlt_trans with x;
[ assumption
| pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
assumption ].
Qed.
Lemma Rplus_le_lt_0_compat : forall r1 r2, 0 <= r1 -> 0 < r2 -> 0 < r1 + r2.
Proof.
intros x y; intros; apply Rle_lt_trans with x;
[ assumption
| pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
assumption ].
Qed.
Lemma Rplus_lt_le_0_compat : forall r1 r2, 0 < r1 -> 0 <= r2 -> 0 < r1 + r2.
Proof.
intros x y; intros; rewrite <- Rplus_comm; apply Rplus_le_lt_0_compat;
assumption.
Qed.
Lemma Rplus_le_le_0_compat : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 + r2.
Proof.
intros x y; intros; apply Rle_trans with x;
[ assumption
| pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
assumption ].
Qed.
(**********)
Lemma sum_inequa_Rle_lt :
forall a x b c y d:R,
a <= x -> x < b -> c < y -> y <= d -> a + c < x + y < b + d.
Proof.
intros; split.
apply Rlt_le_trans with (a + y); auto with real.
apply Rlt_le_trans with (b + y); auto with real.
Qed.
(** *** Cancellation *)
Lemma Rplus_lt_reg_l : forall r r1 r2, r + r1 < r + r2 -> r1 < r2.
Proof.
intros; cut (- r + r + r1 < - r + r + r2).
rewrite Rplus_opp_l.
elim (Rplus_ne r1); elim (Rplus_ne r2); intros; rewrite <- H3; rewrite <- H1;
auto with zarith real.
rewrite Rplus_assoc; rewrite Rplus_assoc;
apply (Rplus_lt_compat_l (- r) (r + r1) (r + r2) H).
Qed.
Lemma Rplus_lt_reg_r : forall r r1 r2, r1 + r < r2 + r -> r1 < r2.
Proof.
intros.
apply (Rplus_lt_reg_l r).
now rewrite 2!(Rplus_comm r).
Qed.
Lemma Rplus_le_reg_l : forall r r1 r2, r + r1 <= r + r2 -> r1 <= r2.
Proof.
unfold Rle; intros; elim H; intro.
left; apply (Rplus_lt_reg_l r r1 r2 H0).
right; apply (Rplus_eq_reg_l r r1 r2 H0).
Qed.
Lemma Rplus_le_reg_r : forall r r1 r2, r1 + r <= r2 + r -> r1 <= r2.
Proof.
intros.
apply (Rplus_le_reg_l r).
now rewrite 2!(Rplus_comm r).
Qed.
Lemma Rplus_gt_reg_l : forall r r1 r2, r + r1 > r + r2 -> r1 > r2.
Proof.
unfold Rgt; intros; apply (Rplus_lt_reg_l r r2 r1 H).
Qed.
Lemma Rplus_ge_reg_l : forall r r1 r2, r + r1 >= r + r2 -> r1 >= r2.
Proof.
intros; apply Rle_ge; apply Rplus_le_reg_l with r; auto with real.
Qed.
(**********)
Lemma Rplus_le_reg_pos_r :
forall r1 r2 r3, 0 <= r2 -> r1 + r2 <= r3 -> r1 <= r3.
Proof.
intros x y z; intros; apply Rle_trans with (x + y);
[ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
assumption
| assumption ].
Qed.
Lemma Rplus_lt_reg_pos_r : forall r1 r2 r3, 0 <= r2 -> r1 + r2 < r3 -> r1 < r3.
Proof.
intros x y z; intros; apply Rle_lt_trans with (x + y);
[ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
assumption
| assumption ].
Qed.
Lemma Rplus_ge_reg_neg_r :
forall r1 r2 r3, 0 >= r2 -> r1 + r2 >= r3 -> r1 >= r3.
Proof.
intros x y z; intros; apply Rge_trans with (x + y);
[ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_ge_compat_l;
assumption
| assumption ].
Qed.
Lemma Rplus_gt_reg_neg_r : forall r1 r2 r3, 0 >= r2 -> r1 + r2 > r3 -> r1 > r3.
Proof.
intros x y z; intros; apply Rge_gt_trans with (x + y);
[ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_ge_compat_l;
assumption
| assumption ].
Qed.
(*********************************************************)
(** ** Order and opposite *)
(*********************************************************)
(** *** Contravariant compatibility *)
Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2.
Proof.
unfold Rgt; intros.
apply (Rplus_lt_reg_l (r2 + r1)).
replace (r2 + r1 + - r1) with r2 by ring.
replace (r2 + r1 + - r2) with r1 by ring.
exact H.
Qed.
Hint Resolve Ropp_gt_lt_contravar : core.
Lemma Ropp_lt_gt_contravar : forall r1 r2, r1 < r2 -> - r1 > - r2.
Proof.
unfold Rgt; auto with real.
Qed.
Hint Resolve Ropp_lt_gt_contravar: real.
(**********)
Lemma Ropp_lt_contravar : forall r1 r2, r2 < r1 -> - r1 < - r2.
Proof.
auto with real.
Qed.
Hint Resolve Ropp_lt_contravar: real.
Lemma Ropp_gt_contravar : forall r1 r2, r2 > r1 -> - r1 > - r2.
Proof. auto with real. Qed.
(**********)
Lemma Ropp_le_ge_contravar : forall r1 r2, r1 <= r2 -> - r1 >= - r2.
