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-rw-r--r--theories/Reals/Alembert.v70
-rw-r--r--theories/Reals/AltSeries.v14
-rw-r--r--theories/Reals/ArithProp.v2
-rw-r--r--theories/Reals/Cos_plus.v6
-rw-r--r--theories/Reals/Cos_rel.v8
-rw-r--r--theories/Reals/DiscrR.v5
-rw-r--r--theories/Reals/Exp_prop.v20
-rw-r--r--theories/Reals/LegacyRfield.v2
-rw-r--r--theories/Reals/MVT.v11
-rw-r--r--theories/Reals/NewtonInt.v39
-rw-r--r--theories/Reals/PSeries_reg.v29
-rw-r--r--theories/Reals/PartSum.v12
-rw-r--r--theories/Reals/RIneq.v1091
-rw-r--r--theories/Reals/R_sqr.v5
-rw-r--r--theories/Reals/R_sqrt.v5
-rw-r--r--theories/Reals/Ranalysis.v5
-rw-r--r--theories/Reals/Ranalysis1.v53
-rw-r--r--theories/Reals/Ranalysis2.v5
-rw-r--r--theories/Reals/Ranalysis3.v13
-rw-r--r--theories/Reals/Ranalysis4.v21
-rw-r--r--theories/Reals/Raxioms.v6
-rw-r--r--theories/Reals/Rbasic_fun.v5
-rw-r--r--theories/Reals/Rcomplete.v6
-rw-r--r--theories/Reals/Rdefinitions.v26
-rw-r--r--theories/Reals/Rderiv.v5
-rw-r--r--theories/Reals/Rfunctions.v12
-rw-r--r--theories/Reals/Rgeom.v5
-rw-r--r--theories/Reals/RiemannInt.v131
-rw-r--r--theories/Reals/RiemannInt_SF.v36
-rw-r--r--theories/Reals/Rlimit.v9
-rw-r--r--theories/Reals/Rlogic.v293
-rw-r--r--theories/Reals/Rpow_def.v10
-rw-r--r--theories/Reals/Rpower.v58
-rw-r--r--theories/Reals/Rprod.v112
-rw-r--r--theories/Reals/Rseries.v6
-rw-r--r--theories/Reals/Rsigma.v2
-rw-r--r--theories/Reals/Rsqrt_def.v30
-rw-r--r--theories/Reals/Rtopology.v6
-rw-r--r--theories/Reals/Rtrigo.v2
-rw-r--r--theories/Reals/Rtrigo_alt.v6
-rw-r--r--theories/Reals/Rtrigo_def.v55
-rw-r--r--theories/Reals/Rtrigo_fun.v5
-rw-r--r--theories/Reals/Rtrigo_reg.v32
-rw-r--r--theories/Reals/SeqProp.v236
-rw-r--r--theories/Reals/SeqSeries.v20
-rw-r--r--theories/Reals/Sqrt_reg.v7
46 files changed, 1502 insertions, 1035 deletions
diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v
index 802bfa71..7625cce6 100644
--- a/theories/Reals/Alembert.v
+++ b/theories/Reals/Alembert.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Alembert.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Alembert.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -25,12 +25,12 @@ Lemma Alembert_C1 :
forall An:nat -> R,
(forall n:nat, 0 < An n) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros An H H0.
cut
- (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)).
+ ({ l:R | is_lub (EUn (fun N:nat => sum_f_R0 An N)) l } ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
intro X; apply X.
apply completeness.
unfold Un_cv in H0; unfold bound in |- *; cut (0 < / 2);
@@ -109,18 +109,18 @@ Proof.
symmetry in |- *; apply tech2; assumption.
exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity.
intro X; elim X; intros.
- apply existT with x; apply tech10;
+ exists x; apply tech10;
[ unfold Un_growing in |- *; intro; rewrite tech5;
pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l; left; apply H
| apply p ].
-Qed.
+Defined.
Lemma Alembert_C2 :
forall An:nat -> R,
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros.
set (Vn := fun i:nat => (2 * Rabs (An i) + An i) / 2).
@@ -133,7 +133,7 @@ Proof.
assert (H6 := Alembert_C1 Wn H2 H4).
elim H5; intros.
elim H6; intros.
- apply existT with (x - x0); unfold Un_cv in |- *; unfold Un_cv in p;
+ exists (x - x0); unfold Un_cv in |- *; unfold Un_cv in p;
unfold Un_cv in p0; intros; cut (0 < eps / 2).
intro; elim (p (eps / 2) H8); clear p; intros.
elim (p0 (eps / 2) H8); clear p0; intros.
@@ -334,21 +334,21 @@ Proof.
rewrite <- Rabs_Ropp; apply RRle_abs.
rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H.
-Qed.
+Defined.
Lemma AlembertC3_step1 :
forall (An:nat -> R) (x:R),
x <> 0 ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
- sigT (fun l:R => Pser An x l).
+ { l:R | Pser An x l }.
Proof.
intros; set (Bn := fun i:nat => An i * x ^ i).
cut (forall n:nat, Bn n <> 0).
intro; cut (Un_cv (fun n:nat => Rabs (Bn (S n) / Bn n)) 0).
intro; assert (H4 := Alembert_C2 Bn H2 H3).
elim H4; intros.
- apply existT with x0; unfold Bn in p; apply tech12; assumption.
+ exists x0; unfold Bn in p; apply tech12; assumption.
unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x).
intro; elim (H1 (eps / Rabs x) H4); intros.
exists x0; intros; unfold R_dist in |- *; unfold Rminus in |- *;
@@ -379,13 +379,13 @@ Proof.
[ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
intro; unfold Bn in |- *; apply prod_neq_R0;
[ apply H0 | apply pow_nonzero; assumption ].
-Qed.
+Defined.
Lemma AlembertC3_step2 :
- forall (An:nat -> R) (x:R), x = 0 -> sigT (fun l:R => Pser An x l).
+ forall (An:nat -> R) (x:R), x = 0 -> { l:R | Pser An x l }.
Proof.
- intros; apply existT with (An 0%nat).
- unfold Pser in |- *; unfold infinit_sum in |- *; intros; exists 0%nat; intros;
+ intros; exists (An 0%nat).
+ unfold Pser in |- *; unfold infinite_sum in |- *; intros; exists 0%nat; intros;
replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat).
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
rewrite Rabs_R0; assumption.
@@ -395,12 +395,12 @@ Proof.
[ rewrite H; simpl in |- *; ring | unfold ge in |- *; apply le_O_n ].
Qed.
-(** An useful criterion of convergence for power series *)
+(** A useful criterion of convergence for power series *)
Theorem Alembert_C3 :
forall (An:nat -> R) (x:R),
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
- sigT (fun l:R => Pser An x l).
+ { l:R | Pser An x l }.
Proof.
intros; case (total_order_T x 0); intro.
elim s; intro.
@@ -411,19 +411,19 @@ Proof.
cut (x <> 0).
intro; apply AlembertC3_step1; assumption.
red in |- *; intro; rewrite H1 in r; elim (Rlt_irrefl _ r).
-Qed.
+Defined.
Lemma Alembert_C4 :
forall (An:nat -> R) (k:R),
0 <= k < 1 ->
(forall n:nat, 0 < An n) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros An k Hyp H H0.
cut
- (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)).
+ ({ l:R | is_lub (EUn (fun N:nat => sum_f_R0 An N)) l } ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
intro X; apply X.
apply completeness.
assert (H1 := tech13 _ _ Hyp H0).
@@ -524,7 +524,7 @@ Proof.
symmetry in |- *; apply tech2; assumption.
exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity.
intro X; elim X; intros.
- apply existT with x; apply tech10;
+ exists x; apply tech10;
[ unfold Un_growing in |- *; intro; rewrite tech5;
pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l; left; apply H
@@ -536,21 +536,19 @@ Lemma Alembert_C5 :
0 <= k < 1 ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros.
cut
- (sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)).
+ ({ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l } ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
intro Hyp0; apply Hyp0.
apply cv_cauchy_2.
apply cauchy_abs.
apply cv_cauchy_1.
cut
- (sigT
- (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l) ->
- sigT
- (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l)).
+ ({ l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l } ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l }).
intro Hyp; apply Hyp.
apply (Alembert_C4 (fun i:nat => Rabs (An i)) k).
assumption.
@@ -568,11 +566,11 @@ Proof.
apply H0.
intro X.
elim X; intros.
- apply existT with x.
+ exists x.
assumption.
intro X.
elim X; intros.
- apply existT with x.
+ exists x.
assumption.
Qed.
@@ -583,14 +581,12 @@ Lemma Alembert_C6 :
0 < k ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
- Rabs x < / k -> sigT (fun l:R => Pser An x l).
+ Rabs x < / k -> { l:R | Pser An x l }.
intros.
- cut
- (sigT
- (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l)).
+ cut { l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l }.
intro X.
elim X; intros.
- apply existT with x0.
+ exists x0.
apply tech12; assumption.
case (total_order_T x 0); intro.
elim s; intro.
@@ -655,7 +651,7 @@ Lemma Alembert_C6 :
assumption.
apply Rinv_0_lt_compat; apply Rabs_pos_lt.
red in |- *; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a).
- apply existT with (An 0%nat).
+ exists (An 0%nat).
unfold Un_cv in |- *.
intros.
exists 0%nat.
diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v
index 581c181f..5c4bbd6a 100644
--- a/theories/Reals/AltSeries.v
+++ b/theories/Reals/AltSeries.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
- (*i $Id: AltSeries.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+ (*i $Id: AltSeries.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -153,14 +153,14 @@ Lemma CV_ALT :
Un_decreasing Un ->
positivity_seq Un ->
Un_cv Un 0 ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.
Proof.
intros.
assert (H2 := CV_ALT_step0 _ H).
assert (H3 := CV_ALT_step4 _ H H0).
assert (X := growing_cv _ H2 H3).
elim X; intros.
- apply existT with x.
+ exists x.
unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1;
unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p.
intros; cut (0 < eps / 2);
@@ -220,7 +220,7 @@ Theorem alternated_series :
forall Un:nat -> R,
Un_decreasing Un ->
Un_cv Un 0 ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.
Proof.
intros; apply CV_ALT.
assumption.
@@ -408,7 +408,7 @@ Proof.
Qed.
Lemma exist_PI :
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l }.
Proof.
apply alternated_series.
apply PI_tg_decreasing.
@@ -416,9 +416,7 @@ Proof.
Qed.
(** Now, PI is defined *)
-Definition PI : R := 4 * match exist_PI with
- | existT a b => a
- end.
+Definition PI : R := 4 * (let (a,_) := exist_PI in a).
(** We can get an approximation of PI with the following inequality *)
Lemma PI_ineq :
diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v
index 7dbbd605..7327c64c 100644
--- a/theories/Reals/ArithProp.v
+++ b/theories/Reals/ArithProp.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
- (*i $Id: ArithProp.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+ (*i $Id: ArithProp.v 9454 2006-12-15 15:30:59Z bgregoir $ i*)
Require Import Rbase.
Require Import Rbasic_fun.
diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v
index 10965951..0de639e8 100644
--- a/theories/Reals/Cos_plus.v
+++ b/theories/Reals/Cos_plus.v
@@ -6,14 +6,16 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
- (*i $Id: Cos_plus.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+ (*i $Id: Cos_plus.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Require Import Cos_rel.
-Require Import Max. Open Local Scope nat_scope. Open Local Scope R_scope.
+Require Import Max.
+Open Local Scope nat_scope.
+Open Local Scope R_scope.
Definition Majxy (x y:R) (n:nat) : R :=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n).
diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v
index d410e14a..aed481c7 100644
--- a/theories/Reals/Cos_rel.v
+++ b/theories/Reals/Cos_rel.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Cos_rel.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Cos_rel.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -263,7 +263,7 @@ assert (H := exist_cos (x * x)).
elim H; intros.
assert (p_i := p).
unfold cos_in in p.
-unfold cos_n, infinit_sum in p.
+unfold cos_n, infinite_sum in p.
unfold R_dist in p.
cut (cos x = x0).
intro.
@@ -295,7 +295,7 @@ assert (H := exist_cos ((x + y) * (x + y))).
elim H; intros.
assert (p_i := p).
unfold cos_in in p.
-unfold cos_n, infinit_sum in p.
+unfold cos_n, infinite_sum in p.
unfold R_dist in p.
cut (cos (x + y) = x0).
intro.
@@ -344,7 +344,7 @@ assert (H0 := exist_sin (x * x)).
elim H0; intros.
assert (p_i := p).
unfold sin_in in p.
-unfold sin_n, infinit_sum in p.
+unfold sin_n, infinite_sum in p.
unfold R_dist in p.
cut (sin x = x * x0).
intro.
diff --git a/theories/Reals/DiscrR.v b/theories/Reals/DiscrR.v
index a16af05c..22a52e67 100644
--- a/theories/Reals/DiscrR.v
+++ b/theories/Reals/DiscrR.v
@@ -6,10 +6,11 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: DiscrR.v 9178 2006-09-26 11:18:22Z barras $ i*)
+(*i $Id: DiscrR.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import RIneq.
-Require Import Omega. Open Local Scope R_scope.
+Require Import Omega.
+Open Local Scope R_scope.
Lemma Rlt_R0_R2 : 0 < 2.
change 2 with (INR 2); apply lt_INR_0; apply lt_O_Sn.
diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v
index beb4b744..bf729526 100644
--- a/theories/Reals/Exp_prop.v
+++ b/theories/Reals/Exp_prop.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Exp_prop.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Exp_prop.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -27,7 +27,7 @@ Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x).
Proof.
intro; unfold exp in |- *; unfold projT1 in |- *.
case (exist_exp x); intro.
- unfold exp_in, Un_cv in |- *; unfold infinit_sum, E1 in |- *; trivial.
+ unfold exp_in, Un_cv in |- *; unfold infinite_sum, E1 in |- *; trivial.
Qed.
Definition Reste_E (x y:R) (N:nat) : R :=
@@ -734,7 +734,7 @@ Proof.
apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply (pow_lt _ n H).
unfold exp in |- *; unfold projT1 in |- *; case (exist_exp x); intro.
- unfold exp_in in |- *; unfold infinit_sum, Un_cv in |- *; trivial.
+ unfold exp_in in |- *; unfold infinite_sum, Un_cv in |- *; trivial.
Qed.
(**********)
@@ -769,7 +769,7 @@ Proof.
unfold derivable_pt_lim in |- *; intros.
set (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))).
cut (CVN_R fn).
- intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)).
+ intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
intro cv; cut (forall n:nat, continuity (fn n)).
intro; cut (continuity (SFL fn cv)).
intro; unfold continuity in H1.
@@ -809,13 +809,12 @@ Proof.
unfold Rdiv in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ].
unfold SFL, exp in |- *.
- unfold projT1 in |- *.
- case (cv h); case (exist_exp h); intros.
+ case (cv h); case (exist_exp h); simpl; intros.
eapply UL_sequence.
apply u.
unfold Un_cv in |- *; intros.
unfold exp_in in e.
- unfold infinit_sum in e.
+ unfold infinite_sum in e.
cut (0 < eps0 * Rabs h).
intro; elim (e _ H9); intros N0 H10.
exists N0; intros.
@@ -871,13 +870,12 @@ Proof.
assert (H0 := Alembert_exp).
unfold CVN_R in |- *.
intro; unfold CVN_r in |- *.
- apply existT with (fun N:nat => r ^ N / INR (fact (S N))).
+ exists (fun N:nat => r ^ N / INR (fact (S N))).
cut
- (sigT
- (fun l:R =>
+ { l:R |
Un_cv
(fun n:nat =>
- sum_f_R0 (fun k:nat => Rabs (r ^ k / INR (fact (S k)))) n) l)).
+ sum_f_R0 (fun k:nat => Rabs (r ^ k / INR (fact (S k)))) n) l }.
intro X.
elim X; intros.
exists x; intros.
diff --git a/theories/Reals/LegacyRfield.v b/theories/Reals/LegacyRfield.v
index b33274af..3f76e77a 100644
--- a/theories/Reals/LegacyRfield.v
+++ b/theories/Reals/LegacyRfield.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id$ i*)
+(*i $Id: LegacyRfield.v 10739 2008-04-01 14:45:20Z herbelin $ i*)
Require Export Raxioms.
Require Export LegacyField.
diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v
index 8bb9298a..f22e49e1 100644
--- a/theories/Reals/MVT.v
+++ b/theories/Reals/MVT.v
@@ -6,12 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: MVT.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: MVT.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
-Require Import Rtopology. Open Local Scope R_scope.
+Require Import Rtopology.
+Open Local Scope R_scope.
(* The Mean Value Theorem *)
Theorem MVT :
@@ -189,7 +190,7 @@ Proof.
intros; apply derivable_pt_id.
intros; apply derivable_continuous_pt; apply X; assumption.
intros; elim H1; intros; apply X; split; left; assumption.
- intros; unfold derivable_pt in |- *; apply existT with (f' c); apply H0;
+ intros; unfold derivable_pt in |- *; exists (f' c); apply H0;
apply H1.
Qed.
@@ -695,11 +696,11 @@ Proof.
unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0;
clear H0; intros H0 _; exists (g1 a - g2 a); intros;
assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x).
- intros; unfold derivable_pt in |- *; apply existT with (f x0); elim (H x0 H3);
+ intros; unfold derivable_pt in |- *; exists (f x0); elim (H x0 H3);
intros; eapply derive_pt_eq_1; symmetry in |- *;
apply H4.
assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x).
- intros; unfold derivable_pt in |- *; apply existT with (f x0);
+ intros; unfold derivable_pt in |- *; exists (f x0);
elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry in |- *;
apply H5.
assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x).
diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v
index 306d5ac4..47ae149e 100644
--- a/theories/Reals/NewtonInt.v
+++ b/theories/Reals/NewtonInt.v
@@ -6,32 +6,31 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: NewtonInt.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: NewtonInt.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo.
-Require Import Ranalysis. Open Local Scope R_scope.
+Require Import Ranalysis.
+Open Local Scope R_scope.
(*******************************************)
(* Newton's Integral *)
(*******************************************)
Definition Newton_integrable (f:R -> R) (a b:R) : Type :=
- sigT (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a).
+ { g:R -> R | antiderivative f g a b \/ antiderivative f g b a }.
Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R :=
- let g := match pr with
- | existT a b => a
- end in g b - g a.
+ let (g,_) := pr in g b - g a.
(* If f is differentiable, then f' is Newton integrable (Tautology ?) *)
Lemma FTCN_step1 :
forall (f:Differential) (a b:R),
Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b.
Proof.
- intros f a b; unfold Newton_integrable in |- *; apply existT with (d1 f);
+ intros f a b; unfold Newton_integrable in |- *; exists (d1 f);
unfold antiderivative in |- *; intros; case (Rle_dec a b);
intro;
[ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ]
@@ -52,7 +51,7 @@ Qed.
Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a.
Proof.
intros f a; unfold Newton_integrable in |- *;
- apply existT with (fct_cte (f a) * id)%F; left;
+ exists (fct_cte (f a) * id)%F; left;
unfold antiderivative in |- *; split.
intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x).
apply derivable_pt_mult.
@@ -82,7 +81,7 @@ Lemma NewtonInt_P3 :
Newton_integrable f b a.
Proof.
unfold Newton_integrable in |- *; intros; elim X; intros g H;
- apply existT with g; tauto.
+ exists g; tauto.
Defined.
(* $\int_a^b f = -\int_b^a f$ *)
@@ -94,7 +93,7 @@ Proof.
unfold NewtonInt in |- *;
case
(NewtonInt_P3 f a b
- (existT
+ (exist
(fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x
p)).
intros; elim o; intro.
@@ -112,7 +111,7 @@ Proof.
unfold NewtonInt in |- *;
case
(NewtonInt_P3 f a b
- (existT
+ (exist
(fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x
p)); intros; elim o; intro.
assert (H1 := antiderivative_Ucte f x x0 b a H H0); elim H1; intros;
@@ -325,7 +324,7 @@ Proof.
| left _ => F0 x
| right _ => F1 x + (F0 b - F1 b)
end) x).
- unfold derivable_pt in |- *; apply existT with (f x); apply H7.
+ unfold derivable_pt in |- *; exists (f x); apply H7.
exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7.
assert (H5 : a <= x <= b).
split; [ assumption | right; assumption ].
@@ -370,7 +369,7 @@ Proof.
| left _ => F0 x
| right _ => F1 x + (F0 b - F1 b)
end) x).
- unfold derivable_pt in |- *; apply existT with (f x); apply H13.
