# Library Coq.Reals.RIneq

``` ```
Basic lemmas for the classical reals numbers
``` Require Export Raxioms. Require Import Rpow_def. Require Import Zpower. Require Export ZArithRing. Require Import Omega. Require Export RealField. Open Local Scope Z_scope. Open Local Scope R_scope. Implicit Type r : R. ```

# Relation between orders and equality

``` Lemma Rlt_irrefl : forall r, ~ r < r. Proof.   generalize Rlt_asym. intuition eauto. Qed. Hint Resolve Rlt_irrefl: real. Lemma Rle_refl : forall r, r <= r. Proof.   intro; right; reflexivity. Qed. Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2. Proof.   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. Proof.   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. Proof.   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. Proof.   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. 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. ```

# Order Lemma : relating `<`, `>`, `<=` and `>=`

``` Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2. Proof.   intros; red in |- *; tauto. Qed. Hint Resolve Rlt_le: real. Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1. Proof.   destruct 1; red in |- *; auto with real. Qed. Hint Immediate Rle_ge: real. Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1. Proof.   destruct 1; red in |- *; auto with real. Qed. Hint Resolve Rge_le: real. Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1. Proof.   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. Proof.   intros; apply Rnot_le_lt; auto with real. Qed. Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2. Proof.   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. Proof.   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. Proof.   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. Proof.   unfold Rle in |- *; tauto. Qed. Hint Immediate Req_le: real. Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2. Proof.   unfold Rge in |- *; tauto. Qed. Hint Immediate Req_ge: real. Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2. Proof.   unfold Rle in |- *; auto. Qed. Hint Immediate Req_le_sym: real. Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2. Proof.   unfold Rge in |- *; auto. Qed. Hint Immediate Req_ge_sym: real. Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2. Proof.   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. Proof.   intuition. Qed. 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 Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3. Proof.   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. Proof.   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. Proof.   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}. 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.   intros; unfold Rgt in |- *; intros; apply Rlt_dec. Qed. Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}. Proof.   intros; generalize (Rle_dec r2 r1); intuition. Qed. Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}. Proof.   intros; generalize (total_order_T r1 r2); intuition. 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 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. ```

# Field Lemmas

``` ```

``` Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r. Proof.   split; ring. Qed. Hint Resolve Rplus_ne: real v62. Lemma Rplus_0_r : forall r, r + 0 = r. Proof.   intro; ring. Qed. Hint Resolve Rplus_0_r: real. 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. Hint Resolve (f_equal (A:=R)): real. Lemma Rplus_eq_compat_l : forall r r1 r2, r1 = r2 -> r + r1 = r + r2. Proof.   auto with real. Qed. Hint Resolve Rplus_eq_compat_l: v62. 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_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0. Proof.   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. 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. Lemma Rinv_r_sym : forall r, r <> 0 -> 1 = r * / r. Proof.   intros; field; trivial. Qed. Hint Resolve Rinv_l_sym Rinv_r_sym: real. Lemma Rmult_0_r : forall r, r * 0 = 0. Proof.   intro; ring. Qed. Hint Resolve Rmult_0_r: real v62. Lemma Rmult_0_l : forall r, 0 * r = 0. Proof.   intro; ring. Qed. Hint Resolve Rmult_0_l: real v62. Lemma Rmult_ne : forall r, r * 1 = r /\ 1 * r = r. Proof.   intro; split; ring. Qed. Hint Resolve Rmult_ne: real v62. 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. Hint Resolve Rmult_eq_compat_l: v62. 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_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 in |- *; intro; apply H; auto with real. Qed. Lemma Rmult_integral_contrapositive :   forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0. Proof.   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. Proof.   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. Proof.   auto with real. Qed. Hint Resolve Ropp_eq_compat: real. Lemma Ropp_0 : -0 = 0. Proof.   ring. Qed. Hint Resolve Ropp_0: real v62. 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 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. Proof.   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). 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_reverse : forall r1 r2, r1 * - r2 = - (r1 * r2). Proof.   intros; ring. Qed. ```

## Substraction

``` 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. Hint Resolve Ropp_minus_distr': real. 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 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. 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 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. Proof.   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. Proof.   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. 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 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. 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. ```

# Field operations and order