Proof.
unfold Rge; intros r1 r2 [H| H]; auto with real.
Qed.
Hint Resolve Ropp_le_ge_contravar: real.
Lemma Ropp_ge_le_contravar : forall r1 r2, r1 >= r2 -> - r1 <= - r2.
Proof.
unfold Rle; intros r1 r2 [H| H]; auto with real.
Qed.
Hint Resolve Ropp_ge_le_contravar: real.
(**********)
Lemma Ropp_le_contravar : forall r1 r2, r2 <= r1 -> - r1 <= - r2.
Proof.
intros r1 r2 H; elim H; auto with real.
Qed.
Hint Resolve Ropp_le_contravar: real.
Lemma Ropp_ge_contravar : forall r1 r2, r2 >= r1 -> - r1 >= - r2.
Proof. auto using Ropp_le_contravar with real. Qed.
(**********)
Lemma Ropp_0_lt_gt_contravar : forall r, 0 < r -> 0 > - r.
Proof.
intros; replace 0 with (-0); auto with real.
Qed.
Hint Resolve Ropp_0_lt_gt_contravar: real.
Lemma Ropp_0_gt_lt_contravar : forall r, 0 > r -> 0 < - r.
Proof.
intros; replace 0 with (-0); auto with real.
Qed.
Hint Resolve Ropp_0_gt_lt_contravar: real.
(**********)
Lemma Ropp_lt_gt_0_contravar : forall r, r > 0 -> - r < 0.
Proof.
intros; rewrite <- Ropp_0; auto with real.
Qed.
Hint Resolve Ropp_lt_gt_0_contravar: real.
Lemma Ropp_gt_lt_0_contravar : forall r, r < 0 -> - r > 0.
Proof.
intros; rewrite <- Ropp_0; auto with real.
Qed.
Hint Resolve Ropp_gt_lt_0_contravar: real.
(**********)
Lemma Ropp_0_le_ge_contravar : forall r, 0 <= r -> 0 >= - r.
Proof.
intros; replace 0 with (-0); auto with real.
Qed.
Hint Resolve Ropp_0_le_ge_contravar: real.
Lemma Ropp_0_ge_le_contravar : forall r, 0 >= r -> 0 <= - r.
Proof.
intros; replace 0 with (-0); auto with real.
Qed.
Hint Resolve Ropp_0_ge_le_contravar: real.
(** *** Cancellation *)
Lemma Ropp_lt_cancel : forall r1 r2, - r2 < - r1 -> r1 < r2.
Proof.
intros x y H'.
rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y);
auto with real.
Qed.
Hint Immediate Ropp_lt_cancel: real.
Lemma Ropp_gt_cancel : forall r1 r2, - r2 > - r1 -> r1 > r2.
Proof. auto using Ropp_lt_cancel with rorders. Qed.
Lemma Ropp_le_cancel : forall r1 r2, - r2 <= - r1 -> r1 <= r2.
Proof.
intros x y H.
elim H; auto with real.
intro H1; rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y);
rewrite H1; auto with real.
Qed.
Hint Immediate Ropp_le_cancel: real.
Lemma Ropp_ge_cancel : forall r1 r2, - r2 >= - r1 -> r1 >= r2.
Proof. auto using Ropp_le_cancel with rorders. Qed.
(*********************************************************)
(** ** Order and multiplication *)
(*********************************************************)
(** Remark: [Rmult_lt_compat_l] is an axiom *)
(** *** Covariant compatibility *)
Lemma Rmult_lt_compat_r : forall r r1 r2, 0 < r -> r1 < r2 -> r1 * r < r2 * r.
Proof.
intros; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); auto with real.
Qed.
Hint Resolve Rmult_lt_compat_r : core.
Lemma Rmult_gt_compat_r : forall r r1 r2, r > 0 -> r1 > r2 -> r1 * r > r2 * r.
Proof. eauto using Rmult_lt_compat_r with rorders. Qed.
Lemma Rmult_gt_compat_l : forall r r1 r2, r > 0 -> r1 > r2 -> r * r1 > r * r2.
Proof. eauto using Rmult_lt_compat_l with rorders. Qed.
(**********)
Lemma Rmult_le_compat_l :
forall r r1 r2, 0 <= r -> r1 <= r2 -> r * r1 <= r * r2.
Proof.
intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle;
auto with real.
right; rewrite <- H; do 2 rewrite Rmult_0_l; reflexivity.
Qed.
Hint Resolve Rmult_le_compat_l: real.
Lemma Rmult_le_compat_r :
forall r r1 r2, 0 <= r -> r1 <= r2 -> r1 * r <= r2 * r.
Proof.
intros r r1 r2 H; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r);
auto with real.
Qed.
Hint Resolve Rmult_le_compat_r: real.
Lemma Rmult_ge_compat_l :
forall r r1 r2, r >= 0 -> r1 >= r2 -> r * r1 >= r * r2.
Proof. eauto using Rmult_le_compat_l with rorders. Qed.
Lemma Rmult_ge_compat_r :
forall r r1 r2, r >= 0 -> r1 >= r2 -> r1 * r >= r2 * r.
Proof. eauto using Rmult_le_compat_r with rorders. Qed.