+ unfold derivable_pt in |- *; exists (f x); apply H13.
exists H14; symmetry in |- *; apply derive_pt_eq_0; apply H13.
assert (H5 : b <= x <= c).
split; [ left; assumption | assumption ].
@@ -417,7 +416,7 @@ Proof.
| left _ => F0 x
| right _ => F1 x + (F0 b - F1 b)
end) x).
- unfold derivable_pt in |- *; apply existT with (f x); apply H7.
+ unfold derivable_pt in |- *; exists (f x); apply H7.
exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7.
Qed.
@@ -482,7 +481,7 @@ Proof.
match Rle_dec x b with
| left _ => F0 x
| right _ => F1 x + (F0 b - F1 b)
- end); apply existT with g; left; unfold g in |- *;
+ end); exists g; left; unfold g in |- *;
apply antiderivative_P2.
elim H0; intro.
assumption.
@@ -508,7 +507,7 @@ Proof.
elim s0; intro.
(* a<b & b<c *)
unfold Newton_integrable in |- *;
- apply existT with
+ exists
(fun x:R =>
match Rle_dec x b with
| left _ => F0 x
@@ -526,7 +525,7 @@ Proof.
(* a<b & b>c *)
case (total_order_T a c); intro.
elim s0; intro.
- unfold Newton_integrable in |- *; apply existT with F0.
+ unfold Newton_integrable in |- *; exists F0.
left.
elim H1; intro.
unfold antiderivative in H; elim H; clear H; intros _ H.
@@ -540,7 +539,7 @@ Proof.
unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)).
rewrite b0; apply NewtonInt_P1.
- unfold Newton_integrable in |- *; apply existT with F1.
+ unfold Newton_integrable in |- *; exists F1.
right.
elim H1; intro.
unfold antiderivative in H; elim H; clear H; intros _ H.
@@ -560,7 +559,7 @@ Proof.
(* a>b & b<c *)
case (total_order_T a c); intro.
elim s0; intro.
- unfold Newton_integrable in |- *; apply existT with F1.
+ unfold Newton_integrable in |- *; exists F1.
left.
elim H1; intro.
(*****************)
@@ -575,7 +574,7 @@ Proof.
unfold antiderivative in H; elim H; clear H; intros _ H.
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)).
rewrite b0; apply NewtonInt_P1.
- unfold Newton_integrable in |- *; apply existT with F0.
+ unfold Newton_integrable in |- *; exists F0.
right.
elim H0; intro.
unfold antiderivative in H; elim H; clear H; intros _ H.
diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v
index 64b8e0af..e122a26a 100644
--- a/theories/Reals/PSeries_reg.v
+++ b/theories/Reals/PSeries_reg.v
@@ -6,14 +6,15 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: PSeries_reg.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: PSeries_reg.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis1.
Require Import Max.
-Require Import Even. Open Local Scope R_scope.
+Require Import Even.
+Open Local Scope R_scope.
Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.
@@ -28,25 +29,21 @@ Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R)
(** Normal convergence *)
Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type :=
- sigT
- (fun An:nat -> R =>
- sigT
- (fun l:R =>
+ { An:nat -> R &
+ { l:R |
Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
- (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n))).
+ (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }.
Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.
Definition SFL (fn:nat -> R -> R)
- (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l))
- (y:R) : R := match cv y with
- | existT a b => a
- end.
+ (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
+ (y:R) : R := let (a,_) := cv y in a.
(** In a complete space, normal convergence implies uniform convergence *)
Lemma CVN_CVU :
forall (fn:nat -> R -> R)
- (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l))
+ (cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r.
Proof.
intros; unfold CVU in |- *; intros.
@@ -193,7 +190,7 @@ Qed.
(** Continuity and normal convergence *)
Lemma SFL_continuity_pt :
forall (fn:nat -> R -> R)
- (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l))
+ (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal),
CVN_r fn r ->
(forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) ->
@@ -210,7 +207,7 @@ Qed.
Lemma SFL_continuity :
forall (fn:nat -> R -> R)
- (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)),
+ (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }),
CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv).
Proof.
intros; unfold continuity in |- *; intro.
@@ -229,7 +226,7 @@ Qed.
(** As R is complete, normal convergence implies that (fn) is simply-uniformly convergent *)
Lemma CVN_R_CVS :
forall fn:nat -> R -> R,
- CVN_R fn -> forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l).
+ CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }.
Proof.
intros; apply R_complete.
unfold SP in |- *; set (An := fun N:nat => fn N x).
@@ -248,7 +245,7 @@ Proof.
rewrite Rminus_0_r.
pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
apply Rlt_0_1.
- apply existT with l.
+ exists l.
cut (forall n:nat, 0 <= Bn n).
intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros.
elim (H3 _ H6); intros.
diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v
index a8f72302..d5ae2aca 100644
--- a/theories/Reals/PartSum.v
+++ b/theories/Reals/PartSum.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: PartSum.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: PartSum.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -153,7 +153,7 @@ Lemma tech12 :
Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l ->
Pser An x l.
Proof.
- intros; unfold Pser in |- *; unfold infinit_sum in |- *; unfold Un_cv in H;
+ intros; unfold Pser in |- *; unfold infinite_sum in |- *; unfold Un_cv in H;
assumption.
Qed.
@@ -218,9 +218,9 @@ Qed.
(* Unicity of the limit defined by convergent series *)
Lemma uniqueness_sum :
forall (An:nat -> R) (l1 l2:R),
- infinit_sum An l1 -> infinit_sum An l2 -> l1 = l2.
+ infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2.
Proof.
- unfold infinit_sum in |- *; intros.
+ unfold infinite_sum in |- *; intros.
case (Req_dec l1 l2); intro.
assumption.
cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ].
@@ -450,7 +450,7 @@ Qed.
(**********)
Lemma cv_cauchy_1 :
forall An:nat -> R,
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l } ->
Cauchy_crit_series An.
Proof.
intros An X.
@@ -481,7 +481,7 @@ Qed.
Lemma cv_cauchy_2 :
forall An:nat -> R,
Cauchy_crit_series An ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros.
apply R_complete.
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v
index 7d98a844..19bdeccd 100644
--- a/theories/Reals/RIneq.v
+++ b/theories/Reals/RIneq.v
@@ -6,11 +6,11 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: RIneq.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: RIneq.v 10762 2008-04-06 16:57:31Z herbelin $ i*)
-(***************************************************************************)
-(** Basic lemmas for the classical reals numbers *)
-(***************************************************************************)
+(*********************************************************)
+(** * Basic lemmas for the classical real numbers *)
+(*********************************************************)
Require Export Raxioms.
Require Import Rpow_def.
@@ -24,21 +24,32 @@ Open Local Scope R_scope.
Implicit Type r : R.
-(**************************************************************************)
-(** * Relation between orders and equality *)
-(**************************************************************************)
+(*********************************************************)
+(** ** Relation between orders and equality *)
+(*********************************************************)
+
+(** Reflexivity of the large order *)
+
+Lemma Rle_refl : forall r, r <= r.
+Proof.
+ intro; right; reflexivity.
+Qed.
+Hint Immediate Rle_refl: rorders.
+
+Lemma Rge_refl : forall r, r <= r.
+Proof. exact Rle_refl. Qed.
+Hint Immediate Rge_refl: rorders.
+
+(** Irreflexivity of the strict order *)
-(**********)
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 Rgt_irrefl : forall r, ~ r > r.
+Proof. exact Rlt_irrefl. Qed.
Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2.
Proof.
@@ -58,7 +69,7 @@ Proof.
Qed.
Hint Resolve Rlt_dichotomy_converse: real.
-(** Reasoning by case on equalities and order *)
+(** Reasoning by case on equality and order *)
(**********)
Lemma Req_dec : forall r1 r2, r1 = r2 \/ r1 <> r2.
@@ -80,58 +91,104 @@ Proof.
intros; generalize (total_order_T r1 r2); tauto.
Qed.
+(*********************************************************)
+(** ** Relating [<], [>], [<=] and [>=] *)
+(*********************************************************)
-(*********************************************************************************)
-(** * Order Lemma : relating [<], [>], [<=] and [>=] *)
-(*********************************************************************************)
+(*********************************************************)
+(** ** Order *)
+(*********************************************************)
+
+(** *** Relating strict and large orders *)
-(**********)
Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2.
Proof.
intros; red in |- *; tauto.
Qed.
Hint Resolve Rlt_le: real.
+Lemma Rgt_ge : forall r1 r2, r1 > r2 -> r1 >= r2.
+Proof.
+ intros; red; tauto.
+Qed.
+
(**********)
Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1.
Proof.
destruct 1; red in |- *; auto with real.
Qed.
-
Hint Immediate Rle_ge: real.
+Hint Resolve Rle_ge: rorders.
-(**********)
Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1.
Proof.
destruct 1; red in |- *; auto with real.
Qed.
-
Hint Resolve Rge_le: real.
+Hint Immediate Rge_le: rorders.
(**********)
+Lemma Rlt_gt : forall r1 r2, r1 < r2 -> r2 > r1.
+Proof.
+ trivial.
+Qed.
+Hint Resolve Rlt_gt: rorders.
+
+Lemma Rgt_lt : forall r1 r2, r1 > r2 -> r2 < r1.
+Proof.
+ trivial.
+Qed.
+Hint Immediate Rgt_lt: rorders.
+
+(**********)
+
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_gt : forall r1 r2, ~ r1 >= r2 -> r2 > r1.
+Proof. intros; red; apply Rnot_le_lt. auto with real. Qed.
+
+Lemma Rnot_le_gt : forall r1 r2, ~ r1 <= r2 -> r1 > r2.
+Proof. intros; red; apply Rnot_le_lt. auto with real. Qed.
+
Lemma Rnot_ge_lt : forall r1 r2, ~ r1 >= r2 -> r1 < r2.
+Proof. intros; apply Rnot_le_lt. auto with real. Qed.
+
+Lemma Rnot_lt_le : forall r1 r2, ~ r1 < r2 -> r2 <= r1.
Proof.
- intros; apply Rnot_le_lt; auto with real.
+ intros r1 r2 H; destruct (Rtotal_order r1 r2) as [ | [ H0 | H0 ] ].
+ contradiction. subst; auto with rorders. auto with real.
Qed.
+Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2.
+Proof. auto using Rnot_lt_le with real. Qed.
+
+Lemma Rnot_gt_ge : forall r1 r2, ~ r1 > r2 -> r2 >= r1.
+Proof. intros; eauto using Rnot_lt_le with rorders. Qed.
+
+Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2.
+Proof. eauto using Rnot_gt_ge with rorders. 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.
+Hint Immediate Rlt_not_le: real.
Lemma Rgt_not_le : forall r1 r2, r1 > r2 -> ~ r1 <= r2.
-Proof Rlt_not_le.
+Proof. exact Rlt_not_le. Qed.
-Hint Immediate Rlt_not_le: real.
+Lemma Rlt_not_ge : forall r1 r2, r1 < r2 -> ~ r1 >= r2.
+Proof. red; intros; eapply Rlt_not_le; eauto with real. Qed.
+Hint Immediate Rlt_not_ge: real.
+
+Lemma Rgt_not_ge : forall r1 r2, r2 > r1 -> ~ r1 >= r2.
+Proof. exact Rlt_not_ge. Qed.
Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2.
Proof.
@@ -139,13 +196,14 @@ Proof.
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.
+Lemma Rge_not_lt : forall r1 r2, r1 >= r2 -> ~ r1 < r2.
+Proof. intros; apply Rle_not_lt; auto with real. Qed.
-Hint Immediate Rlt_not_ge: real.
+Lemma Rle_not_gt : forall r1 r2, r1 <= r2 -> ~ r1 > r2.
+Proof. do 2 intro; apply Rle_not_lt. Qed.
+
+Lemma Rge_not_gt : forall r1 r2, r2 >= r1 -> ~ r1 > r2.
+Proof. do 2 intro; apply Rge_not_lt. Qed.
(**********)
Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2.
@@ -172,25 +230,51 @@ Proof.
Qed.
Hint Immediate Req_ge_sym: real.
+(** *** Asymmetry *)
+
+(** Remark: [Rlt_asym] is an axiom *)
+
+Lemma Rgt_asym : forall r1 r2:R, r1 > r2 -> ~ r2 > r1.
+Proof. do 2 intro; apply Rlt_asym. Qed.
+
+(** *** Antisymmetry *)
+
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 Rge_antisym : forall r1 r2, r1 >= r2 -> r2 >= r1 -> r1 = r2.
+Proof. auto with real. Qed.
+
(**********)
Lemma Rle_le_eq : forall r1 r2, r1 <= r2 /\ r2 <= r1 <-> r1 = r2.
Proof.
intuition.
Qed.
+Lemma Rge_ge_eq : forall r1 r2, r1 >= r2 /\ r2 >= r1 <-> r1 = r2.
+Proof.
+ intuition.
+Qed.
+
+(** *** Compatibility with equality *)
+
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 Rgt_eq_compat :
+ forall r1 r2 r3 r4, r1 = r2 -> r2 > r4 -> r4 = r3 -> r1 > r3.
+Proof. intros; red; apply Rlt_eq_compat with (r2:=r4) (r4:=r2); auto. Qed.
+
+(** *** Transitivity *)
+
+(** Remark: [Rlt_trans] is an axiom *)
+
Lemma Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3.
Proof.
generalize trans_eq Rlt_trans Rlt_eq_compat.
@@ -198,6 +282,12 @@ Proof.
intuition eauto 2.
Qed.
+Lemma Rge_trans : forall r1 r2 r3, r1 >= r2 -> r2 >= r3 -> r1 >= r3.
+Proof. eauto using Rle_trans with rorders. Qed.
+
+Lemma Rgt_trans : forall r1 r2 r3, r1 > r2 -> r2 > r3 -> r1 > r3.
+Proof. eauto using Rlt_trans with rorders. Qed.
+
(**********)
Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3.
Proof.
@@ -206,21 +296,25 @@ Proof.
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.
+Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3.
+Proof. eauto using Rlt_le_trans with rorders. Qed.
+
+Lemma Rgt_ge_trans : forall r1 r2 r3, r1 > r2 -> r2 >= r3 -> r1 > r3.
+Proof. eauto using Rle_lt_trans with rorders. Qed.
+
+(** *** (Classical) decidability *)
-(** 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.
@@ -228,28 +322,44 @@ Proof.
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.
+Proof. do 2 intro; apply Rlt_dec. Qed.
-(**********)
Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}.
+Proof. intros; edestruct Rle_dec; [left|right]; eauto with rorders. Qed.
+
+Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}.
Proof.
- intros; generalize (Rle_dec r2 r1); intuition.
+ intros; generalize (total_order_T r1 r2); intuition.
Qed.
-Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}.
+Lemma Rgt_ge_dec : forall r1 r2, {r1 > r2} + {r2 >= r1}.
+Proof. intros; edestruct Rlt_le_dec; [left|right]; eauto with rorders. Qed.
+
+Lemma Rle_lt_dec : forall r1 r2, {r1 <= r2} + {r2 < r1}.
Proof.
intros; generalize (total_order_T r1 r2); intuition.
Qed.
+Lemma Rge_gt_dec : forall r1 r2, {r1 >= r2} + {r2 > r1}.
+Proof. intros; edestruct Rle_lt_dec; [left|right]; eauto with rorders. Qed.
+
+Lemma Rlt_or_le : forall r1 r2, r1 < r2 \/ r2 <= r1.
+Proof.
+ intros n m; elim (Rle_lt_dec m n); auto with real.
+Qed.
+
+Lemma Rgt_or_ge : forall r1 r2, r1 > r2 \/ r2 >= r1.
+Proof. intros; edestruct Rlt_or_le; [left|right]; eauto with rorders. 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 Rge_or_gt : forall r1 r2, r1 >= r2 \/ r2 > r1.
+Proof. intros; edestruct Rle_or_lt; [left|right]; eauto with rorders. 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.
@@ -262,19 +372,11 @@ Proof.
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 *)
+(** ** Addition *)
(*********************************************************)
-Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r.
-Proof.
- split; ring.
-Qed.
-Hint Resolve Rplus_ne: real v62.
+(** Remark: [Rplus_0_l] is an axiom *)
Lemma Rplus_0_r : forall r, r + 0 = r.
Proof.
@@ -282,14 +384,22 @@ Proof.
Qed.
Hint Resolve Rplus_0_r: real.
+Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r.
+Proof.
+ split; ring.
+Qed.
+Hint Resolve Rplus_ne: real v62.
+
(**********)
+
+(** Remark: [Rplus_opp_r] is an axiom *)
+
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.
@@ -298,7 +408,6 @@ Proof.
rewrite Rplus_assoc; rewrite H; 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.
@@ -325,9 +434,31 @@ Proof.
intros r b; pattern r at 2 in |- *; replace r with (r + 0); eauto with real.
Qed.
-(***********************************************************)
-(** ** Multiplication *)
-(***********************************************************)
+(***********)
+Lemma Rplus_eq_0_l :
+ forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0.
+Proof.
+ intros a b H [H0| H0] H1; auto with real.
+ absurd (0 < a + b).
+ rewrite H1; auto with real.
+ apply Rle_lt_trans with (a + 0).
+ rewrite Rplus_0_r in |- *; assumption.
+ auto using Rplus_lt_compat_l with real.
+ rewrite <- H0, Rplus_0_r in H1; assumption.
+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.
+
+(*********************************************************)
+(** ** Multiplication *)
+(*********************************************************)
(**********)
Lemma Rinv_r : forall r, r <> 0 -> r * / r = 1.
@@ -340,13 +471,13 @@ Lemma Rinv_l_sym : forall r, r <> 0 -> 1 = / r * r.
Proof.
intros; field; trivial.
Qed.
+Hint Resolve Rinv_l_sym: real.
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.
-
+Hint Resolve Rinv_r_sym: real.
(**********)
Lemma Rmult_0_r : forall r, r * 0 = 0.
@@ -382,7 +513,7 @@ Proof.
auto with real.
Qed.
-(*i OLD i*)Hint Resolve Rmult_eq_compat_l: v62.
+(*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.
@@ -423,7 +554,6 @@ Proof.
auto with real.
Qed.
-
(**********)
Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0.
Proof.
@@ -439,6 +569,10 @@ Proof.
Qed.
Hint Resolve Rmult_integral_contrapositive: real.
+Lemma Rmult_integral_contrapositive_currified :
+ forall r1 r2, r1 <> 0 -> r2 <> 0 -> r1 * r2 <> 0.
+Proof. auto using Rmult_integral_contrapositive. Qed.
+
(**********)
Lemma Rmult_plus_distr_r :
forall r1 r2 r3, (r1 + r2) * r3 = r1 * r3 + r2 * r3.
@@ -446,11 +580,15 @@ Proof.
intros; ring.
Qed.
-(** ** Square function *)
+(*********************************************************)
+(** ** Square function *)
+(*********************************************************)
(***********)
Definition Rsqr r : R := r * r.
+Notation "r ²" := (Rsqr r) (at level 1, format "r ²") : R_scope.
+
(***********)
Lemma Rsqr_0 : Rsqr 0 = 0.
unfold Rsqr in |- *; auto with real.
@@ -462,7 +600,7 @@ Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0.
Qed.
(*********************************************************)
-(** ** Opposite *)
+(** ** Opposite *)
(*********************************************************)
(**********)
@@ -509,8 +647,9 @@ Proof.
Qed.
Hint Resolve Ropp_plus_distr: real.
-
-(** ** Opposite and multiplication *)
+(*********************************************************)
+(** ** Opposite and multiplication *)
+(*********************************************************)
Lemma Ropp_mult_distr_l_reverse : forall r1 r2, - r1 * r2 = - (r1 * r2).
Proof.
@@ -530,7 +669,9 @@ Proof.
intros; ring.
Qed.
-(** ** Substraction *)
+(*********************************************************)
+(** ** Substraction *)
+(*********************************************************)
Lemma Rminus_0_r : forall r, r - 0 = r.
Proof.
@@ -555,7 +696,6 @@ 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.
@@ -605,7 +745,6 @@ Proof.
Qed.
Hint Resolve Rminus_not_eq_right: real.
-
(**********)
Lemma Rmult_minus_distr_l :
forall r1 r2 r3, r1 * (r2 - r3) = r1 * r2 - r1 * r3.
@@ -613,7 +752,10 @@ Proof.
intros; ring.
Qed.
-(** ** Inverse *)
+(*********************************************************)
+(** ** Inverse *)
+(*********************************************************)
+
Lemma Rinv_1 : / 1 = 1.
Proof.
field.
@@ -677,28 +819,28 @@ Proof.
ring.
Qed.
