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-rw-r--r--theories/Reals/RiemannInt_SF.v958
1 files changed, 478 insertions, 480 deletions
diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v
index d0d9519c..d523a1f4 100644
--- a/theories/Reals/RiemannInt_SF.v
+++ b/theories/Reals/RiemannInt_SF.v
@@ -1,18 +1,16 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: RiemannInt_SF.v 14641 2011-11-06 11:59:10Z herbelin $ i*)
-
Require Import Rbase.
Require Import Rfunctions.
-Require Import Ranalysis.
+Require Import Ranalysis_reg.
Require Import Classical_Prop.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
Set Implicit Arguments.
@@ -23,7 +21,7 @@ Set Implicit Arguments.
Definition Nbound (I:nat -> Prop) : Prop :=
exists n : nat, (forall i:nat, I i -> (i <= n)%nat).
-Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z_of_nat n}.
+Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}.
Proof.
intros; apply Z_of_nat_complete_inf; assumption.
Qed.
@@ -35,19 +33,19 @@ Lemma Nzorn :
Proof.
intros I H H0; set (E := fun x:R => exists i : nat, I i /\ INR i = x);
assert (H1 : bound E).
- unfold Nbound in H0; elim H0; intros N H1; unfold bound in |- *;
- exists (INR N); unfold is_upper_bound in |- *; intros;
+ unfold Nbound in H0; elim H0; intros N H1; unfold bound;
+ exists (INR N); unfold is_upper_bound; intros;
unfold E in H2; elim H2; intros; elim H3; intros;
rewrite <- H5; apply le_INR; apply H1; assumption.
assert (H2 : exists x : R, E x).
- elim H; intros; exists (INR x); unfold E in |- *; exists x; split;
+ elim H; intros; exists (INR x); unfold E; exists x; split;
[ assumption | reflexivity ].
assert (H3 := completeness E H1 H2); elim H3; intros; unfold is_lub in p;
elim p; clear p; intros; unfold is_upper_bound in H4, H5;
assert (H6 : 0 <= x).
elim H2; intros; unfold E in H6; elim H6; intros; elim H7; intros;
apply Rle_trans with x0;
- [ rewrite <- H9; change (INR 0 <= INR x1) in |- *; apply le_INR;
+ [ rewrite <- H9; change (INR 0 <= INR x1); apply le_INR;
apply le_O_n
| apply H4; assumption ].
assert (H7 := archimed x); elim H7; clear H7; intros;
@@ -90,7 +88,7 @@ Proof.
[ idtac | reflexivity ]; rewrite <- minus_INR.
replace (x0 - 1)%nat with (pred x0);
[ reflexivity
- | case x0; [ reflexivity | intro; simpl in |- *; apply minus_n_O ] ].
+ | case x0; [ reflexivity | intro; simpl; apply minus_n_O ] ].
induction x0 as [| x0 Hrecx0];
[ rewrite p in H7; rewrite <- INR_IZR_INZ in H7; simpl in H7;
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H7))
@@ -101,10 +99,10 @@ Proof.
assert (H16 : INR x0 = INR x1 + 1).
rewrite H15; ring.
rewrite <- S_INR in H16; assert (H17 := INR_eq _ _ H16); rewrite H17;
- simpl in |- *; split.
+ simpl; split.
assumption.
intros; apply INR_le; rewrite H15; rewrite <- H15; elim H12; intros;
- rewrite H20; apply H4; unfold E in |- *; exists i;
+ rewrite H20; apply H4; unfold E; exists i;
split; [ assumption | reflexivity ].
Qed.
@@ -149,7 +147,7 @@ Definition subdivision_val (a b:R) (f:StepFun a b) : Rlist :=
| existT a b => a
end.
-Boxed Fixpoint Int_SF (l k:Rlist) : R :=
+Fixpoint Int_SF (l k:Rlist) : R :=
match l with
| nil => 0
| cons a l' =>
@@ -175,7 +173,7 @@ Lemma StepFun_P1 :
forall (a b:R) (f:StepFun a b),
adapted_couple f a b (subdivision f) (subdivision_val f).
Proof.
- intros a b f; unfold subdivision_val in |- *; case (projT2 (pre f)); intros;
+ intros a b f; unfold subdivision_val; case (projT2 (pre f)); intros;
apply a0.