(**********)
Lemma Rmult_le_compat :
forall r1 r2 r3 r4,
0 <= r1 -> 0 <= r3 -> r1 <= r2 -> r3 <= r4 -> r1 * r3 <= r2 * r4.
Proof.
intros x y z t H' H'0 H'1 H'2.
apply Rle_trans with (r2 := x * t); auto with real.
repeat rewrite (fun x => Rmult_comm x t).
apply Rmult_le_compat_l; auto.
apply Rle_trans with z; auto.
Qed.
Hint Resolve Rmult_le_compat: real.
Lemma Rmult_ge_compat :
forall r1 r2 r3 r4,
r2 >= 0 -> r4 >= 0 -> r1 >= r2 -> r3 >= r4 -> r1 * r3 >= r2 * r4.
Proof. auto with real rorders. Qed.
Lemma Rmult_gt_0_lt_compat :
forall r1 r2 r3 r4,
r3 > 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4.
Proof.
intros; apply Rlt_trans with (r2 * r3); auto with real.
Qed.
(*********)
Lemma Rmult_le_0_lt_compat :
forall r1 r2 r3 r4,
0 <= r1 -> 0 <= r3 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4.
Proof.
intros; apply Rle_lt_trans with (r2 * r3);
[ apply Rmult_le_compat_r; [ assumption | left; assumption ]
| apply Rmult_lt_compat_l;
[ apply Rle_lt_trans with r1; assumption | assumption ] ].
Qed.
(*********)
Lemma Rmult_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 * r2.
Proof. intros; replace 0 with (0 * r2); auto with real. Qed.
Lemma Rmult_gt_0_compat : forall r1 r2, r1 > 0 -> r2 > 0 -> r1 * r2 > 0.
Proof Rmult_lt_0_compat.
(** *** Contravariant compatibility *)
Lemma Rmult_le_compat_neg_l :
forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r2 <= r * r1.
Proof.
intros; replace r with (- - r); auto with real.
do 2 rewrite (Ropp_mult_distr_l_reverse (- r)).
apply Ropp_le_contravar; auto with real.
Qed.
Hint Resolve Rmult_le_compat_neg_l: real.
Lemma Rmult_le_ge_compat_neg_l :
forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r1 >= r * r2.
Proof.
intros; apply Rle_ge; auto with real.
Qed.
Hint Resolve Rmult_le_ge_compat_neg_l: real.
Lemma Rmult_lt_gt_compat_neg_l :
forall r r1 r2, r < 0 -> r1 < r2 -> r * r1 > r * r2.
Proof.
intros; replace r with (- - r); auto with real.
rewrite (Ropp_mult_distr_l_reverse (- r));
rewrite (Ropp_mult_distr_l_reverse (- r)).
apply Ropp_lt_gt_contravar; auto with real.
Qed.
(** *** Cancellation *)
Lemma Rmult_lt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2.
Proof.
intros z x y H H0.
case (Rtotal_order x y); intros Eq0; auto; elim Eq0; clear Eq0; intros Eq0.
rewrite Eq0 in H0; exfalso; apply (Rlt_irrefl (z * y)); auto.
generalize (Rmult_lt_compat_l z y x H Eq0); intro; exfalso;
generalize (Rlt_trans (z * x) (z * y) (z * x) H0 H1);
intro; apply (Rlt_irrefl (z * x)); auto.
Qed.
Lemma Rmult_lt_reg_r : forall r r1 r2 : R, 0 < r -> r1 * r < r2 * r -> r1 < r2.
Proof.
intros.
apply Rmult_lt_reg_l with r.
exact H.
now rewrite 2!(Rmult_comm r).
Qed.
Lemma Rmult_gt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2.
Proof. eauto using Rmult_lt_reg_l with rorders. Qed.
Lemma Rmult_le_reg_l : forall r r1 r2, 0 < r -> r * r1 <= r * r2 -> r1 <= r2.
Proof.
intros z x y H H0; case H0; auto with real.
intros H1; apply Rlt_le.
apply Rmult_lt_reg_l with (r := z); auto.
intros H1; replace x with (/ z * (z * x)); auto with real.
replace y with (/ z * (z * y)).
rewrite H1; auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real.
Qed.
Lemma Rmult_le_reg_r : forall r r1 r2, 0 < r -> r1 * r <= r2 * r -> r1 <= r2.
Proof.
intros.
apply Rmult_le_reg_l with r.
exact H.
now rewrite 2!(Rmult_comm r).
Qed.
(*********************************************************)
(** ** Order and subtraction *)
(*********************************************************)
Lemma Rlt_minus : forall r1 r2, r1 < r2 -> r1 - r2 < 0.
Proof.
intros; apply (Rplus_lt_reg_l r2).
replace (r2 + (r1 - r2)) with r1 by ring.
now rewrite Rplus_0_r.
Qed.
Hint Resolve Rlt_minus: real.
Lemma Rgt_minus : forall r1 r2, r1 > r2 -> r1 - r2 > 0.
Proof.
intros; apply (Rplus_lt_reg_l r2).
replace (r2 + (r1 - r2)) with r1 by ring.
now rewrite Rplus_0_r.
Qed.
Lemma Rlt_Rminus : forall a b:R, a < b -> 0 < b - a.
Proof.
intros a b; apply Rgt_minus.
Qed.
(**********)
Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0.