-(** * Field operations and order *)
+(*********************************************************)
+(** ** Order and addition *)
+(*********************************************************)
+
+(** *** Compatibility *)
-(** ** Order and addition *)
+(** Remark: [Rplus_lt_compat_l] is an axiom *)
+Lemma Rplus_gt_compat_l : forall r r1 r2, r1 > r2 -> r + r1 > r + r2.
+Proof. eauto using Rplus_lt_compat_l with rorders. Qed.
+Hint Resolve Rplus_gt_compat_l: real.
+
+(**********)
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_gt_compat_r : forall r r1 r2, r1 > r2 -> r1 + r > r2 + r.
+Proof. do 3 intro; apply Rplus_lt_compat_r. Qed.
(**********)
Lemma Rplus_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2.
@@ -708,6 +850,10 @@ Proof.
right; rewrite <- H0; auto with zarith real.
Qed.
+Lemma Rplus_ge_compat_l : forall r r1 r2, r1 >= r2 -> r + r1 >= r + r2.
+Proof. auto using Rplus_le_compat_l with rorders. Qed.
+Hint Resolve Rplus_ge_compat_l: real.
+
(**********)
Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r.
Proof.
@@ -718,23 +864,8 @@ 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_ge_compat_r : forall r r1 r2, r1 >= r2 -> r1 + r >= r2 + r.
+Proof. auto using Rplus_le_compat_r with rorders. Qed.
(*********)
Lemma Rplus_lt_compat :
@@ -742,12 +873,22 @@ Lemma Rplus_lt_compat :
Proof.
intros; apply Rlt_trans with (r2 + r3); auto with real.
Qed.
+Hint Immediate Rplus_lt_compat: real.
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.
+Hint Immediate Rplus_le_compat: real.
+
+Lemma Rplus_gt_compat :
+ forall r1 r2 r3 r4, r1 > r2 -> r3 > r4 -> r1 + r3 > r2 + r4.
+Proof. auto using Rplus_lt_compat with rorders. Qed.
+
+Lemma Rplus_ge_compat :
+ forall r1 r2 r3 r4, r1 >= r2 -> r3 >= r4 -> r1 + r3 >= r2 + r4.
+Proof. auto using Rplus_le_compat with rorders. Qed.
(*********)
Lemma Rplus_lt_le_compat :
@@ -756,19 +897,133 @@ 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.
+Hint Immediate Rplus_lt_le_compat Rplus_le_lt_compat: real.
+
+Lemma Rplus_gt_ge_compat :
+ forall r1 r2 r3 r4, r1 > r2 -> r3 >= r4 -> r1 + r3 > r2 + r4.
+Proof. auto using Rplus_lt_le_compat with rorders. Qed.
+
+Lemma Rplus_ge_gt_compat :
+ forall r1 r2 r3 r4, r1 >= r2 -> r3 > r4 -> r1 + r3 > r2 + r4.
+Proof. auto using Rplus_le_lt_compat with rorders. Qed.
+
+(**********)
+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.
-(** ** Order and Opposite *)
+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 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.
+
+(** *** Cancellation *)
+
+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_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 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_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 Rplus_le_reg_pos_r :
+ 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; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+ assumption
+ | assumption ].
+Qed.
+
+Lemma Rplus_lt_reg_pos_r : 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; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+ assumption
+ | assumption ].
+Qed.
+
+Lemma Rplus_ge_reg_neg_r :
+ forall r1 r2 r3, 0 >= r2 -> r1 + r2 >= r3 -> r1 >= r3.
+Proof.
+ intros x y z; intros; apply Rge_trans with (x + y);
+ [ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_ge_compat_l;
+ assumption
+ | assumption ].
+Qed.
+
+Lemma Rplus_gt_reg_neg_r : forall r1 r2 r3, 0 >= r2 -> r1 + r2 > r3 -> r1 > r3.
+Proof.
+ intros x y z; intros; apply Rge_gt_trans with (x + y);
+ [ pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_ge_compat_l;
+ assumption
+ | assumption ].
+Qed.
+
+(*********************************************************)
+(** ** Order and opposite *)
+(*********************************************************)
+
+(** *** Contravariant compatibility *)
+
Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2.
Proof.
unfold Rgt in |- *; intros.
@@ -781,55 +1036,44 @@ Proof.
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_gt_contravar : forall r1 r2, r2 > r1 -> - r1 > - r2.
+Proof. auto with real. Qed.
+
(**********)
Lemma Ropp_le_ge_contravar : forall r1 r2, r1 <= r2 -> - r1 >= - r2.
Proof.
- unfold Rge in |- *; intros r1 r2 [H| H]; auto with real.
+ unfold Rge; 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.
+Lemma Ropp_ge_le_contravar : forall r1 r2, r1 >= r2 -> - 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.
+ unfold Rle; intros r1 r2 [H| H]; auto with real.
Qed.
-Hint Immediate Ropp_le_cancel: real.
+Hint Resolve Ropp_ge_le_contravar: 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_ge_contravar : forall r1 r2, r2 >= r1 -> - r1 >= - r2.
+Proof. auto using Ropp_le_contravar with real. Qed.
(**********)
Lemma Ropp_0_lt_gt_contravar : forall r, 0 < r -> 0 > - r.
@@ -838,7 +1082,6 @@ Proof.
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.
@@ -850,13 +1093,13 @@ Lemma Ropp_lt_gt_0_contravar : forall r, r > 0 -> - r < 0.
Proof.
intros; rewrite <- Ropp_0; auto with real.
Qed.
+Hint Resolve Ropp_lt_gt_0_contravar: real.
-(**********)
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.
+Hint Resolve Ropp_gt_lt_0_contravar: real.
(**********)
Lemma Ropp_0_le_ge_contravar : forall r, 0 <= r -> 0 >= - r.
@@ -865,40 +1108,56 @@ Proof.
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 *)
+(** *** Cancellation *)
-Lemma Rmult_lt_compat_r : forall r r1 r2, 0 < r -> r1 < r2 -> r1 * r < r2 * r.
+Lemma Ropp_lt_cancel : forall r1 r2, - r2 < - r1 -> r1 < r2.
Proof.
- intros; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); auto with real.
+ intros x y H'.
+ rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y);
+ auto with real.
Qed.
-Hint Resolve Rmult_lt_compat_r.
+Hint Immediate Ropp_lt_cancel: real.
-Lemma Rmult_lt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2.
+Lemma Ropp_gt_cancel : forall r1 r2, - r2 > - r1 -> r1 > r2.
+Proof. auto using Ropp_lt_cancel with rorders. Qed.
+
+Lemma Ropp_le_cancel : forall r1 r2, - r2 <= - r1 -> 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.
+ 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_ge_cancel : forall r1 r2, - r2 >= - r1 -> r1 >= r2.
+Proof. auto using Ropp_le_cancel with rorders. Qed.
-Lemma Rmult_lt_gt_compat_neg_l :
- forall r r1 r2, r < 0 -> r1 < r2 -> r * r1 > r * r2.
+(*********************************************************)
+(** ** Order and multiplication *)
+(*********************************************************)
+
+(** Remark: [Rmult_lt_compat_l] is an axiom *)
+
+(** *** Covariant compatibility *)
+
+Lemma Rmult_lt_compat_r : forall r r1 r2, 0 < r -> r1 < r2 -> r1 * r < r2 * r.
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.
+ intros; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); auto with real.
Qed.
+Hint Resolve Rmult_lt_compat_r.
+
+Lemma Rmult_gt_compat_r : forall r r1 r2, r > 0 -> r1 > r2 -> r1 * r > r2 * r.
+Proof. eauto using Rmult_lt_compat_r with rorders. Qed.
+
+Lemma Rmult_gt_compat_l : forall r r1 r2, r > 0 -> r1 > r2 -> r * r1 > r * r2.
+Proof. eauto using Rmult_lt_compat_l with rorders. Qed.
(**********)
Lemma Rmult_le_compat_l :
@@ -918,18 +1177,59 @@ Proof.
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.
+Lemma Rmult_ge_compat_l :
+ forall r r1 r2, r >= 0 -> r1 >= r2 -> r * r1 >= r * r2.
+Proof. eauto using Rmult_le_compat_l with rorders. Qed.
+
+Lemma Rmult_ge_compat_r :
+ forall r r1 r2, r >= 0 -> r1 >= r2 -> r1 * r >= r2 * r.
+Proof. eauto using Rmult_le_compat_r with rorders. Qed.
+
+(**********)
+Lemma Rmult_le_compat :
+ forall r1 r2 r3 r4,
+ 0 <= r1 -> 0 <= r3 -> r1 <= r2 -> r3 <= r4 -> r1 * r3 <= r2 * r4.
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.
+ 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_ge_compat :
+ forall r1 r2 r3 r4,
+ 0 <= r1 -> 0 <= r3 -> r1 <= r2 -> r3 <= r4 -> r1 * r3 <= r2 * r4.
+Proof. auto with real rorders. Qed.
+
+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_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 Rmult_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 * r2.
+Proof. intros; replace 0 with (0 * r2); auto with real. Qed.
+
+Lemma Rmult_gt_0_compat : forall r1 r2, r1 > 0 -> r2 > 0 -> r1 * r2 > 0.
+Proof Rmult_lt_0_compat.
+
+(** *** Contravariant compatibility *)
+
Lemma Rmult_le_compat_neg_l :
forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r2 <= r * r1.
Proof.
@@ -946,35 +1246,45 @@ Proof.
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.
+Lemma Rmult_lt_gt_compat_neg_l :
+ forall r r1 r2, r < 0 -> r1 < r2 -> r * r1 > r * r2.
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.
+ 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.
-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.
+(** *** Cancellation *)
+
+Lemma Rmult_lt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2.
Proof.
- intros; apply Rlt_trans with (r2 * r3); auto with real.
+ 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_ge_0_gt_0_lt_compat :
- forall r1 r2 r3 r4,
- r3 >= 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4.
+Lemma Rmult_gt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2.
+Proof. eauto using Rmult_lt_reg_l with rorders. Qed.
+
+Lemma Rmult_le_reg_l : forall r r1 r2, 0 < r -> r * r1 <= r * r2 -> r1 <= r2.
Proof.
- intros; apply Rle_lt_trans with (r2 * r3); auto with real.
+ 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.
-
-(** ** Order and Substractions *)
+(*********************************************************)
+(** ** Order and substraction *)
+(*********************************************************)
Lemma Rlt_minus : forall r1 r2, r1 < r2 -> r1 - r2 < 0.
Proof.
@@ -985,12 +1295,27 @@ Proof.
Qed.
Hint Resolve Rlt_minus: real.
+Lemma Rgt_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.
+
(**********)
Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0.
Proof.
destruct 1; unfold Rle in |- *; auto with real.
Qed.
+Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0.
+Proof.
+ destruct 1.
+ auto using Rgt_minus, Rgt_ge.
+ right; auto using Rminus_diag_eq with rorders.
+Qed.
+
(**********)
Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2.
Proof.
@@ -999,6 +1324,14 @@ Proof.
ring.
Qed.
+Lemma Rminus_gt : forall r1 r2, r1 - r2 > 0 -> r1 > r2.
+Proof.
+ intros; replace r2 with (0 + r2); auto with real.
+ replace r1 with (r1 - r2 + r2).
+ apply Rplus_gt_compat_r; assumption.
+ ring.
+Qed.
+
(**********)
Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2.
Proof.
@@ -1007,6 +1340,14 @@ Proof.
ring.
Qed.
+Lemma Rminus_ge : forall r1 r2, r1 - r2 >= 0 -> r1 >= r2.
+Proof.
+ intros; replace r2 with (0 + r2); auto with real.
+ replace r1 with (r1 - r2 + r2).
+ apply Rplus_ge_compat_r; assumption.
+ ring.
+Qed.
+
(**********)
Lemma tech_Rplus : forall r (s:R), 0 <= r -> 0 < s -> r + s <> 0.
Proof.
@@ -1015,8 +1356,9 @@ Proof.
Qed.
Hint Immediate tech_Rplus: real.
-
-(** ** Order and the square function *)
+(*********************************************************)
+(** ** Order and square function *)
+(*********************************************************)
Lemma Rle_0_sqr : forall r, 0 <= Rsqr r.
Proof.
@@ -1036,7 +1378,26 @@ Proof.
Qed.
Hint Resolve Rle_0_sqr Rlt_0_sqr: real.
-(** ** Zero is less than one *)
+(***********)
+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.
+
+(*********************************************************)
+(** ** Zero is less than one *)
+(*********************************************************)
+
Lemma Rlt_0_1 : 0 < 1.
Proof.
replace 1 with (Rsqr 1); auto with real.
@@ -1050,7 +1411,10 @@ Proof.
exact Rlt_0_1.
Qed.
-(** ** Order and inverse *)
+(*********************************************************)
+(** ** Order and inverse *)
+(*********************************************************)
+
Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r.
Proof.
intros; apply Rnot_le_lt; red in |- *; intros.
@@ -1099,68 +1463,9 @@ Proof.
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.
+(*********************************************************)
+(** ** Miscellaneous *)
+(*********************************************************)
(**********)
Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1.
@@ -1186,121 +1491,9 @@ Proof.
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] *)
-(**********************************************************)
+(*********************************************************)
+(** ** Injection from [N] to [R] *)
+(*********************************************************)
(**********)
Lemma S_INR : forall n:nat, INR (S n) = INR n + 1.
@@ -1323,6 +1516,7 @@ Proof.
repeat rewrite S_INR.
rewrite Hrecn; ring.
Qed.
+Hint Resolve plus_INR: real.
(**********)
Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m.
@@ -1332,6 +1526,7 @@ Proof.
intros; repeat rewrite S_INR; simpl in |- *.
rewrite H0; ring.
Qed.
+Hint Resolve minus_INR: real.
(*********)
Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m.
@@ -1341,16 +1536,15 @@ Proof.
intros; repeat rewrite S_INR; simpl in |- *.
rewrite plus_INR; rewrite Hrecn; ring.
Qed.
-
-Hint Resolve plus_INR minus_INR mult_INR: real.
+Hint Resolve mult_INR: real.
(*********)
-Lemma lt_INR_0 : forall n:nat, (0 < n)%nat -> 0 < INR n.
+Lemma lt_0_INR : 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.
+Hint Resolve lt_0_INR: real.
Lemma lt_INR : forall n m:nat, (n < m)%nat -> INR n < INR m.
Proof.
@@ -1360,20 +1554,20 @@ Proof.
Qed.
Hint Resolve lt_INR: real.
-Lemma INR_lt_1 : forall n:nat, (1 < n)%nat -> 1 < INR n.
+Lemma lt_1_INR : 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.
+Hint Resolve lt_1_INR: real.
(**********)
-Lemma INR_pos : forall p:positive, 0 < INR (nat_of_P p).
+Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (nat_of_P p).
Proof.
- intro; apply lt_INR_0.
+ intro; apply lt_0_INR.
simpl in |- *; auto with real.
apply lt_O_nat_of_P.
Qed.
-Hint Resolve INR_pos: real.
+Hint Resolve pos_INR_nat_of_P: real.
(**********)
Lemma pos_INR : forall n:nat, 0 <= INR n.
@@ -1410,25 +1604,25 @@ Qed.
Hint Resolve le_INR: real.
(**********)
-Lemma not_INR_O : forall n:nat, INR n <> 0 -> n <> 0%nat.
+Lemma INR_not_0 : 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.
+Hint Immediate INR_not_0: real.
(**********)
-Lemma not_O_INR : forall n:nat, n <> 0%nat -> INR n <> 0.
+Lemma not_0_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.
+Hint Resolve not_0_INR: real.
-Lemma not_nm_INR : forall n m:nat, n <> m -> INR n <> INR m.
+Lemma not_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.
@@ -1436,17 +1630,17 @@ Proof.
elimtype False; auto.
apply sym_not_eq; apply Rlt_dichotomy_converse; auto with real.
Qed.
-Hint Resolve not_nm_INR: real.
+Hint Resolve not_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.
+ intro H3; generalize (not_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.
+ intro H3; generalize (not_INR m n H3); intro H4; elimtype False; auto.
omega.
Qed.
Hint Resolve INR_eq: real.
@@ -1465,9 +1659,9 @@ Proof.
Qed.
Hint Resolve not_1_INR: real.
-(**********************************************************)
-(** * Injection from [Z] to [R] *)
-(**********************************************************)
+(*********************************************************)
+(** ** Injection from [Z] to [R] *)
+(*********************************************************)
(**********)
@@ -1541,6 +1735,12 @@ Proof.
Qed.
(**********)
+Lemma succ_IZR : forall n:Z, IZR (Zsucc n) = IZR n + 1.
+Proof.
+ intro; change 1 with (IZR 1); unfold Zsucc; apply plus_IZR.
+Qed.
+
+(**********)
Lemma Ropp_Ropp_IZR : forall n:Z, IZR (- n) = - IZR n.
Proof.
intro z; case z; simpl in |- *; auto with real.
@@ -1554,7 +1754,7 @@ Proof.
Qed.
(**********)
-Lemma lt_O_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z.
+Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z.
Proof.
intro z; case z; simpl in |- *; intros.
absurd (0 < 0); auto with real.
@@ -1567,7 +1767,7 @@ 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.
+ apply lt_0_IZR.
rewrite <- Z_R_minus.
exact (Rgt_minus (IZR z2) (IZR z1) H).
Qed.
@@ -1578,7 +1778,7 @@ 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.
+ apply Ropp_lt_gt_0_contravar. unfold Rgt in |- *; apply pos_INR_nat_of_P.
Qed.
(**********)
@@ -1590,17 +1790,17 @@ Proof.
Qed.
(**********)
-Lemma not_O_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0.
+Lemma not_0_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.
+Lemma le_0_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.
+ red in |- *; intro; apply (Zlt_le_weak 0 z (lt_0_IZR z H)); assumption.
rewrite (eq_IZR_R0 z); auto with zarith real.
Qed.
@@ -1685,32 +1885,6 @@ Proof.
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.
@@ -1728,67 +1902,18 @@ 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 ].
+ [ apply not_0_INR; discriminate | unfold INR in |- *; ring ].
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.
+(*********************************************************)
+(** ** Other rules about < and <= *)
+(*********************************************************)
-Lemma Rmult_le_0_lt_compat :
+Lemma Rmult_ge_0_gt_0_lt_compat :
forall r1 r2 r3 r4,
- 0 <= r1 -> 0 <= r3 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4.
+ r3 >= 0 -> r2 > 0 -> 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 ] ].
+ intros; apply Rle_lt_trans with (r2 * r3); auto with real.
Qed.
Lemma le_epsilon :
@@ -1811,7 +1936,7 @@ Proof.
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 ].
+ replace 2 with (INR 2); [ apply not_0_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.
@@ -1822,12 +1947,12 @@ Proof.
unfold Rdiv in |- *.
rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
replace 2 with (INR 2).
- apply not_O_INR.
+ apply not_0_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 H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR in |- *;
intro; assumption
| discriminate ].
Qed.
@@ -1839,3 +1964,37 @@ Lemma completeness_weak :
Proof.
intros; elim (completeness E H H0); intros; split with x; assumption.
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}.
+
+(** Compatibility *)
+
+Notation prod_neq_R0 := Rmult_integral_contrapositive_currified (only parsing).
+Notation minus_Rgt := Rminus_gt (only parsing).
+Notation minus_Rge := Rminus_ge (only parsing).
+Notation plus_le_is_le := Rplus_le_reg_pos_r (only parsing).
+Notation plus_lt_is_lt := Rplus_lt_reg_pos_r (only parsing).
+Notation INR_lt_1 := lt_1_INR (only parsing).
+Notation lt_INR_0 := lt_0_INR (only parsing).
+Notation not_nm_INR := not_INR (only parsing).
+Notation INR_pos := pos_INR_nat_of_P (only parsing).
+Notation not_INR_O := INR_not_0 (only parsing).
+Notation not_O_INR := not_0_INR (only parsing).
+Notation not_O_IZR := not_0_IZR (only parsing).
+Notation le_O_IZR := le_0_IZR (only parsing).
+Notation lt_O_IZR := lt_0_IZR (only parsing).
diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v
index 270ea6da..17b6c60d 100644
--- a/theories/Reals/R_sqr.v
+++ b/theories/Reals/R_sqr.v
@@ -6,10 +6,11 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: R_sqr.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: R_sqr.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
-Require Import Rbasic_fun. Open Local Scope R_scope.
+Require Import Rbasic_fun.
+Open Local Scope R_scope.
