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Diffstat (limited to 'theories/Reals/ArithProp.v')
| -rw-r--r-- | theories/Reals/ArithProp.v | 178 |
1 files changed, 178 insertions, 0 deletions
diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v new file mode 100644 index 00000000..ad535a9d --- /dev/null +++ b/theories/Reals/ArithProp.v @@ -0,0 +1,178 @@ +(************************************************************************) +(* 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: ArithProp.v,v 1.11.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rbasic_fun. +Require Import Even. +Require Import Div2. +Open Local Scope Z_scope. +Open Local Scope R_scope. + +Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat. +intros; red in |- *; intro. +cut (forall n m:nat, (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m). +intro; assert (H2 := H1 _ _ (lt_le_weak _ _ H) H0); rewrite H2 in H; + elim (lt_irrefl _ H). +set (R := fun n m:nat => (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m). +cut + ((forall n m:nat, R n m) -> + forall n0 m:nat, (m <= n0)%nat -> (n0 - m)%nat = 0%nat -> n0 = m). +intro; apply H1. +apply nat_double_ind. +unfold R in |- *; intros; inversion H2; reflexivity. +unfold R in |- *; intros; simpl in H3; assumption. +unfold R in |- *; intros; simpl in H4; assert (H5 := le_S_n _ _ H3); + assert (H6 := H2 H5 H4); rewrite H6; reflexivity. +unfold R in |- *; intros; apply H1; assumption. +Qed. + +Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat. +set (R := fun m n:nat => (n <= m)%nat -> (m - n <= m)%nat). +cut + ((forall m n:nat, R m n) -> forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat). +intro; apply H. +apply nat_double_ind. +unfold R in |- *; intros; simpl in |- *; apply le_n. +unfold R in |- *; intros; simpl in |- *; apply le_n. +unfold R in |- *; intros; simpl in |- *; apply le_trans with n. +apply H0; apply le_S_n; assumption. +apply le_n_Sn. +unfold R in |- *; intros; apply H; assumption. +Qed. + +Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat. +intros n m; pattern n, m in |- *; apply nat_double_ind; + [ intros; rewrite <- minus_n_O; assumption + | intros; elim (lt_n_O _ H) + | intros; simpl in |- *; apply H; apply lt_S_n; assumption ]. +Qed. + +Lemma even_odd_cor : + forall n:nat, exists p : nat, n = (2 * p)%nat \/ n = S (2 * p). +intro. +assert (H := even_or_odd n). +exists (div2 n). +assert (H0 := even_odd_double n). +elim H0; intros. +elim H1; intros H3 _. +elim H2; intros H4 _. +replace (2 * div2 n)%nat with (double (div2 n)). +elim H; intro. +left. +apply H3; assumption. +right. +apply H4; assumption. +unfold double in |- *; ring. +Qed. + +(* 2m <= 2n => m<=n *) +Lemma le_double : forall m n:nat, (2 * m <= 2 * n)%nat -> (m <= n)%nat. +intros; apply INR_le. +assert (H1 := le_INR _ _ H). +do 2 rewrite mult_INR in H1. +apply Rmult_le_reg_l with (INR 2). +replace (INR 2) with 2; [ prove_sup0 | reflexivity ]. +assumption. +Qed. + +(* Here, we have the euclidian division *) +(* This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=kPI *) +Lemma euclidian_division : + forall x y:R, + y <> 0 -> + exists k : Z, (exists r : R, x = IZR k * y + r /\ 0 <= r < Rabs y). +intros. +set + (k0 := + match Rcase_abs y with + | left _ => (1 - up (x / - y))%Z + | right _ => (up (x / y) - 1)%Z + end). +exists k0. +exists (x - IZR k0 * y). +split. +ring. +unfold k0 in |- *; case (Rcase_abs y); intro. +assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl in |- *; + unfold Rminus in |- *. +replace (- ((1 + - IZR (up (x / - y))) * y)) with + ((IZR (up (x / - y)) - 1) * y); [ idtac | ring ]. +split. +apply Rmult_le_reg_l with (/ - y). +apply Rinv_0_lt_compat; apply Ropp_0_gt_lt_contravar; exact r. +rewrite Rmult_0_r; rewrite (Rmult_comm (/ - y)); rewrite Rmult_plus_distr_r; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]. +apply Rplus_le_reg_l with (IZR (up (x / - y)) - x / - y). +rewrite Rplus_0_r; unfold Rdiv in |- *; pattern (/ - y) at 4 in |- *; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +replace + (IZR (up (x * / - y)) - x * - / y + + (- (x * / y) + - (IZR (up (x * / - y)) - 1))) with 1; + [ idtac | ring ]. +elim H0; intros _ H1; unfold Rdiv in H1; exact H1. +rewrite (Rabs_left _ r); apply Rmult_lt_reg_l with (/ - y). +apply Rinv_0_lt_compat; apply Ropp_0_gt_lt_contravar; exact r. +rewrite <- Rinv_l_sym. +rewrite (Rmult_comm (/ - y)); rewrite Rmult_plus_distr_r; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]; + apply Rplus_lt_reg_r with (IZR (up (x / - y)) - 1). +replace (IZR (up (x / - y)) - 1 + 1) with (IZR (up (x / - y))); + [ idtac | ring ]. +replace (IZR (up (x / - y)) - 1 + (- (x * / y) + - (IZR (up (x / - y)) - 1))) + with (- (x * / y)); [ idtac | ring ]. +rewrite <- Ropp_mult_distr_r_reverse; rewrite (Ropp_inv_permute _ H); elim H0; + unfold Rdiv in |- *; intros H1 _; exact H1. +apply Ropp_neq_0_compat; assumption. +assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl in |- *; + cut (0 < y). +intro; unfold Rminus in |- *; + replace (- ((IZR (up (x / y)) + -1) * y)) with ((1 - IZR (up (x / y))) * y); + [ idtac | ring ]. +split. +apply Rmult_le_reg_l with (/ y). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_0_r; rewrite (Rmult_comm (/ y)); rewrite Rmult_plus_distr_r; + rewrite Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_r | assumption ]; + apply Rplus_le_reg_l with (IZR (up (x / y)) - x / y); + rewrite Rplus_0_r; unfold Rdiv in |- *; + replace + (IZR (up (x * / y)) - x * / y + (x * / y + (1 - IZR (up (x * / y))))) with + 1; [ idtac | ring ]; elim H0; intros _ H2; unfold Rdiv in H2; + exact H2. +rewrite (Rabs_right _ r); apply Rmult_lt_reg_l with (/ y). +apply Rinv_0_lt_compat; assumption. +rewrite <- (Rinv_l_sym _ H); rewrite (Rmult_comm (/ y)); + rewrite Rmult_plus_distr_r; rewrite Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_r | assumption ]; + apply Rplus_lt_reg_r with (IZR (up (x / y)) - 1); + replace (IZR (up (x / y)) - 1 + 1) with (IZR (up (x / y))); + [ idtac | ring ]; + replace (IZR (up (x / y)) - 1 + (x * / y + (1 - IZR (up (x / y))))) with + (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv in |- *; + intros H2 _; exact H2. +case (total_order_T 0 y); intro. +elim s; intro. +assumption. +elim H; symmetry in |- *; exact b. +assert (H1 := Rge_le _ _ r); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r0)). +Qed. + +Lemma tech8 : forall n i:nat, (n <= S n + i)%nat. +intros; induction i as [| i Hreci]. +replace (S n + 0)%nat with (S n); [ apply le_n_Sn | ring ]. +replace (S n + S i)%nat with (S (S n + i)). +apply le_S; assumption. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. +Qed.
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