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diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v new file mode 100644 index 00000000..a23f53ff --- /dev/null +++ b/theories/Reals/RIneq.v @@ -0,0 +1,1631 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RIneq.v,v 1.23.2.1 2004/07/16 19:31:11 herbelin Exp $ i*) + +(***************************************************************************) +(** Basic lemmas for the classical reals numbers *) +(***************************************************************************) + +Require Export Raxioms. +Require Export ZArithRing. +Require Import Omega. +Require Export Field. + +Open Local Scope Z_scope. +Open Local Scope R_scope. + +Implicit Type r : R. + +(***************************************************************************) +(** Instantiating Ring tactic on reals *) +(***************************************************************************) + +Lemma RTheory : Ring_Theory Rplus Rmult 1 0 Ropp (fun x y:R => false). + split. + exact Rplus_comm. + symmetry in |- *; apply Rplus_assoc. + exact Rmult_comm. + symmetry in |- *; apply Rmult_assoc. + intro; apply Rplus_0_l. + intro; apply Rmult_1_l. + exact Rplus_opp_r. + intros. + rewrite Rmult_comm. + rewrite (Rmult_comm n p). + rewrite (Rmult_comm m p). + apply Rmult_plus_distr_l. + intros; contradiction. +Defined. + +Add Field R Rplus Rmult 1 0 Ropp (fun x y:R => false) Rinv RTheory Rinv_l + with minus := Rminus div := Rdiv. + +(**************************************************************************) +(** Relation between orders and equality *) +(**************************************************************************) + +(**********) +Lemma Rlt_irrefl : forall r, ~ r < r. + generalize Rlt_asym. intuition eauto. +Qed. +Hint Resolve Rlt_irrefl: real. + +Lemma Rle_refl : forall r, r <= r. +intro; right; reflexivity. +Qed. + +Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2. + red in |- *; intros r1 r2 H H0; apply (Rlt_irrefl r1). + pattern r1 at 2 in |- *; rewrite H0; trivial. +Qed. + +Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2. +intros; apply sym_not_eq; apply Rlt_not_eq; auto with real. +Qed. + +(**********) +Lemma Rlt_dichotomy_converse : forall r1 r2, r1 < r2 \/ r1 > r2 -> r1 <> r2. +generalize Rlt_not_eq Rgt_not_eq. intuition eauto. +Qed. +Hint Resolve Rlt_dichotomy_converse: real. + +(** Reasoning by case on equalities and order *) + +(**********) +Lemma Req_dec : forall r1 r2, r1 = r2 \/ r1 <> r2. +intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse; + intuition eauto 3. +Qed. +Hint Resolve Req_dec: real. + +(**********) +Lemma Rtotal_order : forall r1 r2, r1 < r2 \/ r1 = r2 \/ r1 > r2. +intros; generalize (total_order_T r1 r2); tauto. +Qed. + +(**********) +Lemma Rdichotomy : forall r1 r2, r1 <> r2 -> r1 < r2 \/ r1 > r2. +intros; generalize (total_order_T r1 r2); tauto. +Qed. + + +(*********************************************************************************) +(** Order Lemma : relating [<], [>], [<=] and [>=] *) +(*********************************************************************************) + +(**********) +Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2. +intros; red in |- *; tauto. +Qed. +Hint Resolve Rlt_le: real. + +(**********) +Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1. +destruct 1; red in |- *; auto with real. +Qed. + +Hint Immediate Rle_ge: real. + +(**********) +Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1. +destruct 1; red in |- *; auto with real. +Qed. + +Hint Resolve Rge_le: real. + +(**********) +Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1. +intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle in |- *; tauto. +Qed. + +Hint Immediate Rnot_le_lt: real. + +Lemma Rnot_ge_lt : forall r1 r2, ~ r1 >= r2 -> r1 < r2. +intros; apply Rnot_le_lt; auto with real. +Qed. + +(**********) +Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2. +generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle in |- *. +intuition eauto 3. +Qed. + +Lemma Rgt_not_le : forall r1 r2, r1 > r2 -> ~ r1 <= r2. +Proof Rlt_not_le. + +Hint Immediate Rlt_not_le: real. + +Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2. +intros r1 r2. generalize (Rlt_asym r1 r2) (Rlt_dichotomy_converse r1 r2). +unfold Rle in |- *; intuition. +Qed. + +(**********) +Lemma Rlt_not_ge : forall r1 r2, r1 < r2 -> ~ r1 >= r2. +generalize Rlt_not_le. unfold Rle, Rge in |- *. intuition eauto 3. +Qed. + +Hint Immediate Rlt_not_ge: real. + +(**********) +Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2. +unfold Rle in |- *; tauto. +Qed. +Hint Immediate Req_le: real. + +Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2. +unfold Rge in |- *; tauto. +Qed. +Hint Immediate Req_ge: real. + +Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2. +unfold Rle in |- *; auto. +Qed. +Hint Immediate Req_le_sym: real. + +Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2. +unfold Rge in |- *; auto. +Qed. +Hint Immediate Req_ge_sym: real. + +Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2. +intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle in |- *; intuition. +Qed. +Hint Resolve Rle_antisym: real. + +(**********) +Lemma Rle_le_eq : forall r1 r2, r1 <= r2 /\ r2 <= r1 <-> r1 = r2. +intuition. +Qed. + +Lemma Rlt_eq_compat : + forall r1 r2 r3 r4, r1 = r2 -> r2 < r4 -> r4 = r3 -> r1 < r3. +intros x x' y y'; intros; replace x with x'; replace y with y'; assumption. +Qed. + +(**********) +Lemma Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3. +generalize trans_eq Rlt_trans Rlt_eq_compat. +unfold Rle in |- *. +intuition eauto 2. +Qed. + +(**********) +Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3. +generalize Rlt_trans Rlt_eq_compat. +unfold Rle in |- *. +intuition eauto 2. +Qed. + +(**********) +Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3. +generalize Rlt_trans Rlt_eq_compat; unfold Rle in |- *; intuition eauto 2. +Qed. + + +(** Decidability of the order *) +Lemma Rlt_dec : forall r1 r2, {r1 < r2} + {~ r1 < r2}. +intros; generalize (total_order_T r1 r2) (Rlt_dichotomy_converse r1 r2); + intuition. +Qed. + +(**********) +Lemma Rle_dec : forall r1 r2, {r1 <= r2} + {~ r1 <= r2}. +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}. +intros; unfold Rgt in |- *; intros; apply Rlt_dec. +Qed. + +(**********) +Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}. +intros; generalize (Rle_dec r2 r1); intuition. +Qed. + +Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}. +intros; generalize (total_order_T r1 r2); intuition. +Qed. + +Lemma Rle_or_lt : forall r1 r2, r1 <= r2 \/ r2 < r1. +intros n m; elim (Rlt_le_dec m n); auto with real. +Qed. + +Lemma Rle_lt_or_eq_dec : forall r1 r2, r1 <= r2 -> {r1 < r2} + {r1 = r2}. +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}. +intros n m p q; intros; generalize (Rlt_le_dec m q); intuition. +Qed. + +(****************************************************************) +(** Field Lemmas *) +(* This part contains lemma involving the Fields operations *) +(****************************************************************) +(*********************************************************) +(** Addition *) +(*********************************************************) + +Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r. +intro; split; ring. +Qed. +Hint Resolve Rplus_ne: real v62. + +Lemma Rplus_0_r : forall r, r + 0 = r. +intro; ring. +Qed. +Hint Resolve Rplus_0_r: real. + +(**********) +Lemma Rplus_opp_l : forall r, - r + r = 0. + intro; ring. +Qed. +Hint Resolve Rplus_opp_l: real. + + +(**********) +Lemma Rplus_opp_r_uniq : forall r1 r2, r1 + r2 = 0 -> r2 = - r1. + intros x y H; replace y with (- x + x + y); + [ rewrite Rplus_assoc; rewrite H; ring | ring ]. +Qed. + +(*i New i*) +Hint Resolve (f_equal (A:=R)): real. + +Lemma Rplus_eq_compat_l : forall r r1 r2, r1 = r2 -> r + r1 = r + r2. + auto with real. +Qed. + +(*i Old i*)Hint Resolve Rplus_eq_compat_l: v62. + +(**********) +Lemma Rplus_eq_reg_l : forall r r1 r2, r + r1 = r + r2 -> r1 = r2. + 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_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0. + intros r b; pattern r at 2 in |- *; replace r with (r + 0); eauto with real. +Qed. + +(***********************************************************) +(** Multiplication *) +(***********************************************************) + +(**********) +Lemma Rinv_r : forall r, r <> 0 -> r * / r = 1. + intros; rewrite Rmult_comm; auto with real. +Qed. +Hint Resolve Rinv_r: real. + +Lemma Rinv_l_sym : forall r, r <> 0 -> 1 = / r * r. + symmetry in |- *; auto with real. +Qed. + +Lemma Rinv_r_sym : forall r, r <> 0 -> 1 = r * / r. + symmetry in |- *; auto with real. +Qed. +Hint Resolve Rinv_l_sym Rinv_r_sym: real. + + +(**********) +Lemma Rmult_0_r : forall r, r * 0 = 0. +intro; ring. +Qed. +Hint Resolve Rmult_0_r: real v62. + +(**********) +Lemma Rmult_0_l : forall r, 0 * r = 0. +intro; ring. +Qed. +Hint Resolve Rmult_0_l: real v62. + +(**********) +Lemma Rmult_ne : forall r, r * 1 = r /\ 1 * r = r. +intro; split; ring. +Qed. +Hint Resolve Rmult_ne: real v62. + +(**********) +Lemma Rmult_1_r : forall r, r * 1 = r. +intro; ring. +Qed. +Hint Resolve Rmult_1_r: real. + +(**********) +Lemma Rmult_eq_compat_l : forall r r1 r2, r1 = r2 -> r * r1 = r * r2. + auto with real. +Qed. + +(*i OLD i*)Hint Resolve Rmult_eq_compat_l: v62. + +(**********) +Lemma Rmult_eq_reg_l : forall r r1 r2, r * r1 = r * r2 -> r <> 0 -> r1 = r2. + intros; transitivity (/ r * r * r1). + rewrite Rinv_l; auto with real. + transitivity (/ r * r * r2). + repeat rewrite Rmult_assoc; rewrite H; trivial. + rewrite Rinv_l; auto with real. +Qed. + +(**********) +Lemma Rmult_integral : forall r1 r2, r1 * r2 = 0 -> r1 = 0 \/ r2 = 0. + 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. + 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. + auto with real. +Qed. + +(**********) +Lemma Rmult_eq_0_compat_l : forall r1 r2, r2 = 0 -> r1 * r2 = 0. + auto with real. +Qed. + + +(**********) +Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0. +intros r1 r2 H; split; red in |- *; intro; apply H; auto with real. +Qed. + +(**********) +Lemma Rmult_integral_contrapositive : + forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0. +red in |- *; intros r1 r2 [H1 H2] H. +case (Rmult_integral r1 r2); auto with real. +Qed. +Hint Resolve Rmult_integral_contrapositive: real. + +(**********) +Lemma Rmult_plus_distr_r : + forall r1 r2 r3, (r1 + r2) * r3 = r1 * r3 + r2 * r3. +intros; ring. +Qed. + +(** Square function *) + +(***********) +Definition Rsqr r : R := r * r. + +(***********) +Lemma Rsqr_0 : Rsqr 0 = 0. + unfold Rsqr in |- *; auto with real. +Qed. + +(***********) +Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0. +unfold Rsqr in |- *; intros; elim (Rmult_integral r r H); trivial. +Qed. + +(*********************************************************) +(** Opposite *) +(*********************************************************) + +(**********) +Lemma Ropp_eq_compat : forall r1 r2, r1 = r2 -> - r1 = - r2. + auto with real. +Qed. +Hint Resolve Ropp_eq_compat: real. + +(**********) +Lemma Ropp_0 : -0 = 0. + ring. +Qed. +Hint Resolve Ropp_0: real v62. + +(**********) +Lemma Ropp_eq_0_compat : forall r, r = 0 -> - r = 0. + intros; rewrite H; auto with real. +Qed. +Hint Resolve Ropp_eq_0_compat: real. + +(**********) +Lemma Ropp_involutive : forall r, - - r = r. + intro; ring. +Qed. +Hint Resolve Ropp_involutive: real. + +(*********) +Lemma Ropp_neq_0_compat : forall r, r <> 0 -> - r <> 0. +red in |- *; 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. + intros; ring. +Qed. +Hint Resolve Ropp_plus_distr: real. + +(** Opposite and multiplication *) + +Lemma Ropp_mult_distr_l_reverse : forall r1 r2, - r1 * r2 = - (r1 * r2). + intros; ring. +Qed. +Hint Resolve Ropp_mult_distr_l_reverse: real. + +(**********) +Lemma Rmult_opp_opp : forall r1 r2, - r1 * - r2 = r1 * r2. + intros; ring. +Qed. +Hint Resolve Rmult_opp_opp: real. + +Lemma Ropp_mult_distr_r_reverse : forall r1 r2, r1 * - r2 = - (r1 * r2). +intros; rewrite <- Ropp_mult_distr_l_reverse; ring. +Qed. + +(** Substraction *) + +Lemma Rminus_0_r : forall r, r - 0 = r. +intro; ring. +Qed. +Hint Resolve Rminus_0_r: real. + +Lemma Rminus_0_l : forall r, 0 - r = - r. +intro; ring. +Qed. +Hint Resolve Rminus_0_l: real. + +(**********) +Lemma Ropp_minus_distr : forall r1 r2, - (r1 - r2) = r2 - r1. + intros; ring. +Qed. +Hint Resolve Ropp_minus_distr: real. + +Lemma Ropp_minus_distr' : forall r1 r2, - (r2 - r1) = r1 - r2. +intros; ring. +Qed. +Hint Resolve Ropp_minus_distr': real. + +(**********) +Lemma Rminus_diag_eq : forall r1 r2, r1 = r2 -> r1 - r2 = 0. + intros; rewrite H; ring. +Qed. +Hint Resolve Rminus_diag_eq: real. + +(**********) +Lemma Rminus_diag_uniq : forall r1 r2, r1 - r2 = 0 -> r1 = r2. + intros r1 r2; unfold Rminus in |- *; 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. +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. +intros; ring. +Qed. +Hint Resolve Rplus_minus: real. + +(**********) +Lemma Rminus_eq_contra : forall r1 r2, r1 <> r2 -> r1 - r2 <> 0. +red in |- *; 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. +red in |- *; 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. +red in |- *; 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. +intros; ring. +Qed. + +(** Inverse *) +Lemma Rinv_1 : / 1 = 1. +field; auto with real. +Qed. +Hint Resolve Rinv_1: real. + +(*********) +Lemma Rinv_neq_0_compat : forall r, r <> 0 -> / r <> 0. +red in |- *; 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. +intros; field; auto with real. +Qed. +Hint Resolve Rinv_involutive: real. + +(*********) +Lemma Rinv_mult_distr : + forall r1 r2, r1 <> 0 -> r2 <> 0 -> / (r1 * r2) = / r1 * / r2. +intros; field; auto with real. +Qed. + +(*********) +Lemma Ropp_inv_permute : forall r, r <> 0 -> - / r = / - r. +intros; field; auto with real. +Qed. + +Lemma Rinv_r_simpl_r : forall r1 r2, r1 <> 0 -> r1 * / r1 * r2 = r2. +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. +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. +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. +intros a b c; intros. +transitivity (a * / a * (c * / b)); auto with