(****************************************************)
(** Rsqr : some results *)
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v
index 736365a0..63b8940b 100644
--- a/theories/Reals/R_sqrt.v
+++ b/theories/Reals/R_sqrt.v
@@ -6,11 +6,12 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: R_sqrt.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: R_sqrt.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
-Require Import Rsqrt_def. Open Local Scope R_scope.
+Require Import Rsqrt_def.
+Open Local Scope R_scope.
(** * Continuous extension of Rsqrt on R *)
Definition sqrt (x:R) : R :=
diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v
index d712f74b..f48ce563 100644
--- a/theories/Reals/Ranalysis.v
+++ b/theories/Reals/Ranalysis.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis.v 9319 2006-10-30 12:41:21Z barras $ i*)
+(*i $Id: Ranalysis.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -27,7 +27,8 @@ Require Export Rgeom.
Require Export RList.
Require Export Sqrt_reg.
Require Export Ranalysis4.
-Require Export Rpower. Open Local Scope R_scope.
+Require Export Rpower.
+Open Local Scope R_scope.
Axiom AppVar : R.
diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v
index 93a66e70..9414f7c9 100644
--- a/theories/Reals/Ranalysis1.v
+++ b/theories/Reals/Ranalysis1.v
@@ -6,12 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis1.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Ranalysis1.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Export Rlimit.
-Require Export Rderiv. Open Local Scope R_scope.
+Require Export Rderiv.
+Open Local Scope R_scope.
Implicit Type f : R -> R.
(****************************************************)
@@ -269,10 +270,10 @@ Definition derivable_pt_lim f (x l:R) : Prop :=
Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l.
-Definition derivable_pt f (x:R) := sigT (derivable_pt_abs f x).
+Definition derivable_pt f (x:R) := { l:R | derivable_pt_abs f x l }.
Definition derivable f := forall x:R, derivable_pt f x.
-Definition derive_pt f (x:R) (pr:derivable_pt f x) := projT1 pr.
+Definition derive_pt f (x:R) (pr:derivable_pt f x) := proj1_sig pr.
Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x).
Arguments Scope derivable_pt_lim [Rfun_scope R_scope].
@@ -380,9 +381,9 @@ Lemma derive_pt_eq :
derive_pt f x pr = l <-> derivable_pt_lim f x l.
Proof.
intros; split.
- intro; assert (H1 := projT2 pr); unfold derive_pt in H; rewrite H in H1;
+ intro; assert (H1 := proj2_sig pr); unfold derive_pt in H; rewrite H in H1;
assumption.
- intro; assert (H1 := projT2 pr); unfold derivable_pt_abs in H1.
+ intro; assert (H1 := proj2_sig pr); unfold derivable_pt_abs in H1.
assert (H2 := uniqueness_limite _ _ _ _ H H1).
unfold derive_pt in |- *; unfold derivable_pt_abs in |- *.
symmetry in |- *; assumption.
@@ -486,7 +487,7 @@ Qed.
Lemma derivable_derive :
forall f (x:R) (pr:derivable_pt f x), exists l : R, derive_pt f x pr = l.
Proof.
- intros; exists (projT1 pr).
+ intros; exists (proj1_sig pr).
unfold derive_pt in |- *; reflexivity.
Qed.
@@ -714,7 +715,7 @@ Proof.
unfold derivable_pt in |- *; intros f1 f2 x X X0.
elim X; intros.
elim X0; intros.
- apply existT with (x0 + x1).
+ exists (x0 + x1).
apply derivable_pt_lim_plus; assumption.
Qed.
@@ -723,7 +724,7 @@ Lemma derivable_pt_opp :
Proof.
unfold derivable_pt in |- *; intros f x X.
elim X; intros.
- apply existT with (- x0).
+ exists (- x0).
apply derivable_pt_lim_opp; assumption.
Qed.
@@ -734,7 +735,7 @@ Proof.
unfold derivable_pt in |- *; intros f1 f2 x X X0.
elim X; intros.
elim X0; intros.
- apply existT with (x0 - x1).
+ exists (x0 - x1).
apply derivable_pt_lim_minus; assumption.
Qed.
@@ -745,14 +746,14 @@ Proof.
unfold derivable_pt in |- *; intros f1 f2 x X X0.
elim X; intros.
elim X0; intros.
- apply existT with (x0 * f2 x + f1 x * x1).
+ exists (x0 * f2 x + f1 x * x1).
apply derivable_pt_lim_mult; assumption.
Qed.
Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x.
Proof.
intros; unfold derivable_pt in |- *.
- apply existT with 0.
+ exists 0.
apply derivable_pt_lim_const.
Qed.
@@ -761,7 +762,7 @@ Lemma derivable_pt_scal :
Proof.
unfold derivable_pt in |- *; intros f1 a x X.
elim X; intros.
- apply existT with (a * x0).
+ exists (a * x0).
apply derivable_pt_lim_scal; assumption.
Qed.
@@ -774,7 +775,7 @@ Qed.
Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x.
Proof.
- unfold derivable_pt in |- *; intro; apply existT with (2 * x).
+ unfold derivable_pt in |- *; intro; exists (2 * x).
apply derivable_pt_lim_Rsqr.
Qed.
@@ -785,7 +786,7 @@ Proof.
unfold derivable_pt in |- *; intros f1 f2 x X X0.
elim X; intros.
elim X0; intros.
- apply existT with (x1 * x0).
+ exists (x1 * x0).
apply derivable_pt_lim_comp; assumption.
Qed.
@@ -860,9 +861,9 @@ Proof.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
- assert (H4 := projT2 pr2).
+ assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_plus; assumption.
Qed.
@@ -877,7 +878,7 @@ Proof.
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
rewrite H; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
apply derivable_pt_lim_opp; assumption.
Qed.
@@ -896,9 +897,9 @@ Proof.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
- assert (H4 := projT2 pr2).
+ assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_minus; assumption.
Qed.
@@ -917,9 +918,9 @@ Proof.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
- assert (H4 := projT2 pr2).
+ assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_mult; assumption.
Qed.
@@ -944,7 +945,7 @@ Proof.
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
rewrite H; apply derive_pt_eq_0.
- assert (H3 := projT2 pr).
+ assert (H3 := proj2_sig pr).
unfold derive_pt in H; rewrite H in H3.
apply derivable_pt_lim_scal; assumption.
Qed.
@@ -978,9 +979,9 @@ Proof.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
- assert (H4 := projT2 pr2).
+ assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_comp; assumption.
Qed.
@@ -1046,7 +1047,7 @@ Lemma derivable_pt_pow :
forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.
Proof.
intros; unfold derivable_pt in |- *.
- apply existT with (INR n * x ^ pred n).
+ exists (INR n * x ^ pred n).
apply derivable_pt_lim_pow.
Qed.
diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v
index fb89da67..54801eb7 100644
--- a/theories/Reals/Ranalysis2.v
+++ b/theories/Reals/Ranalysis2.v
@@ -6,11 +6,12 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis2.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Ranalysis2.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
-Require Import Ranalysis1. Open Local Scope R_scope.
+Require Import Ranalysis1.
+Open Local Scope R_scope.
(**********)
Lemma formule :
diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v
index f50aa2ad..180cf9d6 100644
--- a/theories/Reals/Ranalysis3.v
+++ b/theories/Reals/Ranalysis3.v
@@ -6,12 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis3.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Ranalysis3.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
-Require Import Ranalysis2. Open Local Scope R_scope.
+Require Import Ranalysis2.
+Open Local Scope R_scope.
(** Division *)
Theorem derivable_pt_lim_div :
@@ -23,7 +24,7 @@ Theorem derivable_pt_lim_div :
Proof.
intros f1 f2 x l1 l2 H H0 H1.
cut (derivable_pt f2 x);
- [ intro X | unfold derivable_pt in |- *; apply existT with l2; exact H0 ].
+ [ intro X | unfold derivable_pt in |- *; exists l2; exact H0 ].
assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1).
elim H2; clear H2; intros eps_f2 H2.
unfold div_fct in |- *.
@@ -761,7 +762,7 @@ Proof.
intros f1 f2 x X X0 H.
elim X; intros.
elim X0; intros.
- apply existT with ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)).
+ exists ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)).
apply derivable_pt_lim_div; assumption.
Qed.
@@ -789,9 +790,9 @@ Proof.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
- assert (H3 := projT2 pr1).
+ assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
- assert (H4 := projT2 pr2).
+ assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_div; assumption.
Qed.
diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v
index 205c06b4..95f6d27e 100644
--- a/theories/Reals/Ranalysis4.v
+++ b/theories/Reals/Ranalysis4.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis4.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Ranalysis4.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -14,7 +14,8 @@ Require Import SeqSeries.
Require Import Rtrigo.
Require Import Ranalysis1.
Require Import Ranalysis3.
-Require Import Exp_prop. Open Local Scope R_scope.
+Require Import Exp_prop.
+Open Local Scope R_scope.
(**********)
Lemma derivable_pt_inv :
@@ -28,7 +29,7 @@ Proof.
assumption.
assumption.
unfold div_fct, inv_fct, fct_cte in |- *; intro X0; elim X0; intros;
- unfold derivable_pt in |- *; apply existT with x0;
+ unfold derivable_pt in |- *; exists x0;
unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *;
unfold derivable_pt_abs in p; unfold derivable_pt_lim in p;
intros; elim (p eps H0); intros; exists x1; intros;
@@ -164,10 +165,10 @@ Proof.
intros.
case (total_order_T x 0); intro.
elim s; intro.
- unfold derivable_pt in |- *; apply existT with (-1).
+ unfold derivable_pt in |- *; exists (-1).
apply (Rabs_derive_2 x a).
elim H; exact b.
- unfold derivable_pt in |- *; apply existT with 1.
+ unfold derivable_pt in |- *; exists 1.
apply (Rabs_derive_1 x r).
Qed.
@@ -294,8 +295,8 @@ Proof.
unfold derivable_pt in |- *.
assert (H := derivable_pt_lim_finite_sum An x N).
induction N as [| N HrecN].
- apply existT with 0; apply H.
- apply existT with
+ exists 0; apply H.
+ exists
(sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N)));
apply H.
Qed.
@@ -352,7 +353,7 @@ Lemma derivable_pt_exp : forall x:R, derivable_pt exp x.
Proof.
intro.
unfold derivable_pt in |- *.
- apply existT with (exp x).
+ exists (exp x).
apply derivable_pt_lim_exp.
Qed.
@@ -360,7 +361,7 @@ Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x.
Proof.
intro.
unfold derivable_pt in |- *.
- apply existT with (sinh x).
+ exists (sinh x).
apply derivable_pt_lim_cosh.
Qed.
@@ -368,7 +369,7 @@ Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x.
Proof.
intro.
unfold derivable_pt in |- *.
- apply existT with (cosh x).
+ exists (cosh x).
apply derivable_pt_lim_sinh.
Qed.
diff --git a/theories/Reals/Raxioms.v b/theories/Reals/Raxioms.v
index aaea59f4..6667d2ec 100644
--- a/theories/Reals/Raxioms.v
+++ b/theories/Reals/Raxioms.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Raxioms.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Raxioms.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
(*********************************************************)
(** Axiomatisation of the classical reals *)
@@ -130,7 +130,7 @@ Definition IZR (z:Z) : R :=
Arguments Scope IZR [Z_scope].
(**********************************************************)
-(** * [R] Archimedian *)
+(** * [R] Archimedean *)
(**********************************************************)
(**********)
@@ -154,4 +154,4 @@ Definition is_lub (E:R -> Prop) (m:R) :=
Axiom
completeness :
forall E:R -> Prop,
- bound E -> (exists x : R, E x) -> sigT (fun m:R => is_lub E m).
+ bound E -> (exists x : R, E x) -> { m:R | is_lub E m }.
diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v
index 98bd607b..a5cc9f19 100644
--- a/theories/Reals/Rbasic_fun.v
+++ b/theories/Reals/Rbasic_fun.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rbasic_fun.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rbasic_fun.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
(*********************************************************)
(** Complements for the real numbers *)
@@ -15,7 +15,8 @@
Require Import Rbase.
Require Import R_Ifp.
-Require Import Fourier. Open Local Scope R_scope.
+Require Import Fourier.
+Open Local Scope R_scope.
Implicit Type r : R.
diff --git a/theories/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v
index 16e12d7f..d7fee9c5 100644
--- a/theories/Reals/Rcomplete.v
+++ b/theories/Reals/Rcomplete.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rcomplete.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rcomplete.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -24,7 +24,7 @@ Open Local Scope R_scope.
(****************************************************)
Theorem R_complete :
- forall Un:nat -> R, Cauchy_crit Un -> sigT (fun l:R => Un_cv Un l).
+ forall Un:nat -> R, Cauchy_crit Un -> { l:R | Un_cv Un l } .
Proof.
intros.
set (Vn := sequence_minorant Un (cauchy_min Un H)).
@@ -37,7 +37,7 @@ Proof.
elim H1; intros.
cut (x = x0).
intros.
- apply existT with x.
+ exists x.
rewrite <- H2 in p0.
unfold Un_cv in |- *.
intros.
diff --git a/theories/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v
index 330c0042..002ce8d6 100644
--- a/theories/Reals/Rdefinitions.v
+++ b/theories/Reals/Rdefinitions.v
@@ -5,7 +5,7 @@
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rdefinitions.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rdefinitions.v 10751 2008-04-04 10:23:35Z herbelin $ i*)
(*********************************************************)
@@ -22,6 +22,8 @@ Delimit Scope R_scope with R.
(* Automatically open scope R_scope for arguments of type R *)
Bind Scope R_scope with R.
+Open Local Scope R_scope.
+
Parameter R0 : R.
Parameter R1 : R.
Parameter Rplus : R -> R -> R.
@@ -38,33 +40,33 @@ Notation "/ x" := (Rinv x) : R_scope.
Infix "<" := Rlt : R_scope.
-(*i*******************************************************i*)
+(***********************************************************)
(**********)
-Definition Rgt (r1 r2:R) : Prop := (r2 < r1)%R.
+Definition Rgt (r1 r2:R) : Prop := r2 < r1.
(**********)
-Definition Rle (r1 r2:R) : Prop := (r1 < r2)%R \/ r1 = r2.
+Definition Rle (r1 r2:R) : Prop := r1 < r2 \/ r1 = r2.
(**********)
Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.
(**********)
-Definition Rminus (r1 r2:R) : R := (r1 + - r2)%R.
+Definition Rminus (r1 r2:R) : R := r1 + - r2.
(**********)
-Definition Rdiv (r1 r2:R) : R := (r1 * / r2)%R.
+Definition Rdiv (r1 r2:R) : R := r1 * / r2.
(**********)
Infix "-" := Rminus : R_scope.
-Infix "/" := Rdiv : R_scope.
+Infix "/" := Rdiv : R_scope.
Infix "<=" := Rle : R_scope.
Infix ">=" := Rge : R_scope.
-Infix ">" := Rgt : R_scope.
+Infix ">" := Rgt : R_scope.
-Notation "x <= y <= z" := ((x <= y)%R /\ (y <= z)%R) : R_scope.
-Notation "x <= y < z" := ((x <= y)%R /\ (y < z)%R) : R_scope.
-Notation "x < y < z" := ((x < y)%R /\ (y < z)%R) : R_scope.
-Notation "x < y <= z" := ((x < y)%R /\ (y <= z)%R) : R_scope.
+Notation "x <= y <= z" := (x <= y /\ y <= z) : R_scope.
+Notation "x <= y < z" := (x <= y /\ y < z) : R_scope.
+Notation "x < y < z" := (x < y /\ y < z) : R_scope.
+Notation "x < y <= z" := (x < y /\ y <= z) : R_scope.
diff --git a/theories/Reals/Rderiv.v b/theories/Reals/Rderiv.v
index e2fd2efe..ba42bad9 100644
--- a/theories/Reals/Rderiv.v
+++ b/theories/Reals/Rderiv.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rderiv.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rderiv.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
(*********************************************************)
(** Definition of the derivative,continuity *)
@@ -19,7 +19,8 @@ Require Import Rlimit.
Require Import Fourier.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
-Require Import Omega. Open Local Scope R_scope.
+Require Import Omega.
+Open Local Scope R_scope.
(*********)
Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.
diff --git a/theories/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v
index 3d1c0375..b9aec1ea 100644
--- a/theories/Reals/Rfunctions.v
+++ b/theories/Reals/Rfunctions.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rfunctions.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rfunctions.v 10762 2008-04-06 16:57:31Z herbelin $ i*)
(*i Some properties about pow and sum have been made with John Harrison i*)
(*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*)
@@ -349,8 +349,7 @@ Proof.
rewrite Rabs_Rinv; auto.
rewrite <- Rinv_pow; auto.
rewrite RPow_abs; auto.
- rewrite H'0; rewrite Rabs_right; auto with real.
- apply Rle_ge; auto with real.
+ rewrite H'0; rewrite Rabs_right; auto with real rorders.
apply Rlt_pow; auto with arith.
rewrite Rabs_Rinv; auto.
apply Rmult_lt_reg_l with (r := Rabs r).
@@ -786,11 +785,14 @@ Proof.
Qed.
(*******************************)
-(** * Infinit Sum *)
+(** * Infinite Sum *)
(*******************************)
(*********)
-Definition infinit_sum (s:nat -> R) (l:R) : Prop :=
+Definition infinite_sum (s:nat -> R) (l:R) : Prop :=
forall eps:R,
eps > 0 ->
exists N : nat,
(forall n:nat, (n >= N)%nat -> R_dist (sum_f_R0 s n) l < eps).
+
+(** Compatibility with previous versions *)
+Notation infinit_sum := infinite_sum (only parsing).
diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v
index 8ac9c07f..c96ae5d6 100644
--- a/theories/Reals/Rgeom.v
+++ b/theories/Reals/Rgeom.v
@@ -6,13 +6,14 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rgeom.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rgeom.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo.
-Require Import R_sqrt. Open Local Scope R_scope.
+Require Import R_sqrt.
+Open Local Scope R_scope.
(** * Distance *)
diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v
index 1cba821e..8d069e2d 100644
--- a/theories/Reals/RiemannInt.v
+++ b/theories/Reals/RiemannInt.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: RiemannInt.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: RiemannInt.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rfunctions.
Require Import SeqSeries.
@@ -15,7 +15,8 @@ Require Import Rbase.
Require Import RiemannInt_SF.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
-Require Import Max. Open Local Scope R_scope.
+Require Import Max.
+Open Local Scope R_scope.
Set Implicit Arguments.
@@ -25,13 +26,11 @@ Set Implicit Arguments.
Definition Riemann_integrable (f:R -> R) (a b:R) : Type :=
forall eps:posreal,
- sigT
- (fun phi:StepFun a b =>
- sigT
- (fun psi:StepFun a b =>
+ { phi:StepFun a b &
+ { psi:StepFun a b |
(forall t:R,
Rmin a b <= t <= Rmax a b -> Rabs (f t - phi t) <= psi t) /\
- Rabs (RiemannInt_SF psi) < eps)).
+ Rabs (RiemannInt_SF psi) < eps } }.
Definition phi_sequence (un:nat -> posreal) (f:R -> R)
(a b:R) (pr:Riemann_integrable f a b) (n:nat) :=
@@ -40,12 +39,11 @@ Definition phi_sequence (un:nat -> posreal) (f:R -> R)
Lemma phi_sequence_prop :
forall (un:nat -> posreal) (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
(N:nat),
- sigT
- (fun psi:StepFun a b =>
+ { psi:StepFun a b |
(forall t:R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr N t) <= psi t) /\
- Rabs (RiemannInt_SF psi) < un N).
+ Rabs (RiemannInt_SF psi) < un N }.
Proof.
intros; apply (projT2 (pr (un N))).
Qed.
@@ -55,8 +53,8 @@ Lemma RiemannInt_P1 :
Riemann_integrable f a b -> Riemann_integrable f b a.
Proof.
unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; intros;
- elim p; clear p; intros; apply existT with (mkStepFun (StepFun_P6 (pre x)));
- apply existT with (mkStepFun (StepFun_P6 (pre x0)));
+ elim p; clear p; intros; exists (mkStepFun (StepFun_P6 (pre x)));
+ exists (mkStepFun (StepFun_P6 (pre x0)));
elim p; clear p; intros; split.
intros; apply (H t); elim H1; clear H1; intros; split;
[ apply Rle_trans with (Rmin b a); try assumption; right;
@@ -90,7 +88,7 @@ Lemma RiemannInt_P2 :
(forall n:nat,
(forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n) ->
- sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l).
+ { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.
Proof.
intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit in |- *;
intros; assert (H3 : 0 < eps / 2).