Qed.
@@ -183,13 +181,13 @@ Lemma StepFun_P2 :
forall (a b:R) (f:R -> R) (l lf:Rlist),
adapted_couple f a b l lf -> adapted_couple f b a l lf.
Proof.
- unfold adapted_couple in |- *; intros; decompose [and] H; clear H;
+ unfold adapted_couple; intros; decompose [and] H; clear H;
repeat split; try assumption.
- rewrite H2; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ rewrite H2; unfold Rmin; case (Rle_dec a b); intro;
case (Rle_dec b a); intro; try reflexivity.
apply Rle_antisym; assumption.
apply Rle_antisym; auto with real.
- rewrite H1; unfold Rmax in |- *; case (Rle_dec a b); intro;
+ rewrite H1; unfold Rmax; case (Rle_dec a b); intro;
case (Rle_dec b a); intro; try reflexivity.
apply Rle_antisym; assumption.
apply Rle_antisym; auto with real.
@@ -200,23 +198,23 @@ Lemma StepFun_P3 :
a <= b ->
adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil).
Proof.
- intros; unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H0; inversion H0;
- [ simpl in |- *; assumption | elim (le_Sn_O _ H2) ].
- simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ intros; unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H0; inversion H0;
+ [ simpl; assumption | elim (le_Sn_O _ H2) ].
+ simpl; unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
- simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro;
+ simpl; unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
- unfold constant_D_eq, open_interval in |- *; intros; simpl in H0;
+ unfold constant_D_eq, open_interval; intros; simpl in H0;
inversion H0; [ reflexivity | elim (le_Sn_O _ H3) ].
Qed.
Lemma StepFun_P4 : forall a b c:R, IsStepFun (fct_cte c) a b.
Proof.
- intros; unfold IsStepFun in |- *; case (Rle_dec a b); intro.
- apply existT with (cons a (cons b nil)); unfold is_subdivision in |- *;
+ intros; unfold IsStepFun; case (Rle_dec a b); intro.
+ apply existT with (cons a (cons b nil)); unfold is_subdivision;
apply existT with (cons c nil); apply (StepFun_P3 c r).
- apply existT with (cons b (cons a nil)); unfold is_subdivision in |- *;
+ apply existT with (cons b (cons a nil)); unfold is_subdivision;
apply existT with (cons c nil); apply StepFun_P2;
apply StepFun_P3; auto with real.
Qed.
@@ -234,7 +232,7 @@ Qed.
Lemma StepFun_P6 :
forall (f:R -> R) (a b:R), IsStepFun f a b -> IsStepFun f b a.
Proof.
- unfold IsStepFun in |- *; intros; elim X; intros; apply existT with x;
+ unfold IsStepFun; intros; elim X; intros; apply existT with x;
apply StepFun_P5; assumption.
Qed.
@@ -244,26 +242,26 @@ Lemma StepFun_P7 :
adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf) ->
adapted_couple f r2 b (cons r2 l) lf.
Proof.
- unfold adapted_couple in |- *; intros; decompose [and] H0; clear H0;
+ unfold adapted_couple; intros; decompose [and] H0; clear H0;
assert (H5 : Rmax a b = b).
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (H7 : r2 <= b).
rewrite H5 in H2; rewrite <- H2; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ].
+ [ assumption | simpl; right; left; reflexivity ].
repeat split.
apply RList_P4 with r1; assumption.
- rewrite H5 in H2; unfold Rmin in |- *; case (Rle_dec r2 b); intro;
+ rewrite H5 in H2; unfold Rmin; case (Rle_dec r2 b); intro;
[ reflexivity | elim n; assumption ].
- unfold Rmax in |- *; case (Rle_dec r2 b); intro;
+ unfold Rmax; case (Rle_dec r2 b); intro;
[ rewrite H5 in H2; rewrite <- H2; reflexivity | elim n; assumption ].
- simpl in H4; simpl in |- *; apply INR_eq; apply Rplus_eq_reg_l with 1;
+ simpl in H4; simpl; apply INR_eq; apply Rplus_eq_reg_l with 1;
do 2 rewrite (Rplus_comm 1); do 2 rewrite <- S_INR;
rewrite H4; reflexivity.