real. +ring. +Qed. + +(** Order and addition *) + +Lemma Rplus_lt_compat_r : forall r r1 r2, r1 < r2 -> r1 + r < r2 + r. +intros. +rewrite (Rplus_comm r1 r); rewrite (Rplus_comm r2 r); auto with real. +Qed. + +Hint Resolve Rplus_lt_compat_r: real. + +(**********) +Lemma Rplus_lt_reg_r : forall r r1 r2, r + r1 < r + r2 -> r1 < r2. +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_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2. +unfold Rle in |- *; intros; elim H; intro. +left; apply (Rplus_lt_compat_l r r1 r2 H0). +right; rewrite <- H0; auto with zarith real. +Qed. + +(**********) +Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r. +unfold Rle in |- *; 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_le_reg_l : forall r r1 r2, r + r1 <= r + r2 -> r1 <= r2. +unfold Rle in |- *; intros; elim H; intro. +left; apply (Rplus_lt_reg_r r r1 r2 H0). +right; apply (Rplus_eq_reg_l r r1 r2 H0). +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. +intros; split. +apply Rlt_le_trans with (a + y); auto with real. +apply Rlt_le_trans with (b + y); auto with real. +Qed. + +(*********) +Lemma Rplus_lt_compat : + forall r1 r2 r3 r4, r1 < r2 -> r3 < r4 -> r1 + r3 < r2 + r4. +intros; apply Rlt_trans with (r2 + r3); auto with real. +Qed. + +Lemma Rplus_le_compat : + forall r1 r2 r3 r4, r1 <= r2 -> r3 <= r4 -> r1 + r3 <= r2 + r4. +intros; apply Rle_trans with (r2 + r3); auto with real. +Qed. + +(*********) +Lemma Rplus_lt_le_compat : + forall r1 r2 r3 r4, r1 < r2 -> r3 <= r4 -> r1 + r3 < r2 + r4. +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. +intros; apply Rle_lt_trans with (r2 + r3); auto with real. +Qed. + +Hint Immediate Rplus_lt_compat Rplus_le_compat Rplus_lt_le_compat + Rplus_le_lt_compat: real. + +(** Order and Opposite *) + +(**********) +Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2. +unfold Rgt in |- *; intros. +apply (Rplus_lt_reg_r (r2 + r1)). +replace (r2 + r1 + - r1) with r2. +replace (r2 + r1 + - r2) with r1. +trivial. +ring. +ring. +Qed. +Hint Resolve Ropp_gt_lt_contravar. + +(**********) +Lemma Ropp_lt_gt_contravar : forall r1 r2, r1 < r2 -> - r1 > - r2. +unfold Rgt in |- *; auto with real. +Qed. +Hint Resolve Ropp_lt_gt_contravar: real. + +Lemma Ropp_lt_cancel : forall r1 r2, - r2 < - r1 -> r1 < r2. +intros x y H'. +rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y); + auto with real. +Qed. +Hint Immediate Ropp_lt_cancel: real. + +Lemma Ropp_lt_contravar : forall r1 r2, r2 < r1 -> - r1 < - r2. +auto with real. +Qed. +Hint Resolve Ropp_lt_contravar: real. + +(**********) +Lemma Ropp_le_ge_contravar : forall r1 r2, r1 <= r2 -> - r1 >= - r2. +unfold Rge in |- *; intros r1 r2 [H| H]; auto with real. +Qed. +Hint Resolve Ropp_le_ge_contravar: real. + +Lemma Ropp_le_cancel : forall r1 r2, - r2 <= - r1 -> r1 <= r2. +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_le_contravar : forall r1 r2, r2 <= r1 -> - r1 <= - r2. +intros r1 r2 H; elim H; auto with real. +Qed. +Hint Resolve Ropp_le_contravar: real. + +(**********) +Lemma Ropp_ge_le_contravar : forall r1 r2, r1 >= r2 -> - r1 <= - r2. +unfold Rge in |- *; intros r1 r2 [H| H]; auto with real. +Qed. +Hint Resolve Ropp_ge_le_contravar: real. + +(**********) +Lemma Ropp_0_lt_gt_contravar : forall r, 0 < r -> 0 > - r. +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. +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. +intros; rewrite <- Ropp_0; auto with real. +Qed. + +(**********) +Lemma Ropp_gt_lt_0_contravar : forall r, r < 0 -> - r > 0. +intros; rewrite <- Ropp_0; auto with real. +Qed. +Hint Resolve Ropp_lt_gt_0_contravar Ropp_gt_lt_0_contravar: real. + +(**********) +Lemma Ropp_0_le_ge_contravar : forall r, 0 <= r -> 0 >= - r. +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. +intros; replace 0 with (-0); auto with real. +Qed. +Hint Resolve Ropp_0_ge_le_contravar: real. + +(** Order and multiplication *) + +Lemma Rmult_lt_compat_r : forall r r1 r2, 0 < r -> r1 < r2 -> r1 * r < r2 * r. +intros; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); auto with real. +Qed. +Hint Resolve Rmult_lt_compat_r. + +Lemma Rmult_lt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2. +intros z x y H H0. +case (Rtotal_order x y); intros Eq0; auto; elim Eq0; clear Eq0; intros Eq0. + rewrite Eq0 in H0; elimtype False; apply (Rlt_irrefl (z * y)); auto. +generalize (Rmult_lt_compat_l z y x H Eq0); intro; elimtype False; + generalize (Rlt_trans (z * x) (z * y) (z * x) H0 H1); + intro; apply (Rlt_irrefl (z * x)); auto. +Qed. + + +Lemma Rmult_lt_gt_compat_neg_l : + forall r r1 r2, r < 0 -> r1 < r2 -> r * r1 > r * r2. +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. + +(**********) +Lemma Rmult_le_compat_l : + forall r r1 r2, 0 <= r -> r1 <= r2 -> r * r1 <= r * r2. +intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle in |- *; + 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. +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_le_reg_l : forall r r1 r2, 0 < r -> r * r1 <= r * r2 -> r1 <= r2. +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_compat_neg_l : + forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r2 <= r * r1. +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. +intros; apply Rle_ge; auto with real. +Qed. +Hint Resolve Rmult_le_ge_compat_neg_l: real. + +Lemma Rmult_le_compat : + forall r1 r2 r3 r4, + 0 <= r1 -> 0 <= r3 -> r1 <= r2 -> r3 <= r4 -> r1 * r3 <= r2 * r4. +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_gt_0_lt_compat : + forall r1 r2 r3 r4, + r3 > 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4. +intros; apply Rlt_trans with (r2 * r3); auto with real. +Qed. + +(*********) +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. +intros; apply Rle_lt_trans with (r2 * r3); auto with real. +Qed. + +(** Order and Substractions *) +Lemma Rlt_minus : forall r1 r2, r1 < r2 -> r1 - r2 < 0. +intros; apply (Rplus_lt_reg_r r2). +replace (r2 + (r1 - r2)) with r1. +replace (r2 + 0) with r2; auto with real. +ring. +Qed. +Hint Resolve Rlt_minus: real. + +(**********) +Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0. +destruct 1; unfold Rle in |- *; auto with real. +Qed. + +(**********) +Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2. +intros; replace r1 with (r1 - r2 + r2). +pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. +ring. +Qed. + +(**********) +Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2. +intros; replace r1 with (r1 - r2 + r2). +pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. +ring. +Qed. + +(**********) +Lemma tech_Rplus : forall r (s:R), 0 <= r -> 0 < s -> r + s <> 0. +intros; apply sym_not_eq; apply Rlt_not_eq. +rewrite Rplus_comm; replace 0 with (0 + 0); auto with real. +Qed. +Hint Immediate tech_Rplus: real. + +(** Order and the square function *) +Lemma Rle_0_sqr : forall r, 0 <= Rsqr r. +intro; case (Rlt_le_dec r 0); unfold Rsqr in |- *; 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. +intros; case (Rdichotomy r 0); trivial; unfold Rsqr in |- *; 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. + +(** Zero is less than one *) +Lemma Rlt_0_1 : 0 < 1. +replace 1 with (Rsqr 1); auto with real. +unfold Rsqr in |- *; auto with real. +Qed. +Hint Resolve Rlt_0_1: real. + +Lemma Rle_0_1 : 0 <= 1. +left. +exact Rlt_0_1. +Qed. + +(** Order and inverse *) +Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r. +intros; apply Rnot_le_lt; red in |- *; 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. +intros; apply Rnot_le_lt; red in |- *; 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. +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 in |- *; auto with real. +symmetry in |- *; auto with real. +Qed. + +Lemma Rinv_1_lt_contravar : forall r1 r2, 1 <= r1 -> r1 < r2 -> / r2 < / r1. +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 in |- *; apply Rlt_trans with (r2 := x); auto with real. +Qed. +Hint Resolve Rinv_1_lt_contravar: real. + +(*********************************************************) +(** Greater *) +(*********************************************************) + +(**********) +Lemma Rge_antisym : forall r1 r2, r1 >= r2 -> r2 >= r1 -> r1 = r2. +intros; apply Rle_antisym; auto with real. +Qed. + +(**********) +Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2. +intros; unfold Rge in |- *; elim (Rtotal_order r1 r2); intro. +absurd (r1 < r2); trivial. +case H0; auto. +Qed. + +(**********) +Lemma Rnot_lt_le : forall r1 r2, ~ r1 < r2 -> r2 <= r1. +intros; apply Rge_le; apply Rnot_lt_ge; assumption. +Qed. + +(**********) +Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2. +intros r1 r2 H; apply Rge_le. +exact (Rnot_lt_ge r2 r1 H). +Qed. + +(**********) +Lemma Rgt_ge : forall r1 r2, r1 > r2 -> r1 >= r2. +red in |- *; auto with real. +Qed. + + +(**********) +Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rlt_le_trans with r2; auto with real. +Qed. + +(**********) +Lemma Rgt_ge_trans : forall r1 r2 r3, r1 > r2 -> r2 >= r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rle_lt_trans with r2; auto with real. +Qed. + +(**********) +Lemma Rgt_trans : forall r1 r2 r3, r1 > r2 -> r2 > r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rlt_trans with r2; auto with real. +Qed. + +(**********) +Lemma Rge_trans : forall r1 r2 r3, r1 >= r2 -> r2 >= r3 -> r1 >= r3. +intros; apply Rle_ge. +apply Rle_trans with r2; auto with real. +Qed. + +(**********) +Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1. +intros. +apply Rlt_le_trans with 1; auto with real. +pattern 1 at 1 in |- *; 