@@ -143,7 +141,7 @@ Lemma RiemannInt_P3 :
(forall n:nat,
(forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n) ->
- sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l).
+ { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.
Proof.
intros; case (Rle_dec a b); intro.
apply RiemannInt_P2 with f un wn; assumption.
@@ -181,7 +179,7 @@ Proof.
rewrite Rabs_Ropp in H4; apply H4.
apply H4.
assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros;
- apply existT with (- x); unfold Un_cv in |- *; unfold Un_cv in p;
+ exists (- x); unfold Un_cv in |- *; unfold Un_cv in p;
intros; elim (p _ H4); intros; exists x0; intros;
generalize (H5 _ H6); unfold R_dist, RiemannInt_SF in |- *;
case (Rle_dec b a); case (Rle_dec a b); intros.
@@ -205,13 +203,12 @@ Lemma RiemannInt_exists :
forall (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
(un:nat -> posreal),
Un_cv un 0 ->
- sigT
- (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr N)) l).
+ { l:R | Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr N)) l }.
Proof.
intros f; intros;
apply RiemannInt_P3 with
- f un (fun n:nat => projT1 (phi_sequence_prop un pr n));
- [ apply H | intro; apply (projT2 (phi_sequence_prop un pr n)) ].
+ f un (fun n:nat => proj1_sig (phi_sequence_prop un pr n));
+ [ apply H | intro; apply (proj2_sig (phi_sequence_prop un pr n)) ].
Qed.
Lemma RiemannInt_P4 :
@@ -411,9 +408,7 @@ Qed.
(**********)
Definition RiemannInt (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) : R :=
- match RiemannInt_exists pr RinvN RinvN_cv with
- | existT a' b' => a'
- end.
+ let (a,_) := RiemannInt_exists pr RinvN RinvN_cv in a.
Lemma RiemannInt_P5 :
forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b),
@@ -433,8 +428,7 @@ Qed.
Lemma maxN :
forall (a b:R) (del:posreal),
- a < b ->
- sigT (fun n:nat => a + INR n * del < b /\ b <= a + INR (S n) * del).
+ a < b -> { n:nat | a + INR n * del < b /\ b <= a + INR (S n) * del }.
Proof.
intros; set (I := fun n:nat => a + INR n * del < b);
assert (H0 : exists n : nat, I n).
@@ -478,9 +472,7 @@ Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) {struct N} : Rlist :=
end.
Definition max_N (a b:R) (del:posreal) (h:a < b) : nat :=
- match maxN del h with
- | existT N H0 => N
- end.
+ let (N,_) := maxN del h in N.
Definition SubEqui (a b:R) (del:posreal) (h:a < b) : Rlist :=
SubEquiN (S (max_N del h)) a b del.
@@ -490,12 +482,11 @@ Lemma Heine_cor1 :
a < b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall eps:posreal,
- sigT
- (fun delta:posreal =>
+ { delta:posreal |
delta <= b - a /\
(forall x y:R,
a <= x <= b ->
- a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps)).
+ a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps) }.
Proof.
intro f; intros;
set
@@ -520,7 +511,7 @@ Proof.
| intros; apply H3; try assumption; apply Rlt_le_trans with (Rmin x (b - a));
[ assumption | apply Rmin_l ] ].
assert (H3 := completeness E H1 H2); elim H3; intros; cut (0 < x <= b - a).
- intro; elim H4; clear H4; intros; apply existT with (mkposreal _ H4); split.
+ intro; elim H4; clear H4; intros; exists (mkposreal _ H4); split.
apply H5.
unfold is_lub in p; elim p; intros; unfold is_upper_bound in H6;
set (D := Rabs (x0 - y)); elim (classic (exists y : R, D < y /\ E y));
@@ -549,22 +540,21 @@ Lemma Heine_cor2 :
forall (f:R -> R) (a b:R),
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall eps:posreal,
- sigT
- (fun delta:posreal =>
+ { delta:posreal |
forall x y:R,
a <= x <= b ->
- a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps).
+ a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps }.
Proof.
intro f; intros; case (total_order_T a b); intro.
elim s; intro.
- assert (H0 := Heine_cor1 a0 H eps); elim H0; intros; apply existT with x;
+ assert (H0 := Heine_cor1 a0 H eps); elim H0; intros; exists x;
elim p; intros; apply H2; assumption.
- apply existT with (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y);
+ exists (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y);
[ elim H0; elim H1; intros; rewrite b0 in H3; rewrite b0 in H5;
apply Rle_antisym; apply Rle_trans with b; assumption
| rewrite H3; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
apply (cond_pos eps) ].
- apply existT with (mkposreal _ Rlt_0_1); intros; elim H0; intros;
+ exists (mkposreal _ Rlt_0_1); intros; elim H0; intros;
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H3 H4) r)).
Qed.
@@ -664,15 +654,14 @@ Qed.
Lemma SubEqui_P9 :
forall (a b:R) (del:posreal) (f:R -> R) (h:a < b),
- sigT
- (fun g:StepFun a b =>
+ { g:StepFun a b |
g b = f b /\
(forall i:nat,
(i < pred (Rlength (SubEqui del h)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (SubEqui del h) i)
(pos_Rl (SubEqui del h) (S i)))
- (f (pos_Rl (SubEqui del h) i)))).
+ (f (pos_Rl (SubEqui del h) i))) }.
Proof.
intros; apply StepFun_P38;
[ apply SubEqui_P7 | apply SubEqui_P1 | apply SubEqui_P2 ].
@@ -1003,11 +992,11 @@ Proof.
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr2 n)); intro;
rewrite Rmin_comm; rewrite RmaxSym;
- apply (projT2 (phi_sequence_prop RinvN pr2 n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr1 n)).
+ apply (proj2_sig (phi_sequence_prop RinvN pr2 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr1 n)).
Qed.
Lemma RiemannInt_P9 :
@@ -1272,11 +1261,11 @@ Proof.
case (RiemannInt_exists pr1 RinvN RinvN_cv); intros;
eapply UL_sequence;
[ apply u0
- | set (psi1 := fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n));
- set (psi2 := fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n));
+ | set (psi1 := fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n));
+ set (psi2 := fun n:nat => proj1_sig (phi_sequence_prop RinvN pr3 n));
apply RiemannInt_P11 with f RinvN (phi_sequence RinvN pr1) psi1 psi2;
[ apply RinvN_cv
- | intro; apply (projT2 (phi_sequence_prop RinvN pr1 n))
+ | intro; apply (proj2_sig (phi_sequence_prop RinvN pr1 n))
| intro;
assert
(H1 :
@@ -1284,7 +1273,7 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n);
- [ apply (projT2 (phi_sequence_prop RinvN pr3 n))
+ [ apply (proj2_sig (phi_sequence_prop RinvN pr3 n))
| elim H1; intros; split; try assumption; intros;
replace (f t) with (f t + l * g t);
[ apply H2; assumption | rewrite H0; ring ] ]
@@ -1360,8 +1349,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr1 n0)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr1 n0)).
assert
(H8 :
exists psi2 : nat -> StepFun a b,
@@ -1370,8 +1359,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (g t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr2 n0)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr2 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr2 n0)).
assert
(H9 :
exists psi3 : nat -> StepFun a b,
@@ -1380,8 +1369,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr3 n0)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr3 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr3 n0)).
elim H7; clear H7; intros psi1 H7; elim H8; clear H8; intros psi2 H8; elim H9;
clear H9; intros psi3 H9;
replace
@@ -1552,8 +1541,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr n)).
elim H1; clear H1; intros psi1 H1;
set (phi2 := fun n:nat => mkStepFun (StepFun_P4 a b c));
set (psi2 := fun n:nat => mkStepFun (StepFun_P4 a b 0));
@@ -1647,8 +1636,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (Rabs (f t) - phi3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr2 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr2 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr2 n)).
assert
(H1 :
exists psi2 : nat -> StepFun a b,
@@ -1664,8 +1653,8 @@ Proof.
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b -> Rabs (f t - phi1 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr1 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr1 n)).
elim H1; clear H1; intros psi2 H1; split with psi2; intros; elim (H1 n);
clear H1; intros; split; try assumption.
intros; unfold phi2 in |- *; simpl in |- *;
@@ -1698,8 +1687,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr1 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr1 n)).
elim H1; clear H1; intros psi1 H1;
set (phi2 := fun N:nat => phi_sequence RinvN pr2 N).
set
@@ -1722,8 +1711,8 @@ Proof.
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b -> Rabs (g t - phi2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr2 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr2 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr2 n)).
elim H2; clear H2; intros psi2 H2;
apply RiemannInt_P11 with f RinvN phi2_m psi2 psi1;
try assumption.
@@ -2378,8 +2367,8 @@ Proof.
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr1 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr1 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr1 n)).
assert
(H2 :
exists psi2 : nat -> StepFun b c,
@@ -2388,8 +2377,8 @@ Proof.
Rmin b c <= t /\ t <= Rmax b c ->
Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr2 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr2 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr2 n)).
assert
(H3 :
exists psi3 : nat -> StepFun a c,
@@ -2398,8 +2387,8 @@ Proof.
Rmin a c <= t /\ t <= Rmax a c ->
Rabs (f t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
- split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro;
- apply (projT2 (phi_sequence_prop RinvN pr3 n)).
+ split with (fun n:nat => proj1_sig (phi_sequence_prop RinvN pr3 n)); intro;
+ apply (proj2_sig (phi_sequence_prop RinvN pr3 n)).
elim H1; clear H1; intros psi1 H1; elim H2; clear H2; intros psi2 H2; elim H3;
clear H3; intros psi3 H3; assert (H := RinvN_cv);
unfold Un_cv in |- *; intros; assert (H4 : 0 < eps / 3).
@@ -3259,7 +3248,7 @@ Lemma RiemannInt_P30 :
forall (f:R -> R) (a b:R),
a <= b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
- sigT (fun g:R -> R => antiderivative f g a b).
+ { g:R -> R | antiderivative f g a b }.
Proof.
intros; split with (primitive H (FTC_P1 H H0)); apply RiemannInt_P29.
Qed.
diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v
index 0f91d006..7a02544e 100644
--- a/theories/Reals/RiemannInt_SF.v
+++ b/theories/Reals/RiemannInt_SF.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: RiemannInt_SF.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: RiemannInt_SF.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -31,7 +31,7 @@ Qed.
Lemma Nzorn :
forall I:nat -> Prop,
(exists n : nat, I n) ->
- Nbound I -> sigT (fun n:nat => I n /\ (forall i:nat, I i -> (i <= n)%nat)).
+ Nbound I -> { n:nat | I n /\ (forall i:nat, I i -> (i <= n)%nat) }.
Proof.
intros I H H0; set (E := fun x:R => exists i : nat, I i /\ INR i = x);
assert (H1 : bound E).
@@ -133,10 +133,10 @@ Definition adapted_couple_opt (f:R -> R) (a b:R) (l lf:Rlist) :=
(forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <> pos_Rl l (S i)).
Definition is_subdivision (f:R -> R) (a b:R) (l:Rlist) : Type :=
- sigT (fun l0:Rlist => adapted_couple f a b l l0).
+ { l0:Rlist & adapted_couple f a b l l0 }.
Definition IsStepFun (f:R -> R) (a b:R) : Type :=
- sigT (fun l:Rlist => is_subdivision f a b l).
+ { l:Rlist & is_subdivision f a b l }.
(** ** Class of step functions *)
Record StepFun (a b:R) : Type := mkStepFun
@@ -1779,13 +1779,12 @@ Lemma StepFun_P38 :
ordered_Rlist l ->
pos_Rl l 0 = a ->
pos_Rl l (pred (Rlength l)) = b ->
- sigT
- (fun g:StepFun a b =>
+ { g:StepFun a b |
g b = f b /\
(forall i:nat,
(i < pred (Rlength l))%nat ->
constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i)))
- (f (pos_Rl l i)))).
+ (f (pos_Rl l i))) }.
Proof.
intros l a b f; generalize a; clear a; induction l.
intros a H H0 H1; simpl in H0; simpl in H1;
@@ -2206,21 +2205,10 @@ Lemma StepFun_P43 :
RiemannInt_SF (mkStepFun pr1) + RiemannInt_SF (mkStepFun pr2) =
RiemannInt_SF (mkStepFun pr3).
Proof.
- intros f; intros;
- assert
- (H1 :
- sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a b l l0))).
- apply pr1.
- assert
- (H2 :
- sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f b c l l0))).
- apply pr2.
- assert
- (H3 :
- sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a c l l0))).
- apply pr3.
- elim H1; clear H1; intros l1 [lf1 H1]; elim H2; clear H2; intros l2 [lf2 H2];
- elim H3; clear H3; intros l3 [lf3 H3].
+ intros f; intros.
+ pose proof pr1 as (l1,(lf1,H1)).
+ pose proof pr2 as (l2,(lf2,H2)).
+ pose proof pr3 as (l3,(lf3,H3)).
replace (RiemannInt_SF (mkStepFun pr1)) with
match Rle_dec a b with
| left _ => Int_SF lf1 l1
@@ -2462,7 +2450,7 @@ Proof.
(forall (l1 lf1:Rlist) (a b c:R) (f:R -> R),
adapted_couple f a b l1 lf1 ->
a <= c <= b ->
- sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a c l l0))).
+ { l:Rlist & { l0:Rlist & adapted_couple f a c l l0 } }).
intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X.
apply H2.
split; assumption.
@@ -2578,7 +2566,7 @@ Proof.
(forall (l1 lf1:Rlist) (a b c:R) (f:R -> R),
adapted_couple f a b l1 lf1 ->
a <= c <= b ->
- sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f c b l l0))).
+ { l:Rlist & { l0:Rlist & adapted_couple f c b l l0 } }).
intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X;
[ apply H2 | split; assumption ].
clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.
diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v
index 76579ccb..1a2fa03a 100644
--- a/theories/Reals/Rlimit.v
+++ b/theories/Reals/Rlimit.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rlimit.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rlimit.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
(*********************************************************)
(** Definition of the limit *)
@@ -16,7 +16,8 @@
Require Import Rbase.
Require Import Rfunctions.
Require Import Classical_Prop.
-Require Import Fourier. Open Local Scope R_scope.
+Require Import Fourier.
+Open Local Scope R_scope.
(*******************************)
(** * Calculus *)
@@ -560,9 +561,9 @@ Proof.
| apply Rlt_le_trans with (Rmin delta1 delta2);
[ assumption | apply Rmin_l ] ].
change (0 < eps * (Rsqr l / 2)) in |- *; unfold Rdiv in |- *;
- repeat rewrite Rmult_assoc; repeat apply Rmult_lt_0_compat.
+ repeat rewrite Rmult_assoc; apply Rmult_lt_0_compat.
assumption.
- apply Rsqr_pos_lt; assumption.
+ apply Rmult_lt_0_compat. apply Rsqr_pos_lt; assumption.
apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
[ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *;
intro; assumption
diff --git a/theories/Reals/Rlogic.v b/theories/Reals/Rlogic.v
new file mode 100644
index 00000000..8aadf8f5
--- /dev/null
+++ b/theories/Reals/Rlogic.v
@@ -0,0 +1,293 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+
+(** * This module proves some logical properties of the axiomatics of Reals
+
+1. Decidablity of arithmetical statements from
+ the axiom that the order of the real numbers is decidable.
+
+2. Derivability of the archimedean "axiom"
+*)
+
+(** 1- Proof of the decidablity of arithmetical statements from
+excluded middle and the axiom that the order of the real numbers is
+decidable. *)
+
+(** Assuming a decidable predicate [P n], A series is constructed whose
+[n]th term is 1/2^n if [P n] holds and 0 otherwise. This sum reaches 2
+only if [P n] holds for all [n], otherwise the sum is less than 2.
+Comparing the sum to 2 decides if [forall n, P n] or [~forall n, P n] *)
+
+(** One can iterate this lemma and use classical logic to decide any
+statement in the arithmetical hierarchy. *)
+
+(** Contributed by Cezary Kaliszyk and Russell O'Connor *)
+
+Require Import ConstructiveEpsilon.
+Require Import Rfunctions.
+Require Import PartSum.
+Require Import SeqSeries.
+Require Import RiemannInt.
+Require Import Fourier.
+
+Section Arithmetical_dec.
+
+Variable P : nat -> Prop.
+Hypothesis HP : forall n, {P n} + {~P n}.
+
+Let ge_fun_sums_ge_lemma : (forall (m n : nat) (f : nat -> R), (lt m n) -> (forall i : nat, 0 <= f i) -> sum_f_R0 f m <= sum_f_R0 f n).
+intros m n f mn fpos.
+replace (sum_f_R0 f m) with (sum_f_R0 f m + 0) by ring.
+rewrite (tech2 f m n mn).
+apply Rplus_le_compat_l.
+ induction (n - S m)%nat; simpl in *.
+ apply fpos.
+replace 0 with (0 + 0) by ring.
+apply (Rplus_le_compat _ _ _ _ IHn0 (fpos (S (m + S n0)%nat))).
+Qed.
+
+Let ge_fun_sums_ge : (forall (m n : nat) (f : nat -> R), (le m n) -> (forall i : nat, 0 <= f i) -> sum_f_R0 f m <= sum_f_R0 f n).
+intros m n f mn pos.
+ elim (le_lt_or_eq _ _ mn).
+ intro; apply ge_fun_sums_ge_lemma; assumption.
+intro H; rewrite H; auto with *.
+Qed.
+
+Let f:=fun n => (if HP n then (1/2)^n else 0)%R.
+
+Lemma cauchy_crit_geometric_dec_fun : Cauchy_crit_series f.
+intros e He.
+assert (X:(Pser (fun n:nat => 1) (1/2) (/ (1 - (1/2))))%R).
+ apply GP_infinite.
+ apply Rabs_def1; fourier.
+assert (He':e/2 > 0) by fourier.
+destruct (X _ He') as [N HN].
+clear X.
+exists N.
+intros n m Hn Hm.
+replace e with (e/2 + e/2)%R by field.
+set (g:=(fun n0 : nat => 1 * (1 / 2) ^ n0)) in *.
+assert (R_dist (sum_f_R0 g n) (sum_f_R0 g m) < e / 2 + e / 2).
+ apply Rle_lt_trans with (R_dist (sum_f_R0 g n) 2+R_dist 2 (sum_f_R0 g m))%R.
+ apply R_dist_tri.
+ replace (/(1 - 1/2)) with 2 in HN by field.
+ cut (forall n, (n >= N)%nat -> R_dist (sum_f_R0 g n) 2 < e/2)%R.
+ intros Z.
+ apply Rplus_lt_compat.
+ apply Z; assumption.
+ rewrite R_dist_sym.
+ apply Z; assumption.
+ clear - HN He.
+ intros n Hn.
+ apply HN.
+ auto.
+eapply Rle_lt_trans;[|apply H].
+clear -ge_fun_sums_ge n.
+cut (forall n m, (m <= n)%nat -> R_dist (sum_f_R0 f n) (sum_f_R0 f m) <= R_dist (sum_f_R0 g n) (sum_f_R0 g m)).
+ intros H.
+ destruct (le_lt_dec m n).
+ apply H; assumption.
+ rewrite R_dist_sym.
+ rewrite (R_dist_sym (sum_f_R0 g n)).
+ apply H; auto with *.
+clear n m.
+intros n m Hnm.
+unfold R_dist.
+cut (forall i : nat, (1 / 2) ^ i >= 0). intro RPosPow.
+rewrite Rabs_pos_eq.
+ rewrite Rabs_pos_eq.
+ cut (sum_f_R0 g m - sum_f_R0 f m <= sum_f_R0 g n - sum_f_R0 f n).
+ intros; fourier.
+ do 2 rewrite <- minus_sum.
+ apply (ge_fun_sums_ge m n (fun i : nat => g i - f i) Hnm).
+ intro i.
+ unfold f, g.
+ elim (HP i); intro; ring_simplify; auto with *.
+ cut (sum_f_R0 g m <= sum_f_R0 g n).
+ intro; fourier.
+ apply (ge_fun_sums_ge m n g Hnm).
+ intro. unfold g.
+ ring_simplify.
+ apply Rge_le.
+ apply RPosPow.
+ cut (sum_f_R0 f m <= sum_f_R0 f n).
+ intro; fourier.
+ apply (ge_fun_sums_ge m n f Hnm).
+ intro; unfold f.
+ elim (HP i); intro; simpl.
+ apply Rge_le.
+ apply RPosPow.
+ auto with *.
+intro i.
+apply Rle_ge.
+apply pow_le.
+fourier.
+Qed.
+
+Lemma forall_dec : {forall n, P n} + {~forall n, P n}.