Proof.
destruct 1; unfold Rle; auto with real.
Qed.
Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0.
Proof.
destruct 1.
auto using Rgt_minus, Rgt_ge.
right; auto using Rminus_diag_eq with rorders.
Qed.
(**********)
Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2.
Proof.
intros; replace r1 with (r1 - r2 + r2).
pattern r2 at 3; replace r2 with (0 + r2); auto with real.
ring.
Qed.
Lemma Rminus_gt : forall r1 r2, r1 - r2 > 0 -> r1 > r2.
Proof.
intros; replace r2 with (0 + r2); auto with real.
replace r1 with (r1 - r2 + r2).
apply Rplus_gt_compat_r; assumption.
ring.
Qed.
Lemma Rminus_gt_0_lt : forall a b, 0 < b - a -> a < b.
Proof. intro; intro; apply Rminus_gt. Qed.
(**********)
Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2.
Proof.
intros; replace r1 with (r1 - r2 + r2).
pattern r2 at 3; replace r2 with (0 + r2); auto with real.
ring.
Qed.
Lemma Rminus_ge : forall r1 r2, r1 - r2 >= 0 -> r1 >= r2.
Proof.
intros; replace r2 with (0 + r2); auto with real.
replace r1 with (r1 - r2 + r2).
apply Rplus_ge_compat_r; assumption.
ring.
Qed.
(**********)
Lemma tech_Rplus : forall r (s:R), 0 <= r -> 0 < s -> r + s <> 0.
Proof.
intros; apply not_eq_sym; apply Rlt_not_eq.
rewrite Rplus_comm; replace 0 with (0 + 0); auto with real.
Qed.
Hint Immediate tech_Rplus: real.
(*********************************************************)
(** ** Order and square function *)
(*********************************************************)
Lemma Rle_0_sqr : forall r, 0 <= Rsqr r.
Proof.
intro; case (Rlt_le_dec r 0); unfold Rsqr; intro.
replace (r * r) with (- r * - r); auto with real.
replace 0 with (- r * 0); auto with real.
replace 0 with (0 * r); auto with real.
Qed.
(***********)
Lemma Rlt_0_sqr : forall r, r <> 0 -> 0 < Rsqr r.
Proof.
intros; case (Rdichotomy r 0); trivial; unfold Rsqr; intro.
replace (r * r) with (- r * - r); auto with real.
replace 0 with (- r * 0); auto with real.
replace 0 with (0 * r); auto with real.
Qed.
Hint Resolve Rle_0_sqr Rlt_0_sqr: real.
(***********)
Lemma Rplus_sqr_eq_0_l : forall r1 r2, Rsqr r1 + Rsqr r2 = 0 -> r1 = 0.
Proof.
intros a b; intros; apply Rsqr_0_uniq; apply Rplus_eq_0_l with (Rsqr b);
auto with real.
Qed.
Lemma Rplus_sqr_eq_0 :
forall r1 r2, Rsqr r1 + Rsqr r2 = 0 -> r1 = 0 /\ r2 = 0.
Proof.
intros a b; split.
apply Rplus_sqr_eq_0_l with b; auto with real.
apply Rplus_sqr_eq_0_l with a; auto with real.
rewrite Rplus_comm; auto with real.
Qed.
(*********************************************************)
(** ** Zero is less than one *)
(*********************************************************)
Lemma Rlt_0_1 : 0 < 1.
Proof.
replace 1 with (Rsqr 1); auto with real.
unfold Rsqr; auto with real.
Qed.
Hint Resolve Rlt_0_1: real.
Lemma Rle_0_1 : 0 <= 1.
Proof.
left.
exact Rlt_0_1.
Qed.
(*********************************************************)
(** ** Order and inverse *)
(*********************************************************)
Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r.
Proof.
intros; apply Rnot_le_lt; red; intros.
absurd (1 <= 0); auto with real.
replace 1 with (r * / r); auto with real.
replace 0 with (r * 0); auto with real.
Qed.
Hint Resolve Rinv_0_lt_compat: real.
(*********)
Lemma Rinv_lt_0_compat : forall r, r < 0 -> / r < 0.
Proof.
intros; apply Rnot_le_lt; red; intros.
absurd (1 <= 0); auto with real.
replace 1 with (r * / r); auto with real.
replace 0 with (r * 0); auto with real.
Qed.
Hint Resolve Rinv_lt_0_compat: real.
(*********)
Lemma Rinv_lt_contravar : forall r1 r2, 0 < r1 * r2 -> r1 < r2 -> / r2 < / r1.
Proof.
intros; apply Rmult_lt_reg_l with (r1 * r2); auto with real.
case (Rmult_neq_0_reg r1 r2); intros; auto with real.
replace (r1 * r2 * / r2) with r1.
replace (r1 * r2 * / r1) with r2; trivial.
symmetry ; auto with real.
symmetry ; auto with real.
Qed.
Lemma Rinv_1_lt_contravar : forall r1 r2, 1 <= r1 -> r1 < r2 -> / r2 < / r1.