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. +intros. +pattern r at 1 in |- *; 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. +red in |- *; unfold Rminus in |- *; intros. +pattern r1 at 2 in |- *; replace r1 with (r1 + 0); auto with real. +Qed. + +(***********) +Lemma Rplus_gt_compat_l : forall r r1 r2, r1 > r2 -> r + r1 > r + r2. +unfold Rgt in |- *; auto with real. +Qed. +Hint Resolve Rplus_gt_compat_l: real. + +(***********) +Lemma Rplus_gt_reg_l : forall r r1 r2, r + r1 > r + r2 -> r1 > r2. +unfold Rgt in |- *; intros; apply (Rplus_lt_reg_r r r2 r1 H). +Qed. + +(***********) +Lemma Rplus_ge_compat_l : forall r r1 r2, r1 >= r2 -> r + r1 >= r + r2. +intros; apply Rle_ge; auto with real. +Qed. +Hint Resolve Rplus_ge_compat_l: real. + +(***********) +Lemma Rplus_ge_reg_l : forall r r1 r2, r + r1 >= r + r2 -> r1 >= r2. +intros; apply Rle_ge; apply Rplus_le_reg_l with r; auto with real. +Qed. + +(***********) +Lemma Rmult_ge_compat_r : + forall r r1 r2, r >= 0 -> r1 >= r2 -> r1 * r >= r2 * r. +intros; apply Rle_ge; apply Rmult_le_compat_r; apply Rge_le; assumption. +Qed. + +(***********) +Lemma Rgt_minus : forall r1 r2, r1 > r2 -> r1 - r2 > 0. +intros; replace 0 with (r2 - r2); auto with real. +unfold Rgt, Rminus in |- *; auto with real. +Qed. + +(*********) +Lemma minus_Rgt : forall r1 r2, r1 - r2 > 0 -> r1 > r2. +intros; replace r2 with (r2 + 0); auto with real. +intros; replace r1 with (r2 + (r1 - r2)); auto with real. +Qed. + +(**********) +Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0. +unfold Rge in |- *; intros; elim H; intro. +left; apply (Rgt_minus r1 r2 H0). +right; apply (Rminus_diag_eq r1 r2 H0). +Qed. + +(*********) +Lemma minus_Rge : forall r1 r2, r1 - r2 >= 0 -> r1 >= r2. +intros; replace r2 with (r2 + 0); auto with real. +intros; replace r1 with (r2 + (r1 - r2)); auto with real. +Qed. + + +(*********) +Lemma Rmult_gt_0_compat : forall r1 r2, r1 > 0 -> r2 > 0 -> r1 * r2 > 0. +unfold Rgt in |- *; intros. +replace 0 with (0 * r2); auto with real. +Qed. + +(*********) +Lemma Rmult_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 * r2. +Proof Rmult_gt_0_compat. + +(***********) +Lemma Rplus_eq_0_l : + forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0. +intros a b [H| H] H0 H1; auto with real. +absurd (0 < a + b). +rewrite H1; auto with real. +replace 0 with (0 + 0); auto with real. +Qed. + + +Lemma Rplus_eq_R0 : + forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0 /\ r2 = 0. +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. + + +(***********) +Lemma Rplus_sqr_eq_0_l : forall r1 r2, Rsqr r1 + Rsqr r2 = 0 -> r1 = 0. +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. +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. + + +(**********************************************************) +(** Injection from [N] to [R] *) +(**********************************************************) + +(**********) +Lemma S_INR : forall n:nat, INR (S n) = INR n + 1. +intro; case n; auto with real. +Qed. + +(**********) +Lemma S_O_plus_INR : forall n:nat, INR (1 + n) = INR 1 + INR n. +intro; simpl in |- *; case n; intros; auto with real. +Qed. + +(**********) +Lemma plus_INR : forall n m:nat, INR (n + m) = INR n + INR m. +intros n m; induction n as [| n Hrecn]. +simpl in |- *; auto with real. +replace (S n + m)%nat with (S (n + m)); auto with arith. +repeat rewrite S_INR. +rewrite Hrecn; ring. +Qed. + +(**********) +Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m. +intros n m le; pattern m, n in |- *; apply le_elim_rel; auto with real. +intros; rewrite <- minus_n_O; auto with real. +intros; repeat rewrite S_INR; simpl in |- *. +rewrite H0; ring. +Qed. + +(*********) +Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m. +intros n m; induction n as [| n Hrecn]. +simpl in |- *; auto with real. +intros; repeat rewrite S_INR; simpl in |- *. +rewrite plus_INR; rewrite Hrecn; ring. +Qed. + +Hint Resolve plus_INR minus_INR mult_INR: real. + +(*********) +Lemma lt_INR_0 : forall n:nat, (0 < n)%nat -> 0 < INR n. +simple induction 1; intros; auto with real. +rewrite S_INR; auto with real. +Qed. +Hint Resolve lt_INR_0: real. + +Lemma lt_INR : forall n m:nat, (n < m)%nat -> INR n < INR m. +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 INR_lt_1 : forall n:nat, (1 < n)%nat -> 1 < INR n. +intros; replace 1 with (INR 1); auto with real. +Qed. +Hint Resolve INR_lt_1: real. + +(**********) +Lemma INR_pos : forall p:positive, 0 < INR (nat_of_P p). +intro; apply lt_INR_0. +simpl in |- *; auto with real. +apply lt_O_nat_of_P. +Qed. +Hint Resolve INR_pos: real. + +(**********) +Lemma pos_INR : forall n:nat, 0 <= INR n. +intro n; case n. +simpl in |- *; 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. +double