+Proof.
+destruct (cv_cauchy_2 _ cauchy_crit_geometric_dec_fun).
+ cut (2 <= x <-> forall n : nat, P n).
+ intro H.
+ elim (Rle_dec 2 x); intro X.
+ left; tauto.
+ right; tauto.
+assert (A:Rabs(1/2) < 1) by (apply Rabs_def1; fourier).
+assert (A0:=(GP_infinite (1/2) A)).
+symmetry.
+ split; intro.
+ replace 2 with (/ (1 - (1 / 2))) by field.
+ unfold Pser, infinite_sum in A0.
+ eapply Rle_cv_lim;[|unfold Un_cv; apply A0 |apply u].
+ intros n.
+ clear -n H.
+ induction n; unfold f;simpl.
+ destruct (HP 0); auto with *.
+ elim n; auto.
+ apply Rplus_le_compat; auto.
+ destruct (HP (S n)); auto with *.
+ elim n0; auto.
+intros n.
+destruct (HP n); auto.
+elim (RIneq.Rle_not_lt _ _ H).
+assert (B:0< (1/2)^n).
+ apply pow_lt.
+ fourier.
+apply Rle_lt_trans with (2-(1/2)^n);[|fourier].
+replace (/(1-1/2))%R with 2 in A0 by field.
+set (g:= fun m => if (eq_nat_dec m n) then (1/2)^n else 0).
+assert (Z: Un_cv (fun N : nat => sum_f_R0 g N) ((1/2)^n)).
+ intros e He.
+ exists n.
+ intros a Ha.
+ replace (sum_f_R0 g a) with ((1/2)^n).
+ rewrite (R_dist_eq); assumption.
+ symmetry.
+ cut (forall a : nat, ((a >= n)%nat -> sum_f_R0 g a = (1 / 2) ^ n) /\ ((a < n)%nat -> sum_f_R0 g a = 0))%R.
+ intros H0.
+ destruct (H0 a).
+ auto.
+ clear - g.
+ induction a.
+ split;
+ intros H;
+ simpl; unfold g;
+ destruct (eq_nat_dec 0 n); try reflexivity.
+ elim f; auto with *.
+ elimtype False; omega.
+ destruct IHa as [IHa0 IHa1].
+ split;
+ intros H;
+ simpl; unfold g at 2;
+ destruct (eq_nat_dec (S a) n).
+ rewrite IHa1.
+ ring.
+ omega.
+ ring_simplify.
+ apply IHa0.
+ omega.
+ elimtype False; omega.
+ ring_simplify.
+ apply IHa1.
+ omega.
+assert (C:=CV_minus _ _ _ _ A0 Z).
+eapply Rle_cv_lim;[|apply u |apply C].
+clear - n0 B.
+intros m.
+simpl.
+induction m.
+ simpl.
+ unfold f, g.
+ destruct (eq_nat_dec 0 n).
+ destruct (HP 0).
+ elim n0.
+ congruence.
+ clear -n.
+ induction n; simpl; fourier.
+ destruct (HP); simpl; fourier.
+cut (f (S m) <= 1 * ((1 / 2) ^ (S m)) - g (S m)).
+ intros L.
+ eapply Rle_trans.
+ simpl.
+ apply Rplus_le_compat.
+ apply IHm.
+ apply L.
+ simpl; fourier.
+unfold f, g.
+destruct (eq_nat_dec (S m) n).
+ destruct (HP (S m)).
+ elim n0.
+ congruence.
+ rewrite e.
+ fourier.
+destruct (HP (S m)).
+ fourier.
+ring_simplify.
+apply pow_le.
+fourier.
+Qed.
+
+Lemma sig_forall_dec : {n | ~P n}+{forall n, P n}.
+destruct forall_dec.
+ right; assumption.
+left.
+apply constructive_indefinite_description_nat; auto.
+ clear - HP.
+ firstorder.
+apply Classical_Pred_Type.not_all_ex_not.
+assumption.
+Qed.
+
+End Arithmetical_dec.
+
+(** 2- Derivability of the Archimedean axiom *)
+
+(* This is a standard proof (it has been taken from PlanetMath). It is
+formulated negatively so as to avoid the need for classical
+logic. Using a proof of {n | ~P n}+{forall n, P n} (the one above or a
+variant of it that does not need classical axioms) , we can in
+principle also derive [up] and its [specification] *)
+
+Theorem not_not_archimedean :
+ forall r : R, ~ (forall n : nat, (INR n <= r)%R).
+intros r H.
+set (E := fun r => exists n : nat, r = INR n).
+assert (exists x : R, E x) by
+ (exists 0%R; simpl; red; exists 0%nat; reflexivity).
+assert (bound E) by (exists r; intros x (m,H2); rewrite H2; apply H).
+destruct (completeness E) as (M,(H3,H4)); try assumption.
+set (M' := (M + -1)%R).
+assert (H2 : ~ is_upper_bound E M').
+ intro H5.
+ assert (M <= M')%R by (apply H4; exact H5).
+ apply (Rlt_not_le M M').
+ unfold M' in |- *.
+ pattern M at 2 in |- *.
+ rewrite <- Rplus_0_l.
+ pattern (0 + M)%R in |- *.
+ rewrite Rplus_comm.
+ rewrite <- (Rplus_opp_r 1).
+ apply Rplus_lt_compat_l.
+ rewrite Rplus_comm.
+ apply Rlt_plus_1.
+ assumption.
+apply H2.
+intros N (n,H7).
+rewrite H7.
+unfold M' in |- *.
+assert (H5 : (INR (S n) <= M)%R) by (apply H3; exists (S n); reflexivity).
+rewrite S_INR in H5.
+assert (H6 : (INR n + 1 + -1 <= M + -1)%R).
+ apply Rplus_le_compat_r.
+ assumption.
+rewrite Rplus_assoc in H6.
+rewrite Rplus_opp_r in H6.
+rewrite (Rplus_comm (INR n) 0) in H6.
+rewrite Rplus_0_l in H6.
+assumption.
+Qed.
diff --git a/theories/Reals/Rpow_def.v b/theories/Reals/Rpow_def.v
index 5bdbb76b..90ea9726 100644
--- a/theories/Reals/Rpow_def.v
+++ b/theories/Reals/Rpow_def.v
@@ -1,3 +1,13 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+
+(* $Id: Rpow_def.v 10923 2008-05-12 18:25:06Z herbelin $ *)
+
Require Import Rdefinitions.
Fixpoint pow (r:R) (n:nat) {struct n} : R :=
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v
index cb6c59d5..adf53ef9 100644
--- a/theories/Reals/Rpower.v
+++ b/theories/Reals/Rpower.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rpower.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rpower.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
(*i Due to L.Thery i*)
(************************************************************)
@@ -22,7 +22,8 @@ Require Import Exp_prop.
Require Import Rsqrt_def.
Require Import R_sqrt.
Require Import MVT.
-Require Import Ranalysis4. Open Local Scope R_scope.
+Require Import Ranalysis4.
+Open Local Scope R_scope.
Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y).
Proof.
@@ -90,7 +91,7 @@ Proof.
replace (/ INR (fact n)) with (1 ^ n / INR (fact n)).
apply (H2 _ H3).
unfold Rdiv in |- *; rewrite pow1; rewrite Rmult_1_l; reflexivity.
- unfold infinit_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0);
+ unfold infinite_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0);
intros; exists x0; intros;
replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with
(sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n).
@@ -150,62 +151,59 @@ Proof.
symmetry in |- *; apply derive_pt_eq_0; apply derivable_pt_lim_exp.
Qed.
-Lemma ln_exists1 : forall y:R, 0 < y -> 1 <= y -> sigT (fun z:R => y = exp z).
+Lemma ln_exists1 : forall y:R, 1 <= y -> { z:R | y = exp z }.
Proof.
- intros; set (f := fun x:R => exp x - y); cut (f 0 <= 0).
- intro; cut (continuity f).
- intro; cut (0 <= f y).
- intro; cut (f 0 * f y <= 0).
- intro; assert (X := IVT_cor f 0 y H2 (Rlt_le _ _ H) H4); elim X; intros t H5;
- apply existT with t; elim H5; intros; unfold f in H7;
- apply Rminus_diag_uniq_sym; exact H7.
+ intros; set (f := fun x:R => exp x - y).
+ assert (H0 : 0 < y) by (apply Rlt_le_trans with 1; auto with real).
+ cut (f 0 <= 0); [intro H1|].
+ cut (continuity f); [intro H2|].
+ cut (0 <= f y); [intro H3|].
+ cut (f 0 * f y <= 0); [intro H4|].
+ pose proof (IVT_cor f 0 y H2 (Rlt_le _ _ H0) H4) as (t,(_,H7));
+ exists t; unfold f in H7; apply Rminus_diag_uniq_sym; exact H7.
pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y));
rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l;
assumption.
unfold f in |- *; apply Rplus_le_reg_l with y; left;
apply Rlt_trans with (1 + y).
rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1.
- replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H) | ring ].
+ replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H0) | ring ].
unfold f in |- *; change (continuity (exp - fct_cte y)) in |- *;
apply continuity_minus;
[ apply derivable_continuous; apply derivable_exp
| apply derivable_continuous; apply derivable_const ].
unfold f in |- *; rewrite exp_0; apply Rplus_le_reg_l with y;
- rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H0 | ring ].
+ rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H | ring ].
Qed.
(**********)
-Lemma ln_exists : forall y:R, 0 < y -> sigT (fun z:R => y = exp z).
+Lemma ln_exists : forall y:R, 0 < y -> { z:R | y = exp z }.
Proof.
intros; case (Rle_dec 1 y); intro.
- apply (ln_exists1 _ H r).
+ apply (ln_exists1 _ r).
assert (H0 : 1 <= / y).
apply Rmult_le_reg_l with y.
apply H.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n).
red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
- assert (H1 : 0 < / y).
- apply Rinv_0_lt_compat; apply H.
- assert (H2 := ln_exists1 _ H1 H0); elim H2; intros; apply existT with (- x);
+ destruct (ln_exists1 _ H0) as (x,p); exists (- x);
apply Rmult_eq_reg_l with (exp x / y).
unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc;
rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0;
rewrite Rmult_1_r; symmetry in |- *; apply p.
- red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H).
+ red in |- *; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H).
unfold Rdiv in |- *; apply prod_neq_R0.
- assert (H3 := exp_pos x); red in |- *; intro; rewrite H4 in H3;
+ assert (H3 := exp_pos x); red in |- *; intro H4; rewrite H4 in H3;
elim (Rlt_irrefl _ H3).
- apply Rinv_neq_0_compat; red in |- *; intro; rewrite H3 in H;
+ apply Rinv_neq_0_compat; red in |- *; intro H3; rewrite H3 in H;
elim (Rlt_irrefl _ H).
Qed.
(* Definition of log R+* -> R *)
Definition Rln (y:posreal) : R :=
- match ln_exists (pos y) (cond_pos y) with
- | existT a b => a
- end.
+ let (a,_) := ln_exists (pos y) (cond_pos y) in a.
(* Extension on R *)
Definition ln (x:R) : R :=
@@ -403,6 +401,16 @@ Infix Local "^R" := Rpower (at level 30, right associativity) : R_scope.
(** * Properties of Rpower *)
(******************************************************************)
+(** Note: [Rpower] is prolongated to [1] on negative real numbers and
+ it thus does not extend integer power. The next two lemmas, which
+ hold for integer power, accidentally hold on negative real numbers
+ as a side effect of the default value taken on negative real
+ numbers. Contrastingly, the lemmas that do not hold for the
+ integer power of a negative number are stated for [Rpower] on the
+ positive numbers only (even if they accidentally hold due to the
+ default value of [Rpower] on the negative side, as it is the case
+ for [Rpower_O]). *)
+
Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y.
Proof.
intros x y z; unfold Rpower in |- *.
@@ -420,7 +428,7 @@ Qed.
Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1.
Proof.
- intros x H; unfold Rpower in |- *.
+ intros x _; unfold Rpower in |- *.
rewrite Rmult_0_l; apply exp_0.
Qed.
diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v
index a84d5149..2113cc8f 100644
--- a/theories/Reals/Rprod.v
+++ b/theories/Reals/Rprod.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rprod.v 9298 2006-10-27 13:05:29Z notin $ i*)
+(*i $Id: Rprod.v 10146 2007-09-27 12:28:12Z herbelin $ i*)
Require Import Compare.
Require Import Rbase.
@@ -16,41 +16,42 @@ Require Import PartSum.
Require Import Binomial.
Open Local Scope R_scope.
-(** TT Ak; 1<=k<=N *)
-Boxed Fixpoint prod_f_SO (An:nat -> R) (N:nat) {struct N} : R :=
+(** TT Ak; 0<=k<=N *)
+Boxed Fixpoint prod_f_R0 (f:nat -> R) (N:nat) {struct N} : R :=
match N with
- | O => 1
- | S p => prod_f_SO An p * An (S p)
+ | O => f O
+ | S p => prod_f_R0 f p * f (S p)
end.
+Notation prod_f_SO := (fun An N => prod_f_R0 (fun n => An (S n)) N).
+
(**********)
Lemma prod_SO_split :
forall (An:nat -> R) (n k:nat),
- (k <= n)%nat ->
- prod_f_SO An n =
- prod_f_SO An k * prod_f_SO (fun l:nat => An (k + l)%nat) (n - k).
+ (k < n)%nat ->
+ prod_f_R0 An n =
+ prod_f_R0 An k * prod_f_R0 (fun l:nat => An (k +1+l)%nat) (n - k -1).
Proof.
intros; induction n as [| n Hrecn].
- cut (k = 0%nat);
- [ intro; rewrite H0; simpl in |- *; ring | inversion H; reflexivity ].
- cut (k = S n \/ (k <= n)%nat).
- intro; elim H0; intro.
- rewrite H1; simpl in |- *; rewrite <- minus_n_n; simpl in |- *; ring.
- replace (S n - k)%nat with (S (n - k)).
+ absurd (k < 0)%nat; omega.
+ cut (k = n \/ (k < n)%nat);[intro; elim H0; intro|omega].
+ replace (S n - k - 1)%nat with O; [rewrite H1; simpl|omega].
+ replace (n+1+0)%nat with (S n); ring.
+ replace (S n - k-1)%nat with (S (n - k-1));[idtac|omega].
simpl in |- *; replace (k + S (n - k))%nat with (S n).
+ replace (k + 1 + S (n - k - 1))%nat with (S n).
rewrite Hrecn; [ ring | assumption ].
omega.
omega.
- omega.
-Qed.
+Qed.
(**********)
Lemma prod_SO_pos :
forall (An:nat -> R) (N:nat),
- (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_SO An N.
+ (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N.
Proof.
intros; induction N as [| N HrecN].
- simpl in |- *; left; apply Rlt_0_1.
+ simpl in |- *; apply H; trivial.
simpl in |- *; apply Rmult_le_pos.
apply HrecN; intros; apply H; apply le_trans with N;
[ assumption | apply le_n_Sn ].
@@ -61,11 +62,11 @@ Qed.
Lemma prod_SO_Rle :
forall (An Bn:nat -> R) (N:nat),
(forall n:nat, (n <= N)%nat -> 0 <= An n <= Bn n) ->
- prod_f_SO An N <= prod_f_SO Bn N.
+ prod_f_R0 An N <= prod_f_R0 Bn N.
Proof.
intros; induction N as [| N HrecN].
- right; reflexivity.
- simpl in |- *; apply Rle_trans with (prod_f_SO An N * Bn (S N)).
+ elim H with O; trivial.
+ simpl in |- *; apply Rle_trans with (prod_f_R0 An N * Bn (S N)).
apply Rmult_le_compat_l.
apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
assumption.
@@ -79,12 +80,17 @@ Qed.
(** Application to factorial *)
Lemma fact_prodSO :
- forall n:nat, INR (fact n) = prod_f_SO (fun k:nat => INR k) n.
+ forall n:nat, INR (fact n) = prod_f_R0 (fun k:nat =>
+ (match (eq_nat_dec k 0) with
+ | left _ => 1%R
+ | right _ => INR k
+ end)) n.
Proof.
intro; induction n as [| n Hrecn].
reflexivity.
- change (INR (S n * fact n) = prod_f_SO (fun k:nat => INR k) (S n)) in |- *.
- rewrite mult_INR; rewrite Rmult_comm; rewrite Hrecn; reflexivity.
+ simpl; rewrite <- Hrecn.
+ case n; auto with real.
+ intros; repeat rewrite plus_INR;rewrite mult_INR;ring.
Qed.
Lemma le_n_2n : forall n:nat, (n <= 2 * n)%nat.
@@ -104,40 +110,58 @@ Lemma RfactN_fact2N_factk :
(k <= 2 * N)%nat ->
Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k).
Proof.
+ assert (forall (n:nat), 0 <= (if eq_nat_dec n 0 then 1 else INR n)).
+ intros; case (eq_nat_dec n 0); auto with real.
+ assert (forall (n:nat), (0 < n)%nat ->
+ (if eq_nat_dec n 0 then 1 else INR n) = INR n).
+ intros n; case (eq_nat_dec n 0); auto with real.
+ intros; absurd (0 < n)%nat; omega.
intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO.
- cut ((k <= N)%nat \/ (N <= k)%nat).
- intro; elim H0; intro.
- rewrite (prod_SO_split (fun l:nat => INR l) (2 * N - k) N).
+ cut ((k=N)%nat \/ (k < N)%nat \/ (N < k)%nat).
+ intro H2; elim H2; intro H3.
+ rewrite H3; replace (2*N-N)%nat with N;[right; ring|omega].
+ case H3; intro; clear H2 H3.
+ rewrite (prod_SO_split (fun l:nat => if eq_nat_dec l 0 then 1 else INR l) (2 * N - k) N).
rewrite Rmult_assoc; apply Rmult_le_compat_l.
- apply prod_SO_pos; intros; apply pos_INR.
- replace (2 * N - k - N)%nat with (N - k)%nat.
- rewrite Rmult_comm; rewrite (prod_SO_split (fun l:nat => INR l) N k).
+ apply prod_SO_pos; intros; auto.
+ replace (2 * N - k - N-1)%nat with (N - k-1)%nat.
+ rewrite Rmult_comm; rewrite (prod_SO_split
+ (fun l:nat => if eq_nat_dec l 0 then 1 else INR l) N k).
apply Rmult_le_compat_l.
- apply prod_SO_pos; intros; apply pos_INR.
- apply prod_SO_Rle; intros; split.
- apply pos_INR.
- apply le_INR; apply plus_le_compat_r; assumption.
+ apply prod_SO_pos; intros; auto.
+ apply prod_SO_Rle; intros; split; auto.
+ rewrite H0.
+ rewrite H0.
+ apply le_INR; omega.
+ omega.
+ omega.
assumption.
omega.
omega.
- rewrite <- (Rmult_comm (prod_f_SO (fun l:nat => INR l) k));
- rewrite (prod_SO_split (fun l:nat => INR l) k N).
+ rewrite <- (Rmult_comm (prod_f_R0 (fun l:nat =>
+ if eq_nat_dec l 0 then 1 else INR l) k));
+ rewrite (prod_SO_split (fun l:nat =>
+ if eq_nat_dec l 0 then 1 else INR l) k N).
rewrite Rmult_assoc; apply Rmult_le_compat_l.
- apply prod_SO_pos; intros; apply pos_INR.
+ apply prod_SO_pos; intros; auto.
rewrite Rmult_comm;
- rewrite (prod_SO_split (fun l:nat => INR l) N (2 * N - k)).
+ rewrite (prod_SO_split (fun l:nat =>
+ if eq_nat_dec l 0 then 1 else INR l) N (2 * N - k)).
apply Rmult_le_compat_l.
- apply prod_SO_pos; intros; apply pos_INR.
- replace (N - (2 * N - k))%nat with (k - N)%nat.
- apply prod_SO_Rle; intros; split.
- apply pos_INR.
- apply le_INR; apply plus_le_compat_r.
+ apply prod_SO_pos; intros; auto.
+ replace (N - (2 * N - k)-1)%nat with (k - N-1)%nat.
+ apply prod_SO_Rle; intros; split; auto.
+ rewrite H0.
+ rewrite H0.
+ apply le_INR; omega.
+ omega.
omega.
omega.
omega.
assumption.
omega.
-Qed.
+Qed.
+
(**********)
Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n).
diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v
index 38c39bae..702aafa4 100644
--- a/theories/Reals/Rseries.v
+++ b/theories/Reals/Rseries.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rseries.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rseries.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -194,14 +194,14 @@ Section Isequence.