Proof.
intros x y H' H'0.
cut (0 < x); [ intros Lt0 | apply Rlt_le_trans with (r2 := 1) ];
auto with real.
apply Rmult_lt_reg_l with (r := x); auto with real.
rewrite (Rmult_comm x (/ x)); rewrite Rinv_l; auto with real.
apply Rmult_lt_reg_l with (r := y); auto with real.
apply Rlt_trans with (r2 := x); auto.
cut (y * (x * / y) = x).
intro H1; rewrite H1; rewrite (Rmult_1_r y); auto.
rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite (Rmult_comm y (/ y));
rewrite Rinv_l; auto with real.
apply Rlt_dichotomy_converse; right.
red; apply Rlt_trans with (r2 := x); auto with real.
Qed.
Hint Resolve Rinv_1_lt_contravar: real.
(*********************************************************)
(** ** Miscellaneous *)
(*********************************************************)
(**********)
Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1.
Proof.
intros.
apply Rlt_le_trans with 1; auto with real.
pattern 1 at 1; replace 1 with (0 + 1); auto with real.
Qed.
Hint Resolve Rle_lt_0_plus_1: real.
(**********)
Lemma Rlt_plus_1 : forall r, r < r + 1.
Proof.
intros.
pattern r at 1; replace r with (r + 0); auto with real.
Qed.
Hint Resolve Rlt_plus_1: real.
(**********)
Lemma tech_Rgt_minus : forall r1 r2, 0 < r2 -> r1 > r1 - r2.
Proof.
red; unfold Rminus; intros.
pattern r1 at 2; replace r1 with (r1 + 0); auto with real.
Qed.
(*********************************************************)
(** ** Injection from [N] to [R] *)
(*********************************************************)
(**********)
Lemma S_INR : forall n:nat, INR (S n) = INR n + 1.
Proof.
intro; case n; auto with real.
Qed.
(**********)
Lemma S_O_plus_INR : forall n:nat, INR (1 + n) = INR 1 + INR n.
Proof.
intro; simpl; case n; intros; auto with real.
Qed.
(**********)
Lemma plus_INR : forall n m:nat, INR (n + m) = INR n + INR m.
Proof.
intros n m; induction n as [| n Hrecn].
simpl; auto with real.
replace (S n + m)%nat with (S (n + m)); auto with arith.
repeat rewrite S_INR.
rewrite Hrecn; ring.
Qed.
Hint Resolve plus_INR: real.
(**********)
Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m.
Proof.
intros n m le; pattern m, n; apply le_elim_rel; auto with real.
intros; rewrite <- minus_n_O; auto with real.
intros; repeat rewrite S_INR; simpl.
rewrite H0; ring.
Qed.
Hint Resolve minus_INR: real.
(*********)
Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m.
Proof.
intros n m; induction n as [| n Hrecn].
simpl; auto with real.
intros; repeat rewrite S_INR; simpl.
rewrite plus_INR; rewrite Hrecn; ring.
Qed.
Hint Resolve mult_INR: real.
Lemma pow_INR (m n: nat) : INR (m ^ n) = pow (INR m) n.
Proof. now induction n as [|n IHn];[ | simpl; rewrite mult_INR, IHn]. Qed.
(*********)
Lemma lt_0_INR : forall n:nat, (0 < n)%nat -> 0 < INR n.
Proof.
simple induction 1; intros; auto with real.
rewrite S_INR; auto with real.
Qed.
Hint Resolve lt_0_INR: real.
Lemma lt_INR : forall n m:nat, (n < m)%nat -> INR n < INR m.
Proof.
simple induction 1; intros; auto with real.
rewrite S_INR; auto with real.
rewrite S_INR; apply Rlt_trans with (INR m0); auto with real.
Qed.
Hint Resolve lt_INR: real.
Lemma lt_1_INR : forall n:nat, (1 < n)%nat -> 1 < INR n.
Proof.
apply lt_INR.
Qed.
Hint Resolve lt_1_INR: real.
(**********)
Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (Pos.to_nat p).
Proof.
intro; apply lt_0_INR.
simpl; auto with real.
apply Pos2Nat.is_pos.
Qed.
Hint Resolve pos_INR_nat_of_P: real.
(**********)
Lemma pos_INR : forall n:nat, 0 <= INR n.
Proof.
intro n; case n.
simpl; auto with real.
auto with arith real.
Qed.
Hint Resolve pos_INR: real.
Lemma INR_lt : forall n m:nat, INR n < INR m -> (n < m)%nat.
Proof.
intros n m. revert n.
induction m ; intros n H.
- elim (Rlt_irrefl 0).
apply Rle_lt_trans with (2 := H).
apply pos_INR.
- destruct n as [|n].
apply Nat.lt_0_succ.
apply lt_n_S, IHm.
rewrite 2!S_INR in H.
apply Rplus_lt_reg_r with (1 := H).
Qed.
Hint Resolve INR_lt: real.
(*********)
Lemma le_INR : forall n m:nat, (n <= m)%nat -> INR n <= INR m.
Proof.
simple induction 1; intros; auto with real.
rewrite S_INR.
apply Rle_trans with (INR m0); auto with real.
Qed.
Hint Resolve le_INR: real.
(**********)
Lemma INR_not_0 : forall n:nat, INR n <> 0 -> n <> 0%nat.
Proof.
red; intros n H H1.
apply H.
rewrite H1; trivial.
Qed.
Hint Immediate INR_not_0: real.
(**********)
Lemma not_0_INR : forall n:nat, n <> 0%nat -> INR n <> 0.