induction n m; intros. +simpl in |- *; elimtype False; apply (Rlt_irrefl 0); auto. +auto with arith. +generalize (pos_INR (S n0)); intro; cut (INR 0 = 0); + [ intro H2; rewrite H2 in H0; idtac | simpl in |- *; trivial ]. +generalize (Rle_lt_trans 0 (INR (S n0)) 0 H1 H0); intro; elimtype False; + apply (Rlt_irrefl 0); auto. +do 2 rewrite S_INR in H1; cut (INR n1 < INR n0). +intro H2; generalize (H0 n0 H2); intro; auto with arith. +apply (Rplus_lt_reg_r 1 (INR n1) (INR n0)). +rewrite Rplus_comm; rewrite (Rplus_comm 1 (INR n0)); trivial. +Qed. +Hint Resolve INR_lt: real. + +(*********) +Lemma le_INR : forall n m:nat, (n <= m)%nat -> INR n <= INR m. +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 not_INR_O : forall n:nat, INR n <> 0 -> n <> 0%nat. +red in |- *; intros n H H1. +apply H. +rewrite H1; trivial. +Qed. +Hint Immediate not_INR_O: real. + +(**********) +Lemma not_O_INR : forall n:nat, n <> 0%nat -> INR n <> 0. +intro n; case n. +intro; absurd (0%nat = 0%nat); trivial. +intros; rewrite S_INR. +apply Rgt_not_eq; red in |- *; auto with real. +Qed. +Hint Resolve not_O_INR: real. + +Lemma not_nm_INR : forall n m:nat, n <> m -> INR n <> INR m. +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. +elimtype False; auto. +apply sym_not_eq; apply Rlt_dichotomy_converse; auto with real. +Qed. +Hint Resolve not_nm_INR: real. + +Lemma INR_eq : forall n m:nat, INR n = INR m -> n = m. +intros; case (le_or_lt n m); intros H1. +case (le_lt_or_eq _ _ H1); intros H2; auto. +cut (n <> m). +intro H3; generalize (not_nm_INR n m H3); intro H4; elimtype False; auto. +omega. +symmetry in |- *; cut (m <> n). +intro H3; generalize (not_nm_INR m n H3); intro H4; elimtype False; auto. +omega. +Qed. +Hint Resolve INR_eq: real. + +Lemma INR_le : forall n m:nat, INR n <= INR m -> (n <= m)%nat. +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. +replace 1 with (INR 1); auto with real. +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. +intros z; idtac; apply Z_of_nat_complete; assumption. +Qed. + +(**********) +Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n). +simple induction n; auto with real. +intros; simpl in |- *; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; + auto with real. +Qed. + +Lemma plus_IZR_NEG_POS : + forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q). +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. +Qed. + +(**********) +Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m. +intro z; destruct z; intro t; destruct t; intros; auto with real. +simpl in |- *; intros; rewrite nat_of_P_plus_morphism; 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; + auto with real. +Qed. + +(**********) +Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m. +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. +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. +rewrite Ropp_mult_distr_l_reverse; auto with real. +intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. +rewrite Rmult_opp_opp; auto with real. +Qed. + +(**********) +Lemma Ropp_Ropp_IZR : forall n:Z, IZR (- n) = - IZR n. +intro z; case z; simpl in |- *; auto with real. +Qed. + +(**********) +Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m). +intros z1 z2; unfold Rminus in |- *; unfold Zminus in |- *. +rewrite <- (Ropp_Ropp_IZR z2); symmetry in |- *; apply plus_IZR. +Qed. + +(**********) +Lemma lt_O_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z. +intro z; case z; simpl in |- *; intros. +absurd (0 < 0); auto with real. +unfold Zlt in |- *; simpl in |- *; trivial. +case Rlt_not_le with (1 := H). +replace 0 with (-0); auto with real. +Qed. + +(**********) +Lemma lt_IZR : forall n m:Z, IZR n < IZR m -> (n < m)%Z. +intros z1 z2 H; apply Zlt_O_minus_lt. +apply lt_O_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. +intro z; destruct z; simpl in |- *; intros; auto with zarith. +case (Rlt_not_eq 0 (INR (nat_of_P p))); auto with real. +case (Rlt_not_eq (- INR (nat_of_P p)) 0); auto with real. +apply Ropp_lt_gt_0_contravar. unfold Rgt in |- *; apply INR_pos. +Qed. + +(**********) +Lemma eq_IZR : forall n m:Z, IZR n = IZR m -> n = m. +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_O_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0. +intros z H; red in |- *; intros H0; case H. +apply eq_IZR; auto. +Qed. + +(*********) +Lemma le_O_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z. +unfold Rle in |- *; intros z [H| H]. +red in |- *; intro; apply (Zlt_le_weak 0 z (lt_O_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. +unfold Rle in |- *; intros z1 z2 [H| H]. +apply (Zlt_le_weak 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. +pattern 1 at 1 in |- *; replace 1 with (IZR 1); intros; auto. +apply