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-rw-r--r--theories/Reals/R_Ifp.v80
1 files changed, 32 insertions, 48 deletions
diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v
index b6d07283..77e2a1e0 100644
--- a/theories/Reals/R_Ifp.v
+++ b/theories/Reals/R_Ifp.v
@@ -1,9 +1,11 @@
(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
+(* * The Coq Proof Assistant / The Coq Development Team *)
+(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
+(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(**********************************************************)
@@ -42,28 +44,23 @@ Qed.
Lemma up_tech :
forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r.
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; rewrite H; intro; clear H H1;
- rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1));
- auto with zarith real.
+ intros.
+ apply tech_up with (1 := H0).
+ rewrite plus_IZR.
+ now apply Rplus_le_compat_r.
Qed.
(**********)
Lemma fp_R0 : frac_part 0 = 0.
Proof.
- 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) (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);
- intro; clear H1; generalize (le_IZR_R1 (up 0) H0);
- intro; clear H H0; omega.
+ unfold frac_part, Int_part.
+ replace (up 0) with 1%Z.
+ now rewrite <- minus_IZR.
+ destruct (archimed 0) as [H1 H2].
+ apply lt_IZR in H1.
+ rewrite <- minus_IZR in H2.
+ apply le_IZR in H2.
+ omega.
Qed.
(**********)
@@ -112,21 +109,12 @@ Lemma base_Int_part :
Proof.
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;
- rewrite <- (Rplus_assoc (IZR (up r)) (-1) (- r)) in H1;
- fold (IZR (up r) - 1) in H1; fold (IZR (up r) - 1 - r) in H1;
- apply Rminus_le; auto with zarith real.
- generalize (Rplus_gt_compat_l (-1) (IZR (up r)) r H); intro;
- rewrite (Rplus_comm (-1) (IZR (up r))) in H1;
- generalize (Rplus_gt_compat_l (- r) (IZR (up r) + -1) (-1 + r) H1);
- intro; clear H H0 H1; rewrite (Rplus_comm (- r) (IZR (up r) + -1)) in H2;
- fold (IZR (up r) - 1) in H2; fold (IZR (up r) - 1 - r) in H2;
- rewrite (Rplus_comm (- r) (-1 + r)) in H2;
- rewrite (Rplus_assoc (-1) r (- r)) in H2; rewrite (Rplus_opp_r r) in H2;
- elim (Rplus_ne (-1)); intros a b; rewrite a in H2;
- clear a b; auto with zarith real.
+ apply Rminus_le.
+ replace (IZR (up r) - 1 - r) with (IZR (up r) - r - 1) by ring.
+ now apply Rle_minus.
+ apply Rminus_gt.
+ replace (IZR (up r) - 1 - r - -1) with (IZR (up r) - r) by ring.
+ now apply Rgt_minus.
Qed.
(**********)
@@ -238,9 +226,7 @@ Proof.
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H;
clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
- rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
- cut (1 = IZR 1); auto with zarith real.
- intro; rewrite H1 in H; clear H1;
+ rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H.
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;
@@ -324,12 +310,12 @@ Proof.
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H0;
clear a b; rewrite <- (Rplus_opp_l 1) in H0;
- rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-1) 1)
+ rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-(1)) 1)
in H0; fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
- cut (1 = IZR 1); auto with zarith real.
- intro; rewrite H1 in H; rewrite H1 in H0; clear H1;
+ auto with zarith real.
+ change (_ + -1) with (IZR (Int_part r1 - Int_part r2) - 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 - Int_part r2 - 1) 1) in H0;
@@ -442,9 +428,9 @@ Proof.
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H0;
clear a b;
+ change 2 with (1 + 1) in H0;
rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0;
- cut (1 = IZR 1); auto with zarith real.
- intro; rewrite H1 in H0; rewrite H1 in H; clear H1;
+ auto with zarith real.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H;
@@ -507,9 +493,7 @@ Proof.
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2)));
intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2));
- intros a b; rewrite b in H0; clear a b; cut (1 = IZR 1);
- auto with zarith real.
- intro; rewrite H in H1; clear H;
+ intros a b; rewrite b in H0; clear a b.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1;
@@ -536,7 +520,7 @@ Proof.
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
unfold Rminus;
rewrite
- (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1))
+ (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);
trivial with zarith real.
Qed.
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-07 18:10:34 +0000