Proof.
intro n; case n.
intro; absurd (0%nat = 0%nat); trivial.
intros; rewrite S_INR.
apply Rgt_not_eq; red; auto with real.
Qed.
Hint Resolve not_0_INR: real.
Lemma not_INR : forall n m:nat, n <> m -> INR n <> INR m.
Proof.
intros n m H; case (le_or_lt n m); intros H1.
case (le_lt_or_eq _ _ H1); intros H2.
apply Rlt_dichotomy_converse; auto with real.
exfalso; auto.
apply not_eq_sym; apply Rlt_dichotomy_converse; auto with real.
Qed.
Hint Resolve not_INR: real.
Lemma INR_eq : forall n m:nat, INR n = INR m -> n = m.
Proof.
intros n m HR.
destruct (dec_eq_nat n m) as [H|H].
exact H.
now apply not_INR in H.
Qed.
Hint Resolve INR_eq: real.
Lemma INR_le : forall n m:nat, INR n <= INR m -> (n <= m)%nat.
Proof.
intros; elim H; intro.
generalize (INR_lt n m H0); intro; auto with arith.
generalize (INR_eq n m H0); intro; rewrite H1; auto.
Qed.
Hint Resolve INR_le: real.
Lemma not_1_INR : forall n:nat, n <> 1%nat -> INR n <> 1.
Proof.
intros n.
apply not_INR.
Qed.
Hint Resolve not_1_INR: real.
(*********************************************************)
(** ** Injection from [Z] to [R] *)
(*********************************************************)
(**********)
Lemma IZN : forall n:Z, (0 <= n)%Z -> exists m : nat, n = Z.of_nat m.
Proof.
intros z; idtac; apply Z_of_nat_complete; assumption.
Qed.
Lemma INR_IPR : forall p, INR (Pos.to_nat p) = IPR p.
Proof.
assert (H: forall p, 2 * INR (Pos.to_nat p) = IPR_2 p).
induction p as [p|p|] ; simpl IPR_2.
rewrite Pos2Nat.inj_xI, S_INR, mult_INR, <- IHp.
now rewrite (Rplus_comm (2 * _)).
now rewrite Pos2Nat.inj_xO, mult_INR, <- IHp.
apply Rmult_1_r.
intros [p|p|] ; unfold IPR.
rewrite Pos2Nat.inj_xI, S_INR, mult_INR, <- H.
apply Rplus_comm.
now rewrite Pos2Nat.inj_xO, mult_INR, <- H.
easy.
Qed.
(**********)
Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z.of_nat n).
Proof.
intros [|n].
easy.
simpl Z.of_nat. unfold IZR.
now rewrite <- INR_IPR, SuccNat2Pos.id_succ.
Qed.
Lemma plus_IZR_NEG_POS :
forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q).
Proof.
intros p q; simpl. rewrite Z.pos_sub_spec.
case Pos.compare_spec; intros H; unfold IZR.
subst. ring.
rewrite <- 3!INR_IPR, Pos2Nat.inj_sub by trivial.
rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_lt).
ring.
rewrite <- 3!INR_IPR, Pos2Nat.inj_sub by trivial.
rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_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. unfold IZR. rewrite <- 3!INR_IPR, Pos2Nat.inj_add. apply plus_INR.
apply plus_IZR_NEG_POS.
rewrite Z.add_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS.
simpl. unfold IZR. rewrite <- 3!INR_IPR, Pos2Nat.inj_add, plus_INR.
apply Ropp_plus_distr.
Qed.
(**********)
Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m.
Proof.
intros z t; case z; case t; simpl; auto with real;
unfold IZR; intros m n; rewrite <- 3!INR_IPR, Pos2Nat.inj_mul, mult_INR; ring.
Qed.
Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Z.pow z (Z.of_nat n)).
Proof.
intros z [|n];simpl;trivial.
rewrite Zpower_pos_nat.
rewrite SuccNat2Pos.id_succ. unfold Zpower_nat;simpl.
rewrite mult_IZR.
induction n;simpl;trivial.
rewrite mult_IZR;ring[IHn].
Qed.
(**********)
Lemma succ_IZR : forall n:Z, IZR (Z.succ n) = IZR n + 1.
Proof.
intro; unfold Z.succ; apply plus_IZR.
Qed.
(**********)
Lemma opp_IZR : forall n:Z, IZR (- n) = - IZR n.
Proof.
intros [|z|z]; unfold IZR; simpl; auto with real.
Qed.
Definition Ropp_Ropp_IZR := opp_IZR.
Lemma minus_IZR : forall n m:Z, IZR (n - m) = IZR n - IZR m.
Proof.
intros; unfold Z.sub, Rminus.
rewrite <- opp_IZR.
apply plus_IZR.
Qed.
(**********)
Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m).
Proof.
intros z1 z2; unfold Rminus; unfold Z.sub.
rewrite <- (Ropp_Ropp_IZR z2); symmetry ; apply plus_IZR.
Qed.
(**********)
Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z.
Proof.
intro z; case z; simpl; intros.
elim (Rlt_irrefl _ H).
easy.
elim (Rlt_not_le _ _ H).
unfold IZR.
rewrite <- INR_IPR.
auto with real.
Qed.
(**********)
Lemma lt_IZR : forall n m:Z, IZR n < IZR m -> (n < m)%Z.
