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-rw-r--r--theories/Reals/R_Ifp.v80
1 files changed, 40 insertions, 40 deletions
diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v
index 9e04a7da..8364e986 100644
--- a/theories/Reals/R_Ifp.v
+++ b/theories/Reals/R_Ifp.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -13,7 +13,7 @@
Require Import Rbase.
Require Import Omega.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
(*********************************************************)
(** * Fractional part *)
@@ -45,7 +45,7 @@ Proof.
intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H;
rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1;
cut (1 = IZR 1); auto with zarith real.
- intro; generalize H1; pattern 1 at 1 in |- *; rewrite H; intro; clear H H1;
+ intro; generalize H1; pattern 1 at 1; rewrite H; intro; clear H H1;
rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1));
auto with zarith real.
Qed.
@@ -53,12 +53,12 @@ Qed.
(**********)
Lemma fp_R0 : frac_part 0 = 0.
Proof.
- unfold frac_part in |- *; unfold Int_part in |- *; elim (archimed 0); intros;
- unfold Rminus in |- *; elim (Rplus_ne (- IZR (up 0 - 1)));
+ unfold frac_part; unfold Int_part; elim (archimed 0); intros;
+ unfold Rminus; elim (Rplus_ne (- IZR (up 0 - 1)));
intros a b; rewrite b; clear a b; rewrite <- Z_R_minus;
cut (up 0 = 1%Z).
intro; rewrite H1;
- rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (refl_equal (IZR 1)));
+ rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (eq_refl (IZR 1)));
apply Ropp_0.
elim (archimed 0); intros; clear H2; unfold Rgt in H1;
rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1);
@@ -81,21 +81,21 @@ Qed.
(**********)
Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1.
Proof.
- intro; unfold frac_part in |- *; unfold Int_part in |- *; split.
+ intro; unfold frac_part; unfold Int_part; split.
(*sup a O*)
cut (r - IZR (up r) >= -1).
- rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *;
+ rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
- fold (r - IZR (up r)) in |- *; fold (r - IZR (up r) - -1) in |- *;
+ fold (r - IZR (up r)); fold (r - IZR (up r) - -1);
apply Rge_minus; auto with zarith real.
rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r);
auto with zarith real.
(*inf a 1*)
cut (r - IZR (up r) < 0).
- rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *;
+ rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
- fold (r - IZR (up r)) in |- *; rewrite Ropp_involutive;
- elim (Rplus_ne 1); intros a b; pattern 1 at 2 in |- *;
+ fold (r - IZR (up r)); rewrite Ropp_involutive;
+ elim (Rplus_ne 1); intros a b; pattern 1 at 2;
rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1);
apply Rplus_lt_compat_l; auto with zarith real.
elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr;
@@ -110,8 +110,8 @@ Qed.
Lemma base_Int_part :
forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.
Proof.
- intro; unfold Int_part in |- *; elim (archimed r); intros.
- split; rewrite <- (Z_R_minus (up r) 1); simpl in |- *.
+ intro; unfold Int_part; elim (archimed r); intros.
+ split; rewrite <- (Z_R_minus (up r) 1); simpl.
generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1;
rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1;
rewrite (Rplus_comm (- r) (-1)) in H1;
@@ -130,31 +130,31 @@ Proof.
Qed.
(**********)
-Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z_of_nat n.
+Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n.
Proof.
- intros n; unfold Int_part in |- *.
- cut (up (INR n) = (Z_of_nat n + Z_of_nat 1)%Z).
- intros H'; rewrite H'; simpl in |- *; ring.
- apply sym_equal; apply tech_up; auto.
- replace (Z_of_nat n + Z_of_nat 1)%Z with (Z_of_nat (S n)).
+ intros n; unfold Int_part.
+ cut (up (INR n) = (Z.of_nat n + Z.of_nat 1)%Z).
+ intros H'; rewrite H'; simpl; ring.
+ symmetry; apply tech_up; auto.
+ replace (Z.of_nat n + Z.of_nat 1)%Z with (Z.of_nat (S n)).
repeat rewrite <- INR_IZR_INZ.
apply lt_INR; auto.