Variable An : nat -> R.
(*********)
- Definition Pser (x l:R) : Prop := infinit_sum (fun n:nat => An n * x ^ n) l.
+ Definition Pser (x l:R) : Prop := infinite_sum (fun n:nat => An n * x ^ n) l.
End Isequence.
Lemma GP_infinite :
forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)).
Proof.
- intros; unfold Pser in |- *; unfold infinit_sum in |- *; intros;
+ intros; unfold Pser in |- *; unfold infinite_sum in |- *; intros;
elim (Req_dec x 0).
intros; exists 0%nat; intros; rewrite H1; rewrite Rminus_0_r; rewrite Rinv_1;
cut (sum_f_R0 (fun n0:nat => 1 * 0 ^ n0) n = 1).
diff --git a/theories/Reals/Rsigma.v b/theories/Reals/Rsigma.v
index cb31d3b2..7cdd4d02 100644
--- a/theories/Reals/Rsigma.v
+++ b/theories/Reals/Rsigma.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rsigma.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rsigma.v 9454 2006-12-15 15:30:59Z bgregoir $ i*)
Require Import Rbase.
Require Import Rfunctions.
diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v
index 0a9f7754..0a3af6ca 100644
--- a/theories/Reals/Rsqrt_def.v
+++ b/theories/Reals/Rsqrt_def.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rsqrt_def.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rsqrt_def.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Sumbool.
Require Import Rbase.
@@ -192,7 +192,7 @@ Qed.
Lemma dicho_lb_cv :
forall (x y:R) (P:R -> bool),
- x <= y -> sigT (fun l:R => Un_cv (dicho_lb x y P) l).
+ x <= y -> { l:R | Un_cv (dicho_lb x y P) l }.
Proof.
intros.
apply growing_cv.
@@ -202,7 +202,7 @@ Qed.
Lemma dicho_up_cv :
forall (x y:R) (P:R -> bool),
- x <= y -> sigT (fun l:R => Un_cv (dicho_up x y P) l).
+ x <= y -> { l:R | Un_cv (dicho_up x y P) l }.
Proof.
intros.
apply decreasing_cv.
@@ -466,7 +466,7 @@ Qed.
Lemma IVT :
forall (f:R -> R) (x y:R),
continuity f ->
- x < y -> f x < 0 -> 0 < f y -> sigT (fun z:R => x <= z <= y /\ f z = 0).
+ x < y -> f x < 0 -> 0 < f y -> { z:R | x <= z <= y /\ f z = 0 }.
Proof.
intros.
cut (x <= y).
@@ -478,7 +478,7 @@ Proof.
elim X0; intros.
assert (H4 := cv_dicho _ _ _ _ _ H3 p0 p).
rewrite H4 in p0.
- apply existT with x0.
+ exists x0.
split.
split.
apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0).
@@ -602,7 +602,7 @@ Qed.
Lemma IVT_cor :
forall (f:R -> R) (x y:R),
continuity f ->
- x <= y -> f x * f y <= 0 -> sigT (fun z:R => x <= z <= y /\ f z = 0).
+ x <= y -> f x * f y <= 0 -> { z:R | x <= z <= y /\ f z = 0 }.
Proof.
intros.
case (total_order_T 0 (f x)); intro.
@@ -628,7 +628,7 @@ Proof.
cut (0 < (- f)%F y).
intros.
elim (H3 H5 H4); intros.
- apply existT with x0.
+ exists x0.
elim p; intros.
split.
assumption.
@@ -643,7 +643,7 @@ Proof.
assumption.
rewrite H2 in a.
elim (Rlt_irrefl _ (Rlt_trans _ _ _ r a)).
- apply existT with x.
+ exists x.
split.
split; [ right; reflexivity | assumption ].
symmetry in |- *; assumption.
@@ -656,7 +656,7 @@ Proof.
assumption.
rewrite H2 in r.
elim (Rlt_irrefl _ (Rlt_trans _ _ _ r a)).
- apply existT with y.
+ exists y.
split.
split; [ assumption | right; reflexivity ].
symmetry in |- *; assumption.
@@ -670,7 +670,7 @@ Qed.
(** We can now define the square root function as the reciprocal
transformation of the square root function *)
Lemma Rsqrt_exists :
- forall y:R, 0 <= y -> sigT (fun z:R => 0 <= z /\ y = Rsqr z).
+ forall y:R, 0 <= y -> { z:R | 0 <= z /\ y = Rsqr z }.
Proof.
intros.
set (f := fun x:R => Rsqr x - y).
@@ -686,7 +686,7 @@ Proof.
intro.
assert (X := IVT_cor f 0 1 H1 (Rlt_le _ _ Rlt_0_1) H3).
elim X; intros t H4.
- apply existT with t.
+ exists t.
elim H4; intros.
split.
elim H5; intros; assumption.
@@ -700,7 +700,7 @@ Proof.
rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *;
rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
left; assumption.
- apply existT with 1.
+ exists 1.
split.
left; apply Rlt_0_1.
rewrite b; symmetry in |- *; apply Rsqr_1.
@@ -710,7 +710,7 @@ Proof.
intro.
assert (X := IVT_cor f 0 y H1 H H3).
elim X; intros t H4.
- apply existT with t.
+ exists t.
elim H4; intros.
split.
elim H5; intros; assumption.
@@ -739,9 +739,7 @@ Qed.
(* Definition of the square root: R+->R *)
Definition Rsqrt (y:nonnegreal) : R :=
- match Rsqrt_exists (nonneg y) (cond_nonneg y) with
- | existT a b => a
- end.
+ let (a,_) := Rsqrt_exists (nonneg y) (cond_nonneg y) in a.
(**********)
Lemma Rsqrt_positivity : forall x:nonnegreal, 0 <= Rsqrt x.
diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v
index aa47d72f..9501bc1e 100644
--- a/theories/Reals/Rtopology.v
+++ b/theories/Reals/Rtopology.v
@@ -6,15 +6,15 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtopology.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rtopology.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import RList.
Require Import Classical_Prop.
-Require Import Classical_Pred_Type. Open Local Scope R_scope.
-
+Require Import Classical_Pred_Type.
+Open Local Scope R_scope.
(** * General definitions and propositions *)
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v
index b744c788..0baece39 100644
--- a/theories/Reals/Rtrigo.v
+++ b/theories/Reals/Rtrigo.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rtrigo.v 9454 2006-12-15 15:30:59Z bgregoir $ i*)
Require Import Rbase.
Require Import Rfunctions.
diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v
index 89ee1745..d82bafc6 100644
--- a/theories/Reals/Rtrigo_alt.v
+++ b/theories/Reals/Rtrigo_alt.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo_alt.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rtrigo_alt.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -137,7 +137,7 @@ Proof.
ring.
assert (X := exist_sin (Rsqr a)); elim X; intros.
cut (x = sin a / a).
- intro; rewrite H3 in p; unfold sin_in in p; unfold infinit_sum in p;
+ intro; rewrite H3 in p; unfold sin_in in p; unfold infinite_sum in p;
unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *;
intros.
cut (0 < eps / Rabs a).
@@ -327,7 +327,7 @@ Proof.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
assert (X := exist_cos (Rsqr a0)); elim X; intros.
cut (x = cos a0).
- intro; rewrite H4 in p; unfold cos_in in p; unfold infinit_sum in p;
+ intro; rewrite H4 in p; unfold cos_in in p; unfold infinite_sum in p;
unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *;
intros.
elim (p _ H5); intros N H6.
diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v
index b2aeb766..e94d7448 100644
--- a/theories/Reals/Rtrigo_def.v
+++ b/theories/Reals/Rtrigo_def.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo_def.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rtrigo_def.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -19,7 +19,7 @@ Open Local Scope R_scope.
(** * Definition of exponential *)
(********************************)
Definition exp_in (x l:R) : Prop :=
- infinit_sum (fun i:nat => / INR (fact i) * x ^ i) l.
+ infinite_sum (fun i:nat => / INR (fact i) * x ^ i) l.
Lemma exp_cof_no_R0 : forall n:nat, / INR (fact n) <> 0.
Proof.
@@ -28,7 +28,7 @@ Proof.
apply INR_fact_neq_0.
Qed.
-Lemma exist_exp : forall x:R, sigT (fun l:R => exp_in x l).
+Lemma exist_exp : forall x:R, { l:R | exp_in x l }.
Proof.
intro;
generalize
@@ -37,7 +37,7 @@ Proof.
trivial.
Defined.
-Definition exp (x:R) : R := projT1 (exist_exp x).
+Definition exp (x:R) : R := proj1_sig (exist_exp x).
Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0.
Proof.
@@ -45,11 +45,10 @@ Proof.
red in |- *; intro; rewrite H0 in H; elim (lt_irrefl _ H).
Qed.
-(*i Calculus of $e^0$ *)
-Lemma exist_exp0 : sigT (fun l:R => exp_in 0 l).
+Lemma exist_exp0 : { l:R | exp_in 0 l }.
Proof.
- apply existT with 1.
- unfold exp_in in |- *; unfold infinit_sum in |- *; intros.
+ exists 1.
+ unfold exp_in in |- *; unfold infinite_sum in |- *; intros.
exists 0%nat.
intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1.
unfold R_dist in |- *; replace (1 - 1) with 0;
@@ -63,6 +62,7 @@ Proof.
unfold ge in |- *; apply le_O_n.
Defined.
+(* Value of [exp 0] *)
Lemma exp_0 : exp 0 = 1.
Proof.
cut (exp_in 0 (exp 0)).
@@ -70,8 +70,8 @@ Proof.
unfold exp_in in |- *; intros; eapply uniqueness_sum.
apply H0.
apply H.
- exact (projT2 exist_exp0).
- exact (projT2 (exist_exp 0)).
+ exact (proj2_sig exist_exp0).
+ exact (proj2_sig (exist_exp 0)).
Qed.
(*****************************************)
@@ -235,21 +235,17 @@ Qed.
(**********)
Definition cos_in (x l:R) : Prop :=
- infinit_sum (fun i:nat => cos_n i * x ^ i) l.
+ infinite_sum (fun i:nat => cos_n i * x ^ i) l.
(**********)
-Lemma exist_cos : forall x:R, sigT (fun l:R => cos_in x l).
+Lemma exist_cos : forall x:R, { l:R | cos_in x l }.
intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos).
unfold Pser, cos_in in |- *; trivial.
Qed.
(** Definition of cosinus *)
-Definition cos (x:R) : R :=
- match exist_cos (Rsqr x) with
- | existT a b => a
- end.
-
+Definition cos (x:R) : R := let (a,_) := exist_cos (Rsqr x) in a.
Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).
@@ -348,7 +344,7 @@ Proof.
apply INR_eq; repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR;
rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
replace (INR 0) with 0; [ ring | reflexivity ].
-Qed.
+Defined.
Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.
Proof.
@@ -359,21 +355,18 @@ Qed.
(**********)
Definition sin_in (x l:R) : Prop :=
- infinit_sum (fun i:nat => sin_n i * x ^ i) l.
+ infinite_sum (fun i:nat => sin_n i * x ^ i) l.
(**********)
-Lemma exist_sin : forall x:R, sigT (fun l:R => sin_in x l).
+Lemma exist_sin : forall x:R, { l:R | sin_in x l }.
Proof.
intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin).
unfold Pser, sin_n in |- *; trivial.
-Qed.
+Defined.
(***********************)
(* Definition of sinus *)
-Definition sin (x:R) : R :=
- match exist_sin (Rsqr x) with
- | existT a b => x * a
- end.
+Definition sin (x:R) : R := let (a,_) := exist_sin (Rsqr x) in x * a.
(*********************************************)
(** * Properties *)
@@ -399,10 +392,10 @@ Proof.
intros; ring.
Qed.
-Lemma exist_cos0 : sigT (fun l:R => cos_in 0 l).
+Lemma exist_cos0 : { l:R | cos_in 0 l }.
Proof.
- apply existT with 1.
- unfold cos_in in |- *; unfold infinit_sum in |- *; intros; exists 0%nat.
+ exists 1.
+ unfold cos_in in |- *; unfold infinite_sum in |- *; intros; exists 0%nat.
intros.
unfold R_dist in |- *.
induction n as [| n Hrecn].
@@ -417,7 +410,7 @@ Proof.
simpl in |- *; ring.
Defined.
-(* Calculus of (cos 0) *)
+(* Value of [cos 0] *)
Lemma cos_0 : cos 0 = 1.
Proof.
cut (cos_in 0 (cos 0)).
@@ -425,7 +418,7 @@ Proof.
unfold cos_in in |- *; intros; eapply uniqueness_sum.
apply H0.
apply H.
- exact (projT2 exist_cos0).
- assert (H := projT2 (exist_cos (Rsqr 0))); unfold cos in |- *;
+ exact (proj2_sig exist_cos0).
+ assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos in |- *;
pattern 0 at 1 in |- *; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ].
Qed.
diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v
index 78ef847f..6eec0329 100644
--- a/theories/Reals/Rtrigo_fun.v
+++ b/theories/Reals/Rtrigo_fun.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo_fun.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Rtrigo_fun.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -15,8 +15,7 @@ Open Local Scope R_scope.
(*****************************************************************)
(** To define transcendental functions *)
-(** for exponential function *)
-(* *)
+(** and exponential function *)
(*****************************************************************)
(*********)
diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v
index b105ca69..139563bf 100644
--- a/theories/Reals/Rtrigo_reg.v
+++ b/theories/Reals/Rtrigo_reg.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Rtrigo_reg.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: Rtrigo_reg.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -25,16 +25,15 @@ Proof.
unfold CVN_R in |- *; intros.
cut ((r:R) <> 0).
intro hyp_r; unfold CVN_r in |- *.
- apply existT with (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)).
+ exists (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)).
cut
- (sigT
- (fun l:R =>
+ { l:R |
Un_cv
(fun n:nat =>
sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
- n) l)).
+ n) l }.
intro X; elim X; intros.
- apply existT with x.
+ exists x.
split.
apply p.
intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult.
@@ -124,7 +123,7 @@ Lemma continuity_cos : continuity cos.
Proof.
set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)).
cut (CVN_R fn).
- intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)).
+ intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
intro cv; cut (forall n:nat, continuity (fn n)).
intro; cut (forall x:R, cos x = SFL fn cv x).
intro; cut (continuity (SFL fn cv) -> continuity cos).
@@ -144,7 +143,7 @@ Proof.
case (cv x); case (exist_cos (Rsqr x)); intros.
symmetry in |- *; eapply UL_sequence.
apply u.
- unfold cos_in in c; unfold infinit_sum in c; unfold Un_cv in |- *; intros.
+ unfold cos_in in c; unfold infinite_sum in c; unfold Un_cv in |- *; intros.
elim (c _ H0); intros N0 H1.
exists N0; intros.
unfold R_dist in H1; unfold R_dist, SP in |- *.
@@ -200,17 +199,16 @@ Lemma CVN_R_sin :
CVN_R fn.
Proof.
unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r.
- apply existT with (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)).
+ exists (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)).
cut
- (sigT
- (fun l:R =>
+ { l:R |
Un_cv
(fun n:nat =>
sum_f_R0
(fun k:nat => Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n)
- l)).
+ l }.
intro X; elim X; intros.
- apply existT with x.
+ exists x.
split.
apply p.
intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult;
@@ -305,7 +303,7 @@ Proof.
set
(fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)).
cut (CVN_R fn).
- intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)).
+ intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
intro cv.
set (r := mkposreal _ Rlt_0_1).
cut (CVN_r fn r).
@@ -331,7 +329,7 @@ Proof.
unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6).
eapply UL_sequence.
apply u.
- unfold sin_in in s; unfold sin_n, infinit_sum in s;
+ unfold sin_in in s; unfold sin_n, infinite_sum in s;
unfold SP, fn, Un_cv in |- *; intros.
elim (s _ H10); intros N0 H11.
exists N0; intros.
@@ -584,14 +582,14 @@ Qed.
Lemma derivable_pt_sin : forall x:R, derivable_pt sin x.
Proof.
unfold derivable_pt in |- *; intro.
- apply existT with (cos x).
+ exists (cos x).
apply derivable_pt_lim_sin.
Qed.
Lemma derivable_pt_cos : forall x:R, derivable_pt cos x.
Proof.
unfold derivable_pt in |- *; intro.
- apply existT with (- sin x).
+ exists (- sin x).
apply derivable_pt_lim_cos.
Qed.
diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v
index 96351618..56088a2e 100644
--- a/theories/Reals/SeqProp.v
+++ b/theories/Reals/SeqProp.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: SeqProp.v 9551 2007-01-29 15:13:35Z bgregoir $ i*)
+(*i $Id: SeqProp.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -15,6 +15,10 @@ Require Import Classical.
Require Import Max.
Open Local Scope R_scope.
+(*****************************************************************)
+(** Convergence properties of sequences *)
+(*****************************************************************)
+
Definition Un_decreasing (Un:nat -> R) : Prop :=
forall n:nat, Un (S n) <= Un n.
Definition opp_seq (Un:nat -> R) (n:nat) : R := - Un n.
@@ -23,8 +27,7 @@ Definition has_lb (Un:nat -> R) : Prop := bound (EUn (opp_seq Un)).
(**********)
Lemma growing_cv :
- forall Un:nat -> R,
- Un_growing Un -> has_ub Un -> sigT (fun l:R => Un_cv Un l).
+ forall Un:nat -> R, Un_growing Un -> has_ub Un -> { l:R | Un_cv Un l }.
Proof.
unfold Un_growing, Un_cv in |- *; intros;
destruct (completeness (EUn Un) H0 (EUn_noempty Un)) as [x [H2 H3]].
@@ -64,11 +67,10 @@ Proof.
Qed.
Lemma decreasing_cv :
- forall Un:nat -> R,
- Un_decreasing Un -> has_lb Un -> sigT (fun l:R => Un_cv Un l).
+ forall Un:nat -> R, Un_decreasing Un -> has_lb Un -> { l:R | Un_cv Un l }.
Proof.
intros.
- cut (sigT (fun l:R => Un_cv (opp_seq Un) l) -> sigT (fun l:R => Un_cv Un l)).
+ cut ({ l:R | Un_cv (opp_seq Un) l } -> { l:R | Un_cv Un l }).
intro X.
apply X.
apply growing_cv.
@@ -76,7 +78,7 @@ Proof.
exact H0.
intro X.
elim X; intros.
- apply existT with (- x).
+ exists (- x).
unfold Un_cv in p.
unfold R_dist in p.
unfold opp_seq in p.
@@ -91,8 +93,8 @@ Proof.
Qed.
(***********)
-Lemma maj_sup :
- forall Un:nat -> R, has_ub Un -> sigT (fun l:R => is_lub (EUn Un) l).
+Lemma ub_to_lub :
+ forall Un:nat -> R, has_ub Un -> { l:R | is_lub (EUn Un) l }.
Proof.
intros.
unfold has_ub in H.
@@ -104,9 +106,8 @@ Proof.
Qed.
(**********)
-Lemma min_inf :
- forall Un:nat -> R,
- has_lb Un -> sigT (fun l:R => is_lub (EUn (opp_seq Un)) l).
+Lemma lb_to_glb :
+ forall Un:nat -> R, has_lb Un -> { l:R | is_lub (EUn (opp_seq Un)) l }.
Proof.
intros; unfold has_lb in H.
apply completeness.
@@ -116,15 +117,17 @@ Proof.
reflexivity.
Qed.
-Definition majorant (Un:nat -> R) (pr:has_ub Un) : R :=
- match maj_sup Un pr with
- | existT a b => a
- end.
+Definition lub (Un:nat -> R) (pr:has_ub Un) : R :=
+ let (a,_) := ub_to_lub Un pr in a.
-Definition minorant (Un:nat -> R) (pr:has_lb Un) : R :=
- match min_inf Un pr with
- | existT a b => - a
- end.
+Definition glb (Un:nat -> R) (pr:has_lb Un) : R :=
+ let (a,_) := lb_to_glb Un pr in - a.
+
+(* Compatibility with previous unappropriate terminology *)
+Notation maj_sup := ub_to_lub (only parsing).
+Notation min_inf := lb_to_glb (only parsing).
+Notation majorant := lub (only parsing).
+Notation minorant := glb (only parsing).
Lemma maj_ss :
forall (Un:nat -> R) (k:nat),
@@ -162,26 +165,30 @@ Proof.
exists (k + x1)%nat; assumption.
Qed.
-Definition sequence_majorant (Un:nat -> R) (pr:has_ub Un)
- (i:nat) : R := majorant (fun k:nat => Un (i + k)%nat) (maj_ss Un i pr).