Proof.
intros z1 z2 H; apply Z.lt_0_sub.
apply lt_0_IZR.
rewrite <- Z_R_minus.
exact (Rgt_minus (IZR z2) (IZR z1) H).
Qed.
(**********)
Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z.
Proof.
intro z; destruct z; simpl; intros; auto with zarith.
elim Rgt_not_eq with (2 := H).
unfold IZR. rewrite <- INR_IPR.
apply lt_0_INR, Pos2Nat.is_pos.
elim Rlt_not_eq with (2 := H).
unfold IZR. rewrite <- INR_IPR.
apply Ropp_lt_gt_0_contravar, lt_0_INR, Pos2Nat.is_pos.
Qed.
(**********)
Lemma eq_IZR : forall n m:Z, IZR n = IZR m -> n = m.
Proof.
intros z1 z2 H; generalize (Rminus_diag_eq (IZR z1) (IZR z2) H);
rewrite (Z_R_minus z1 z2); intro; generalize (eq_IZR_R0 (z1 - z2) H0);
intro; omega.
Qed.
(**********)
Lemma not_0_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0.
Proof.
intros z H; red; intros H0; case H.
apply eq_IZR; auto.
Qed.
(*********)
Lemma le_0_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z.
Proof.
unfold Rle; intros z [H| H].
red; intro; apply (Z.lt_le_incl 0 z (lt_0_IZR z H)); assumption.
rewrite (eq_IZR_R0 z); auto with zarith real.
Qed.
(**********)
Lemma le_IZR : forall n m:Z, IZR n <= IZR m -> (n <= m)%Z.
Proof.
unfold Rle; intros z1 z2 [H| H].
apply (Z.lt_le_incl z1 z2); auto with real.
apply lt_IZR; trivial.
rewrite (eq_IZR z1 z2); auto with zarith real.
Qed.
(**********)
Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z.
Proof.
intros n.
apply le_IZR.
Qed.
(**********)
Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m.
Proof.
intros m n H; apply Rnot_lt_ge; red; intro.
generalize (lt_IZR m n H0); intro; omega.
Qed.
Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m.
Proof.
intros m n H; apply Rnot_gt_le; red; intro.
unfold Rgt in H0; generalize (lt_IZR n m H0); intro; omega.
Qed.
Lemma IZR_lt : forall n m:Z, (n < m)%Z -> IZR n < IZR m.
Proof.
intros m n H; cut (m <= n)%Z.
intro H0; elim (IZR_le m n H0); intro; auto.
generalize (eq_IZR m n H1); intro; exfalso; omega.
omega.
Qed.
Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2.
Proof.
intros; red; intro; elim H; apply eq_IZR; assumption.
Qed.
Hint Extern 0 (IZR _ <= IZR _) => apply IZR_le, Zle_bool_imp_le, eq_refl : real.
Hint Extern 0 (IZR _ >= IZR _) => apply Rle_ge, IZR_le, Zle_bool_imp_le, eq_refl : real.
Hint Extern 0 (IZR _ < IZR _) => apply IZR_lt, eq_refl : real.
Hint Extern 0 (IZR _ > IZR _) => apply IZR_lt, eq_refl : real.
Hint Extern 0 (IZR _ <> IZR _) => apply IZR_neq, Zeq_bool_neq, eq_refl : real.
Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z.
Proof.
intros z [H1 H2].
apply Z.le_antisymm.
apply Z.lt_succ_r; apply lt_IZR; trivial.
change 0%Z with (Z.succ (-1)).
apply Z.le_succ_l; apply lt_IZR; trivial.
Qed.
Lemma one_IZR_r_R1 :
forall r (n m:Z), r < IZR n <= r + 1 -> r < IZR m <= r + 1 -> n = m.
Proof.
intros r z x [H1 H2] [H3 H4].
cut ((z - x)%Z = 0%Z); auto with zarith.
apply one_IZR_lt1.
rewrite <- Z_R_minus; split.
replace (-1) with (r - (r + 1)).
unfold Rminus; apply Rplus_lt_le_compat; auto with real.
ring.
replace 1 with (r + 1 - r).
unfold Rminus; apply Rplus_le_lt_compat; auto with real.
ring.
Qed.
(**********)
Lemma single_z_r_R1 :
forall r (n m:Z),
r < IZR n -> IZR n <= r + 1 -> r < IZR m -> IZR m <= r + 1 -> n = m.
Proof.
intros; apply one_IZR_r_R1 with r; auto.
Qed.
(**********)
Lemma tech_single_z_r_R1 :
forall r (n:Z),
r < IZR n ->
IZR n <= r + 1 ->
(exists s : Z, s <> n /\ r < IZR s /\ IZR s <= r + 1) -> False.
Proof.
intros r z H1 H2 [s [H3 [H4 H5]]].
apply H3; apply single_z_r_R1 with r; trivial.
Qed.
(*********)
Lemma Rmult_le_pos : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 * r2.
Proof.
intros x y H H0; rewrite <- (Rmult_0_l x); rewrite <- (Rmult_comm x);
apply (Rmult_le_compat_l x 0 y H H0).
Qed.
Lemma Rinv_le_contravar :
forall x y, 0 < x -> x <= y -> / y <= / x.