- rewrite Zplus_comm; rewrite <- Znat.inj_plus; simpl in |- *; auto.
- rewrite plus_IZR; simpl in |- *; auto with real.
+ rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl; auto.
+ rewrite plus_IZR; simpl; auto with real.
repeat rewrite <- INR_IZR_INZ; auto with real.
Qed.
(**********)
Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z, r = IZR c.
Proof.
- unfold frac_part in |- *; intros; split with (Int_part r);
+ unfold frac_part; intros; split with (Int_part r);
apply Rminus_diag_uniq; auto with zarith real.
Qed.
(**********)
Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r.
Proof.
- red in |- *; intros; rewrite <- H0 in H; generalize fp_R0; intro;
+ red; intros; rewrite <- H0 in H; generalize fp_R0; intro;
auto with zarith real.
Qed.
@@ -243,7 +243,7 @@ Proof.
intro; rewrite H1 in H; clear H1;
rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H);
- intros; clear H H0; unfold Int_part at 1 in |- *;
+ intros; clear H H0; unfold Int_part at 1;
omega.
Qed.
@@ -336,7 +336,7 @@ Proof.
generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H);
intro; clear H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0);
- intros; clear H0 H1; unfold Int_part at 1 in |- *;
+ intros; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
@@ -346,9 +346,9 @@ Lemma Rminus_fp1 :
frac_part r1 >= frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2.
Proof.
- intros; unfold frac_part in |- *; generalize (Rminus_Int_part1 r1 r2 H);
+ intros; unfold frac_part; generalize (Rminus_Int_part1 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
- unfold Rminus in |- *;
+ unfold Rminus;
rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2)));
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR (Int_part r2)));
@@ -366,17 +366,17 @@ Lemma Rminus_fp2 :
frac_part r1 < frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1.
Proof.
- intros; unfold frac_part in |- *; generalize (Rminus_Int_part2 r1 r2 H);
+ intros; unfold frac_part; generalize (Rminus_Int_part2 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1);
rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
- unfold Rminus in |- *;
+ unfold Rminus;
rewrite
(Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1))
; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR 1));
rewrite (Ropp_involutive (IZR (Int_part r2)));
rewrite (Ropp_plus_distr (IZR (Int_part r1)));
- rewrite (Ropp_involutive (IZR (Int_part r2))); simpl in |- *;
+ rewrite (Ropp_involutive (IZR (Int_part r2))); simpl;
rewrite <-
(Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1)
; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
@@ -451,7 +451,7 @@ Proof.
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0);
- intro; clear H H0; unfold Int_part at 1 in |- *; omega.
+ intro; clear H H0; unfold Int_part at 1; omega.
Qed.
(**********)
@@ -514,7 +514,7 @@ Proof.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1);
- intro; clear H0 H1; unfold Int_part at 1 in |- *;
+ intro; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
@@ -524,17 +524,17 @@ Lemma plus_frac_part1 :
frac_part r1 + frac_part r2 >= 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1.
Proof.
- intros; unfold frac_part in |- *; generalize (plus_Int_part1 r1 r2 H); intro;
+ intros; unfold frac_part; generalize (plus_Int_part1 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1);
- rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl in |- *;
- unfold Rminus at 3 4 in |- *;
+ rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl;
+ unfold Rminus at 3 4;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
- unfold Rminus in |- *;
+ unfold Rminus;
rewrite
(Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1))
; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1);
@@ -547,14 +547,14 @@ Lemma plus_frac_part2 :
frac_part r1 + frac_part r2 < 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2.
Proof.
- intros; unfold frac_part in |- *; generalize (plus_Int_part2 r1 r2 H); intro;
+ intros; unfold frac_part; generalize (plus_Int_part2 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2));
- unfold Rminus at 2 3 in |- *;
+ unfold Rminus at 2 3;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
- unfold Rminus in |- *; trivial with zarith real.
+ unfold Rminus; trivial with zarith real.
Qed.
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-17 05:02:26 +0000