``` ```

``` 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_lt_reg_r : 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_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. 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_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. Proof.   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. Proof.   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. 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_lt_contravar : forall r1 r2, r2 < r1 -> - r1 < - r2. Proof.   auto with real. Qed. Hint Resolve Ropp_lt_contravar: real. Lemma Ropp_le_ge_contravar : forall r1 r2, r1 <= r2 -> - r1 >= - r2. Proof.   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. 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_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_le_contravar : forall r1 r2, r1 >= r2 -> - r1 <= - r2. Proof.   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. 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. Lemma Ropp_gt_lt_0_contravar : forall r, r < 0 -> - r > 0. Proof.   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. 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. ```

## Order and multiplication

``` 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. 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; 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. 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. 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 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. 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_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_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_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_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_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. ```

## Order and Substractions

``` Lemma Rlt_minus : forall r1 r2, r1 < r2 -> r1 - r2 < 0. Proof.   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. Proof.   destruct 1; unfold Rle in |- *; auto with real. Qed. Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. 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 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. Proof.   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. 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 in |- *; auto with real.   symmetry in |- *; 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 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. Proof.   intros; apply Rle_antisym; auto with real. Qed. Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2. Proof.   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. Proof.   intros; apply Rge_le; apply Rnot_lt_ge; assumption. Qed. Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2. Proof.   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. Proof.   red in |- *; auto with real. Qed. Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. 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. 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. ```

# 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 in |- *; 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 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. Proof.   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. Proof.   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. Proof.   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. 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 INR_lt_1 : forall n:nat, (1 < n)%nat -> 1 < INR n. Proof.   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). Proof.   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. Proof.   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. Proof.   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. 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 not_INR_O : forall n:nat, INR n <> 0 -> n <> 0%nat. Proof.   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. Proof.   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. 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.   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. Proof.   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. 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.   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. Proof.   intros z; idtac; apply Z_of_nat_complete; assumption. Qed. Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n). Proof.   simple induction n; auto with real.   intros; simpl in |- *; rewrite nat_of_P_o_P_of_succ_nat_eq_succ;     auto with real. Qed. Lemma plus_IZR_NEG_POS :   forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q). Proof.   intros.   case (lt_eq_lt_dec (nat_of_P p) (nat_of_P q)).   intros [H| H]; simpl in |- *.   rewrite nat_of_P_lt_Lt_compare_complement_morphism; simpl in |- *; trivial.   rewrite (nat_of_P_minus_morphism q p).   rewrite minus_INR; auto with arith; ring.   apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial.   rewrite (nat_of_P_inj p q); trivial.   rewrite Pcompare_refl; simpl in |- *; auto with real.   intro H; simpl in |- *.   rewrite nat_of_P_gt_Gt_compare_complement_morphism; simpl in |- *;     auto with arith.   rewrite (nat_of_P_minus_morphism p q).   rewrite minus_INR; auto with arith; ring.   apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial. Qed. Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m. Proof.   intro z; destruct z; intro t; destruct t; intros; auto with real.   simpl in |- *; intros; rewrite nat_of_P_plus_morphism; auto with real.   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. Proof.   intros z t; case z; case t; simpl in |- *; auto with real.   intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.   intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real.   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 pow_IZR : forall z n, pow (IZR z) n = IZR (Zpower z (Z_of_nat n)). Proof.  intros z [|n];simpl;trivial.  rewrite Zpower_pos_nat.  rewrite nat_of_P_o_P_of_succ_nat_eq_succ. unfold Zpower_nat;simpl.  rewrite mult_IZR.  induction n;simpl;trivial.  rewrite mult_IZR;ring[IHn]. Qed. Lemma Ropp_Ropp_IZR : forall n:Z, IZR (- n) = - IZR n. Proof.   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). Proof.   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. Proof.   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. Proof.   intros z1 z2 H; apply Zlt_0_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. Proof.   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. 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_O_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. 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; elimtype False; omega.   omega. Qed. Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z. Proof.   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. 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 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. 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. ```

# 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. Proof.   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. 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 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 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. Proof.   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. Proof.   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. 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 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. Proof.   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. Proof.   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. 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 le_epsilon :   forall r1 r2, (forall eps:R, 0 < eps -> r1 <= r2 + eps) -> r1 <= r2. Proof.   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 | reflexivity ].   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. Proof.   intros; elim (completeness E H H0); intros; split with x; assumption. Qed. ```