Proof.
intros x y H1 [H2|H2].
apply Rlt_le.
apply Rinv_lt_contravar with (2 := H2).
apply Rmult_lt_0_compat with (1 := H1).
now apply Rlt_trans with x.
rewrite H2.
apply Rle_refl.
Qed.
Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x.
Proof.
intros x y H _.
apply Rinv_le_contravar with (1 := H).
Qed.
Lemma Ropp_div : forall x y, -x/y = - (x / y).
intros x y; unfold Rdiv; ring.
Qed.
Lemma double : forall r1, 2 * r1 = r1 + r1.
Proof.
intro; ring.
Qed.
Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2.
Proof.
intro; rewrite <- double; unfold Rdiv; rewrite <- Rmult_assoc;
symmetry ; apply Rinv_r_simpl_m.
now apply not_0_IZR.
Qed.
Lemma R_rm : ring_morph
0%R 1%R Rplus Rmult Rminus Ropp eq
0%Z 1%Z Zplus Zmult Zminus Z.opp Zeq_bool IZR.
Proof.
constructor ; try easy.
exact plus_IZR.
exact minus_IZR.
exact mult_IZR.
exact opp_IZR.
intros x y H.
apply f_equal.
now apply Zeq_bool_eq.
Qed.
Lemma Zeq_bool_IZR x y :
IZR x = IZR y -> Zeq_bool x y = true.
Proof.
intros H.
apply Zeq_is_eq_bool.
now apply eq_IZR.
Qed.
Add Field RField : Rfield
(completeness Zeq_bool_IZR, morphism R_rm, constants [IZR_tac], power_tac R_power_theory [Rpow_tac]).
(*********************************************************)
(** ** Other rules about < and <= *)
(*********************************************************)
Lemma Rmult_ge_0_gt_0_lt_compat :
forall r1 r2 r3 r4,
r3 >= 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4.
Proof.
intros; apply Rle_lt_trans with (r2 * r3); auto with real.
Qed.
Lemma le_epsilon :
forall r1 r2, (forall eps:R, 0 < eps -> r1 <= r2 + eps) -> r1 <= r2.
Proof.
intros x y H.
destruct (Rle_or_lt x y) as [H1|H1].
exact H1.
apply Rplus_le_reg_r with x.
replace (y + x) with (2 * (y + (x - y) * / 2)) by field.
replace (x + x) with (2 * x) by ring.
apply Rmult_le_compat_l.
now apply (IZR_le 0 2).
apply H.
apply Rmult_lt_0_compat.
now apply Rgt_minus.
apply Rinv_0_lt_compat, Rlt_0_2.
Qed.
(**********)
Lemma completeness_weak :
forall E:R -> Prop,
bound E -> (exists x : R, E x) -> exists m : R, is_lub E m.
Proof.
intros; elim (completeness E H H0); intros; split with x; assumption.
Qed.
Lemma Rdiv_lt_0_compat : forall a b, 0 < a -> 0 < b -> 0 < a/b.
Proof.
intros; apply Rmult_lt_0_compat;[|apply Rinv_0_lt_compat]; assumption.
Qed.
Lemma Rdiv_plus_distr : forall a b c, (a + b)/c = a/c + b/c.
intros a b c; apply Rmult_plus_distr_r.
Qed.
Lemma Rdiv_minus_distr : forall a b c, (a - b)/c = a/c - b/c.
intros a b c; unfold Rminus, Rdiv; rewrite Rmult_plus_distr_r; ring.
Qed.
(* A test for equality function. *)
Lemma Req_EM_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}.
Proof.
intros; destruct (total_order_T r1 r2) as [[H|]|H].
- right; red; intros ->; elim (Rlt_irrefl r2 H).
- left; assumption.
- right; red; intros ->; elim (Rlt_irrefl r2 H).
Qed.
(*********************************************************)
(** * Definitions of new types *)
(*********************************************************)
Record nonnegreal : Type := mknonnegreal
{nonneg :> R; cond_nonneg : 0 <= nonneg}.
Record posreal : Type := mkposreal {pos :> R; cond_pos : 0 < pos}.
Record nonposreal : Type := mknonposreal
{nonpos :> R; cond_nonpos : nonpos <= 0}.
Record negreal : Type := mknegreal {neg :> R; cond_neg : neg < 0}.
Record nonzeroreal : Type := mknonzeroreal
{nonzero :> R; cond_nonzero : nonzero <> 0}.
(** Compatibility *)
Notation prod_neq_R0 := Rmult_integral_contrapositive_currified (only parsing).
Notation minus_Rgt := Rminus_gt (only parsing).
Notation minus_Rge := Rminus_ge (only parsing).
Notation plus_le_is_le := Rplus_le_reg_pos_r (only parsing).
Notation plus_lt_is_lt := Rplus_lt_reg_pos_r (only parsing).
Notation INR_lt_1 := lt_1_INR (only parsing).
Notation lt_INR_0 := lt_0_INR (only parsing).
Notation not_nm_INR := not_INR (only parsing).
Notation INR_pos := pos_INR_nat_of_P (only parsing).
Notation not_INR_O := INR_not_0 (only parsing).
Notation not_O_INR := not_0_INR (only parsing).
Notation not_O_IZR := not_0_IZR (only parsing).
Notation le_O_IZR := le_0_IZR (only parsing).
Notation lt_O_IZR := lt_0_IZR (only parsing).
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