le_IZR; trivial. +Qed. + +(**********) +Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m. +intros m n H; apply Rnot_lt_ge; red in |- *; intro. +generalize (lt_IZR m n H0); intro; omega. +Qed. + +Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m. +intros m n H; apply Rnot_gt_le; red in |- *; 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. +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; elimtype False; omega. +omega. +Qed. + +Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z. +intros z [H1 H2]. +apply Zle_antisym. +apply Zlt_succ_le; apply lt_IZR; trivial. +replace 0%Z with (Zsucc (-1)); trivial. +apply Zlt_le_succ; 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. +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 in |- *; apply Rplus_lt_le_compat; auto with real. +ring. +replace 1 with (r + 1 - r). +unfold Rminus in |- *; 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. +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. +intros r z H1 H2 [s [H3 [H4 H5]]]. +apply H3; apply single_z_r_R1 with r; trivial. +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}. + +(**********) +Lemma prod_neq_R0 : forall r1 r2, r1 <> 0 -> r2 <> 0 -> r1 * r2 <> 0. +intros x y; intros; red in |- *; intro; generalize (Rmult_integral x y H1); + intro; elim H2; intro; + [ rewrite H3 in H; elim H | rewrite H3 in H0; elim H0 ]; + reflexivity. +Qed. + +(*********) +Lemma Rmult_le_pos : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 * r2. +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 double : forall r1, 2 * r1 = r1 + r1. +intro; ring. +Qed. + +Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2. +intro; rewrite <- double; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + symmetry in |- *; apply Rinv_r_simpl_m. +replace 2 with (INR 2); + [ apply not_O_INR; discriminate | unfold INR in |- *; ring ]. +Qed. + +(**********************************************************) +(** Other rules about < and <= *) +(**********************************************************) + +Lemma Rplus_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 + r2. +intros x y; intros; apply Rlt_trans with x; + [ assumption + | pattern x at 1 in |- *; 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. +intros x y; intros; apply Rle_lt_trans with x; + [ assumption + | pattern x at 1 in |- *; 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. +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. +intros x y; intros; apply Rle_trans with x; + [ assumption + | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption ]. +Qed. + +Lemma plus_le_is_le : forall r1 r2 r3, 0 <= r2 -> r1 + r2 <= r3 -> r1 <= r3. +intros x y z; intros; apply Rle_trans with (x + y); + [ pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption + | assumption ]. +Qed. + +Lemma plus_lt_is_lt : forall r1 r2 r3, 0 <= r2 -> r1 + r2 < r3 -> r1 < r3. +intros x y z; intros; apply Rle_lt_trans with (x + y); + [ pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption + | assumption ]. +Qed. + +Lemma Rmult_le_0_lt_compat : + forall r1 r2 r3 r4, + 0 <= r1 -> 0 <= r3 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4. +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 le_epsilon : + forall r1 r2, (forall eps:R, 0 < eps -> r1 <= r2 + eps) -> r1 <= r2. +intros x y; intros; elim (Rtotal_order x y); intro. +left; assumption. +elim H0; intro. +right; assumption. +clear H0; generalize (Rgt_minus x y H1); intro H2; change (0 < x - y) in H2. +cut (0 < 2). +intro. +generalize (Rmult_lt_0_compat (x - y) (/ 2) H2 (Rinv_0_lt_compat 2 H0)); + intro H3; generalize (H ((x - y) * / 2) H3); + replace (y + (x - y) * / 2) with ((y + x) * / 2). +intro H4; + generalize (Rmult_le_compat_l 2 x ((y + x) * / 2) (Rlt_le 0 2 H0) H4); + rewrite <- (Rmult_comm ((y + x) * / 2)); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; replace (2 * x) with (x + x). +rewrite (Rplus_comm y); intro H5; apply Rplus_le_reg_l with x; assumption. +ring. +replace 2 with (INR 2); [ apply not_O_INR; discriminate | ring ]. +pattern y at 2 in |- *; replace y with (y / 2 + y / 2). +unfold Rminus, Rdiv in |- *. +repeat rewrite Rmult_plus_distr_r. +ring. +cut (forall z:R, 2 * z = z + z). +intro. +rewrite <- (H4 (y / 2)). +unfold Rdiv in |- *. +rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +replace 2 with (INR 2). +apply not_O_INR. +discriminate. +unfold INR in |- *; reflexivity. +intro; ring. +cut (0%nat <> 2%nat); + [ intro H0; generalize (lt_INR_0 2 (neq_O_lt 2 H0)); unfold INR in |- *; + intro; assumption + | discriminate ]. +Qed. + +(**********) +Lemma completeness_weak : + forall E:R -> Prop, + bound E -> (exists x : R, E x) -> exists m : R, is_lub E m. +intros; elim (completeness E H H0); intros; split with x; assumption. +Qed. |