+Definition sequence_ub (Un:nat -> R) (pr:has_ub Un)
+ (i:nat) : R := lub (fun k:nat => Un (i + k)%nat) (maj_ss Un i pr).
+
+Definition sequence_lb (Un:nat -> R) (pr:has_lb Un)
+ (i:nat) : R := glb (fun k:nat => Un (i + k)%nat) (min_ss Un i pr).
-Definition sequence_minorant (Un:nat -> R) (pr:has_lb Un)
- (i:nat) : R := minorant (fun k:nat => Un (i + k)%nat) (min_ss Un i pr).
+(* Compatibility *)
+Notation sequence_majorant := sequence_ub (only parsing).
+Notation sequence_minorant := sequence_lb (only parsing).
Lemma Wn_decreasing :
- forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_majorant Un pr).
+ forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_ub Un pr).
Proof.
intros.
unfold Un_decreasing in |- *.
intro.
- unfold sequence_majorant in |- *.
- assert (H := maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
- assert (H0 := maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
+ unfold sequence_ub in |- *.
+ assert (H := ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
+ assert (H0 := ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
elim H; intros.
elim H0; intros.
- cut (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr) = x);
+ cut (lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr) = x);
[ intro Maj1; rewrite Maj1 | idtac ].
- cut (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr) = x0);
+ cut (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr) = x0);
[ intro Maj2; rewrite Maj2 | idtac ].
unfold is_lub in p.
unfold is_lub in p0.
@@ -199,47 +206,47 @@ Proof.
replace (S n) with (1 + n)%nat; [ ring | ring ].
cut
(is_lub (EUn (fun k:nat => Un (n + k)%nat))
- (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr))).
+ (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr))).
intro.
unfold is_lub in p0; unfold is_lub in H1.
elim p0; intros; elim H1; intros.
assert (H6 := H5 x0 H2).
assert
- (H7 := H3 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4).
+ (H7 := H3 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4).
apply Rle_antisym; assumption.
- unfold majorant in |- *.
- case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
+ unfold lub in |- *.
+ case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
trivial.
cut
(is_lub (EUn (fun k:nat => Un (S n + k)%nat))
- (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr))).
+ (lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr))).
intro.
unfold is_lub in p; unfold is_lub in H1.
elim p; intros; elim H1; intros.
assert (H6 := H5 x H2).
assert
(H7 :=
- H3 (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4).
+ H3 (lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4).
apply Rle_antisym; assumption.
- unfold majorant in |- *.
- case (maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
+ unfold lub in |- *.
+ case (ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
trivial.
Qed.
Lemma Vn_growing :
- forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_minorant Un pr).
+ forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_lb Un pr).
Proof.
intros.
unfold Un_growing in |- *.
intro.
- unfold sequence_minorant in |- *.
- assert (H := min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)).
- assert (H0 := min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)).
+ unfold sequence_lb in |- *.
+ assert (H := lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)).
+ assert (H0 := lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)).
elim H; intros.
elim H0; intros.
- cut (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr) = - x);
+ cut (glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr) = - x);
[ intro Maj1; rewrite Maj1 | idtac ].
- cut (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr) = - x0);
+ cut (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr) = - x0);
[ intro Maj2; rewrite Maj2 | idtac ].
unfold is_lub in p.
unfold is_lub in p0.
@@ -260,38 +267,38 @@ Proof.
replace (S n) with (1 + n)%nat; [ ring | ring ].
cut
(is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat)))
- (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))).
+ (- glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))).
intro.
unfold is_lub in p0; unfold is_lub in H1.
elim p0; intros; elim H1; intros.
assert (H6 := H5 x0 H2).
assert
- (H7 := H3 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)) H4).
+ (H7 := H3 (- glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)) H4).
rewrite <-
- (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)))
+ (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)))
.
apply Ropp_eq_compat; apply Rle_antisym; assumption.
- unfold minorant in |- *.
- case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)).
+ unfold glb in |- *.
+ case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)); simpl.
intro; rewrite Ropp_involutive.
trivial.
cut
(is_lub (EUn (opp_seq (fun k:nat => Un (S n + k)%nat)))
- (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))).
+ (- glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))).
intro.
unfold is_lub in p; unfold is_lub in H1.
elim p; intros; elim H1; intros.
assert (H6 := H5 x H2).
assert
(H7 :=
- H3 (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)) H4).
+ H3 (- glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)) H4).
rewrite <-
(Ropp_involutive
- (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)))
+ (glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)))
.
apply Ropp_eq_compat; apply Rle_antisym; assumption.
- unfold minorant in |- *.
- case (min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)).
+ unfold glb in |- *.
+ case (lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)); simpl.
intro; rewrite Ropp_involutive.
trivial.
Qed.
@@ -299,16 +306,15 @@ Qed.
(**********)
Lemma Vn_Un_Wn_order :
forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un)
- (n:nat), sequence_minorant Un pr2 n <= Un n <= sequence_majorant Un pr1 n.
+ (n:nat), sequence_lb Un pr2 n <= Un n <= sequence_ub Un pr1 n.
Proof.
intros.
split.
- unfold sequence_minorant in |- *.
- cut
- (sigT (fun l:R => is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l)).
+ unfold sequence_lb in |- *.
+ cut { l:R | is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l }.
intro X.
elim X; intros.
- replace (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) with (- x).
+ replace (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) with (- x).
unfold is_lub in p.
elim p; intros.
unfold is_upper_bound in H.
@@ -320,28 +326,28 @@ Proof.
replace (n + 0)%nat with n; [ reflexivity | ring ].
cut
(is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat)))
- (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))).
+ (- glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))).
intro.
unfold is_lub in p; unfold is_lub in H.
elim p; intros; elim H; intros.
assert (H4 := H3 x H0).
assert
- (H5 := H1 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) H2).
+ (H5 := H1 (- glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) H2).
rewrite <-
- (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)))
+ (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)))
.
apply Ropp_eq_compat; apply Rle_antisym; assumption.
- unfold minorant in |- *.
- case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)).
+ unfold glb in |- *.
+ case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)); simpl.
intro; rewrite Ropp_involutive.
trivial.
- apply min_inf.
+ apply lb_to_glb.
apply min_ss; assumption.
- unfold sequence_majorant in |- *.
- cut (sigT (fun l:R => is_lub (EUn (fun i:nat => Un (n + i)%nat)) l)).
+ unfold sequence_ub in |- *.
+ cut { l:R | is_lub (EUn (fun i:nat => Un (n + i)%nat)) l }.
intro X.
elim X; intros.
- replace (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) with x.
+ replace (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) with x.
unfold is_lub in p.
elim p; intros.
unfold is_upper_bound in H.
@@ -350,24 +356,24 @@ Proof.
replace (n + 0)%nat with n; [ reflexivity | ring ].
cut
(is_lub (EUn (fun k:nat => Un (n + k)%nat))
- (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1))).
+ (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1))).
intro.
unfold is_lub in p; unfold is_lub in H.
elim p; intros; elim H; intros.
assert (H4 := H3 x H0).
assert
- (H5 := H1 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2).
+ (H5 := H1 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2).
apply Rle_antisym; assumption.
- unfold majorant in |- *.
- case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)).
+ unfold lub in |- *.
+ case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)).
intro; trivial.
- apply maj_sup.
+ apply ub_to_lub.
apply maj_ss; assumption.
Qed.
Lemma min_maj :
forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un),
- has_ub (sequence_minorant Un pr2).
+ has_ub (sequence_lb Un pr2).
Proof.
intros.
assert (H := Vn_Un_Wn_order Un pr1 pr2).
@@ -390,7 +396,7 @@ Qed.
Lemma maj_min :
forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un),
- has_lb (sequence_majorant Un pr1).
+ has_lb (sequence_ub Un pr1).
Proof.
intros.
assert (H := Vn_Un_Wn_order Un pr1 pr2).
@@ -451,7 +457,7 @@ Qed.
(**********)
Lemma maj_cv :
forall (Un:nat -> R) (pr:Cauchy_crit Un),
- sigT (fun l:R => Un_cv (sequence_majorant Un (cauchy_maj Un pr)) l).
+ { l:R | Un_cv (sequence_ub Un (cauchy_maj Un pr)) l }.
Proof.
intros.
apply decreasing_cv.
@@ -464,7 +470,7 @@ Qed.
(**********)
Lemma min_cv :
forall (Un:nat -> R) (pr:Cauchy_crit Un),
- sigT (fun l:R => Un_cv (sequence_minorant Un (cauchy_min Un pr)) l).
+ { l:R | Un_cv (sequence_lb Un (cauchy_min Un pr)) l }.
Proof.
intros.
apply growing_cv.
@@ -510,40 +516,40 @@ Qed.
(**********)
Lemma approx_maj :
forall (Un:nat -> R) (pr:has_ub Un) (eps:R),
- 0 < eps -> exists k : nat, Rabs (majorant Un pr - Un k) < eps.
+ 0 < eps -> exists k : nat, Rabs (lub Un pr - Un k) < eps.
Proof.
intros.
- set (P := fun k:nat => Rabs (majorant Un pr - Un k) < eps).
+ set (P := fun k:nat => Rabs (lub Un pr - Un k) < eps).
unfold P in |- *.
cut
((exists k : nat, P k) ->
- exists k : nat, Rabs (majorant Un pr - Un k) < eps).
+ exists k : nat, Rabs (lub Un pr - Un k) < eps).
intros.
apply H0.
apply not_all_not_ex.
red in |- *; intro.
2: unfold P in |- *; trivial.
unfold P in H1.
- cut (forall n:nat, Rabs (majorant Un pr - Un n) >= eps).
+ cut (forall n:nat, Rabs (lub Un pr - Un n) >= eps).
intro.
- cut (is_lub (EUn Un) (majorant Un pr)).
+ cut (is_lub (EUn Un) (lub Un pr)).
intro.
unfold is_lub in H3.
unfold is_upper_bound in H3.
elim H3; intros.
- cut (forall n:nat, eps <= majorant Un pr - Un n).
+ cut (forall n:nat, eps <= lub Un pr - Un n).
intro.
- cut (forall n:nat, Un n <= majorant Un pr - eps).
+ cut (forall n:nat, Un n <= lub Un pr - eps).
intro.
- cut (forall x:R, EUn Un x -> x <= majorant Un pr - eps).
+ cut (forall x:R, EUn Un x -> x <= lub Un pr - eps).
intro.
- assert (H9 := H5 (majorant Un pr - eps) H8).
+ assert (H9 := H5 (lub Un pr - eps) H8).
cut (eps <= 0).
intro.
elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)).
- apply Rplus_le_reg_l with (majorant Un pr - eps).
+ apply Rplus_le_reg_l with (lub Un pr - eps).
rewrite Rplus_0_r.
- replace (majorant Un pr - eps + eps) with (majorant Un pr);
+ replace (lub Un pr - eps + eps) with (lub Un pr);
[ assumption | ring ].
intros.
unfold EUn in H8.
@@ -553,7 +559,7 @@ Proof.
assert (H7 := H6 n).
apply Rplus_le_reg_l with (eps - Un n).
replace (eps - Un n + Un n) with eps.
- replace (eps - Un n + (majorant Un pr - eps)) with (majorant Un pr - Un n).
+ replace (eps - Un n + (lub Un pr - eps)) with (lub Un pr - Un n).
assumption.
ring.
ring.
@@ -565,11 +571,11 @@ Proof.
apply Rle_ge.
apply Rplus_le_reg_l with (Un n).
rewrite Rplus_0_r;
- replace (Un n + (majorant Un pr - Un n)) with (majorant Un pr);
+ replace (Un n + (lub Un pr - Un n)) with (lub Un pr);
[ apply H4 | ring ].
exists n; reflexivity.
- unfold majorant in |- *.
- case (maj_sup Un pr).
+ unfold lub in |- *.
+ case (ub_to_lub Un pr).
trivial.
intro.
assert (H2 := H1 n).
@@ -579,40 +585,40 @@ Qed.
(**********)
Lemma approx_min :
forall (Un:nat -> R) (pr:has_lb Un) (eps:R),
- 0 < eps -> exists k : nat, Rabs (minorant Un pr - Un k) < eps.
+ 0 < eps -> exists k : nat, Rabs (glb Un pr - Un k) < eps.
Proof.
intros.
- set (P := fun k:nat => Rabs (minorant Un pr - Un k) < eps).
+ set (P := fun k:nat => Rabs (glb Un pr - Un k) < eps).
unfold P in |- *.
cut
((exists k : nat, P k) ->
- exists k : nat, Rabs (minorant Un pr - Un k) < eps).
+ exists k : nat, Rabs (glb Un pr - Un k) < eps).
intros.
apply H0.
apply not_all_not_ex.
red in |- *; intro.
2: unfold P in |- *; trivial.
unfold P in H1.
- cut (forall n:nat, Rabs (minorant Un pr - Un n) >= eps).
+ cut (forall n:nat, Rabs (glb Un pr - Un n) >= eps).
intro.
- cut (is_lub (EUn (opp_seq Un)) (- minorant Un pr)).
+ cut (is_lub (EUn (opp_seq Un)) (- glb Un pr)).
intro.
unfold is_lub in H3.
unfold is_upper_bound in H3.
elim H3; intros.
- cut (forall n:nat, eps <= Un n - minorant Un pr).
+ cut (forall n:nat, eps <= Un n - glb Un pr).
intro.
- cut (forall n:nat, opp_seq Un n <= - minorant Un pr - eps).
+ cut (forall n:nat, opp_seq Un n <= - glb Un pr - eps).
intro.
- cut (forall x:R, EUn (opp_seq Un) x -> x <= - minorant Un pr - eps).
+ cut (forall x:R, EUn (opp_seq Un) x -> x <= - glb Un pr - eps).
intro.
- assert (H9 := H5 (- minorant Un pr - eps) H8).
+ assert (H9 := H5 (- glb Un pr - eps) H8).
cut (eps <= 0).
intro.
elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)).
- apply Rplus_le_reg_l with (- minorant Un pr - eps).
+ apply Rplus_le_reg_l with (- glb Un pr - eps).
rewrite Rplus_0_r.
- replace (- minorant Un pr - eps + eps) with (- minorant Un pr);
+ replace (- glb Un pr - eps + eps) with (- glb Un pr);
[ assumption | ring ].
intros.
unfold EUn in H8.
@@ -623,7 +629,7 @@ Proof.
unfold opp_seq in |- *.
apply Rplus_le_reg_l with (eps + Un n).
replace (eps + Un n + - Un n) with eps.
- replace (eps + Un n + (- minorant Un pr - eps)) with (Un n - minorant Un pr).
+ replace (eps + Un n + (- glb Un pr - eps)) with (Un n - glb Un pr).
assumption.
ring.
ring.
@@ -631,16 +637,16 @@ Proof.
assert (H6 := H2 n).
rewrite Rabs_left1 in H6.
apply Rge_le.
- replace (Un n - minorant Un pr) with (- (minorant Un pr - Un n));
+ replace (Un n - glb Un pr) with (- (glb Un pr - Un n));
[ assumption | ring ].
- apply Rplus_le_reg_l with (- minorant Un pr).
+ apply Rplus_le_reg_l with (- glb Un pr).
rewrite Rplus_0_r;
- replace (- minorant Un pr + (minorant Un pr - Un n)) with (- Un n).
+ replace (- glb Un pr + (glb Un pr - Un n)) with (- Un n).
apply H4.
exists n; reflexivity.
ring.
- unfold minorant in |- *.
- case (min_inf Un pr).
+ unfold glb in |- *.
+ case (lb_to_glb Un pr); simpl.
intro.
rewrite Ropp_involutive.
trivial.
@@ -711,7 +717,7 @@ Qed.
(**********)
Lemma CV_Cauchy :
- forall Un:nat -> R, sigT (fun l:R => Un_cv Un l) -> Cauchy_crit Un.
+ forall Un:nat -> R, { l:R | Un_cv Un l } -> Cauchy_crit Un.
Proof.
intros Un X; elim X; intros.
unfold Cauchy_crit in |- *; intros.
@@ -734,11 +740,11 @@ Qed.
(**********)
Lemma maj_by_pos :
forall Un:nat -> R,
- sigT (fun l:R => Un_cv Un l) ->
+ { l:R | Un_cv Un l } ->
exists l : R, 0 < l /\ (forall n:nat, Rabs (Un n) <= l).
Proof.
intros Un X; elim X; intros.
- cut (sigT (fun l:R => Un_cv (fun k:nat => Rabs (Un k)) l)).
+ cut { l:R | Un_cv (fun k:nat => Rabs (Un k)) l }.
intro X0.
assert (H := CV_Cauchy (fun k:nat => Rabs (Un k)) X0).
assert (H0 := cauchy_bound (fun k:nat => Rabs (Un k)) H).
@@ -760,7 +766,7 @@ Proof.
unfold is_upper_bound in H1.
apply H1.
exists 0%nat; reflexivity.
- apply existT with (Rabs x).
+ exists (Rabs x).
apply cv_cvabs; assumption.
Qed.
@@ -770,7 +776,7 @@ Lemma CV_mult :
Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i * Bn i) (l1 * l2).
Proof.
intros.
- cut (sigT (fun l:R => Un_cv An l)).
+ cut { l:R | Un_cv An l }.
intro X.
assert (H1 := maj_by_pos An X).
elim H1; intros M H2.
@@ -881,7 +887,7 @@ Proof.
[ assumption
| apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
[ prove_sup0 | assumption ] ].
- apply existT with l1; assumption.
+ exists l1; assumption.
Qed.
Lemma tech9 :
diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v
index bc17cd43..9680b75e 100644
--- a/theories/Reals/SeqSeries.v
+++ b/theories/Reals/SeqSeries.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: SeqSeries.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: SeqSeries.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -33,15 +33,9 @@ Lemma sum_maj1 :
Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N.
Proof.
intros;
- cut
- (sigT
- (fun l:R =>
- Un_cv (fun n:nat => sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) l)).
+ cut { l:R | Un_cv (fun n => sum_f_R0 (fun l => fn (S N + l)%nat x) n) l }.
intro X;
- cut
- (sigT
- (fun l:R =>
- Un_cv (fun n:nat => sum_f_R0 (fun l:nat => An (S N + l)%nat) n) l)).
+ cut { l:R | Un_cv (fun n => sum_f_R0 (fun l => An (S N + l)%nat) n) l }.
intro X0; elim X; intros l1N H2.
elim X0; intros l2N H3.
cut (l1 - SP fn N x = l1N).
@@ -131,7 +125,7 @@ Proof.
apply le_lt_n_Sm.
apply le_plus_l.
apply le_O_n.
- apply existT with (l2 - sum_f_R0 An N).
+ exists (l2 - sum_f_R0 An N).
unfold Un_cv in H0; unfold Un_cv in |- *; intros.
elim (H0 eps H2); intros N0 H3.
unfold R_dist in H3; exists N0; intros.
@@ -167,7 +161,7 @@ Proof.
apply le_lt_n_Sm.
apply le_plus_l.
apply le_O_n.
- apply existT with (l1 - SP fn N x).
+ exists (l1 - SP fn N x).
unfold Un_cv in H; unfold Un_cv in |- *; intros.
elim (H eps H2); intros N0 H3.
unfold R_dist in H3; exists N0; intros.
@@ -216,8 +210,8 @@ Qed.
Lemma Rseries_CV_comp :
forall An Bn:nat -> R,
(forall n:nat, 0 <= An n <= Bn n) ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 Bn N) l) ->
- sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l).
+ { l:R | Un_cv (fun N:nat => sum_f_R0 Bn N) l } ->
+ { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
Proof.
intros An Bn H X; apply cv_cauchy_2.
assert (H0 := cv_cauchy_1 _ X).
diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v
index ff0a72e8..13be46da 100644
--- a/theories/Reals/Sqrt_reg.v
+++ b/theories/Reals/Sqrt_reg.v
@@ -6,12 +6,13 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Sqrt_reg.v 9245 2006-10-17 12:53:34Z notin $ i*)
+(*i $Id: Sqrt_reg.v 10710 2008-03-23 09:24:09Z herbelin $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
-Require Import R_sqrt. Open Local Scope R_scope.
+Require Import R_sqrt.
+Open Local Scope R_scope.
(**********)
Lemma sqrt_var_maj :
@@ -309,7 +310,7 @@ Qed.
Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x.
Proof.
unfold derivable_pt in |- *; intros.
- apply existT with (/ (2 * sqrt x)).
+ exists (/ (2 * sqrt x)).
apply derivable_pt_lim_sqrt; assumption.
Qed.