- intros; unfold constant_D_eq, open_interval in |- *; intros;
+ intros; unfold constant_D_eq, open_interval; intros;
unfold constant_D_eq, open_interval in H6;
assert (H9 : (S i < pred (Rlength (cons r1 (cons r2 l))))%nat).
- simpl in |- *; simpl in H0; apply lt_n_S; assumption.
+ simpl; simpl in H0; apply lt_n_S; assumption.
assert (H10 := H6 _ H9); apply H10; assumption.
Qed.
@@ -280,19 +278,19 @@ Proof.
discriminate.
intros; induction lf1 as [| r3 lf1 Hreclf1].
reflexivity.
- simpl in |- *; cut (r = r1).
+ simpl; cut (r = r1).
intro; rewrite H3; rewrite (H0 lf1 r b).
ring.
rewrite H3; apply StepFun_P7 with a r r3; [ right; assumption | assumption ].
clear H H0 Hreclf1 r0; unfold adapted_couple in H1; decompose [and] H1;
- intros; simpl in H4; rewrite H4; unfold Rmin in |- *;
+ intros; simpl in H4; rewrite H4; unfold Rmin;
case (Rle_dec a b); intro; [ assumption | reflexivity ].
unfold adapted_couple in H1; decompose [and] H1; intros; apply Rle_antisym.
- apply (H3 0%nat); simpl in |- *; apply lt_O_Sn.
+ apply (H3 0%nat); simpl; apply lt_O_Sn.
simpl in H5; rewrite H2 in H5; rewrite H5; replace (Rmin b b) with (Rmax a b);
[ rewrite <- H4; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ]
- | unfold Rmin, Rmax in |- *; case (Rle_dec b b); case (Rle_dec a b); intros;
+ [ assumption | simpl; right; left; reflexivity ]
+ | unfold Rmin, Rmax; case (Rle_dec b b); case (Rle_dec a b); intros;
try assumption || reflexivity ].
Qed.
@@ -305,10 +303,10 @@ Proof.
[ simpl in H4; discriminate
| induction l as [| r0 l Hrecl0];
[ simpl in H3; simpl in H2; generalize H3; generalize H2;
- unfold Rmin, Rmax in |- *; case (Rle_dec a b);
+ unfold Rmin, Rmax; case (Rle_dec a b);
intros; elim H0; rewrite <- H5; rewrite <- H7;
reflexivity
- | simpl in |- *; do 2 apply le_n_S; apply le_O_n ] ].
+ | simpl; do 2 apply le_n_S; apply le_O_n ] ].
Qed.
Lemma StepFun_P10 :
@@ -322,12 +320,12 @@ Proof.
intros; unfold adapted_couple in H0; decompose [and] H0; simpl in H4;
discriminate.
intros; case (Req_dec a b); intro.
- exists (cons a nil); exists nil; unfold adapted_couple_opt in |- *;
- unfold adapted_couple in |- *; unfold ordered_Rlist in |- *;
+ exists (cons a nil); exists nil; unfold adapted_couple_opt;
+ unfold adapted_couple; unfold ordered_Rlist;
repeat split; try (intros; simpl in H3; elim (lt_n_O _ H3)).
- simpl in |- *; rewrite <- H2; unfold Rmin in |- *; case (Rle_dec a a); intro;
+ simpl; rewrite <- H2; unfold Rmin; case (Rle_dec a a); intro;
reflexivity.
- simpl in |- *; rewrite <- H2; unfold Rmax in |- *; case (Rle_dec a a); intro;
+ simpl; rewrite <- H2; unfold Rmax; case (Rle_dec a a); intro;
reflexivity.
elim (RList_P20 _ (StepFun_P9 H1 H2)); intros t1 [t2 [t3 H3]];
induction lf as [| r1 lf Hreclf].
@@ -342,32 +340,32 @@ Proof.
apply H6.
rewrite <- Hyp_eq; rewrite H3 in H1; unfold adapted_couple in H1;
decompose [and] H1; clear H1; simpl in H9; rewrite H9;
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
elim H6; clear H6; intros l' [lf' H6]; case (Req_dec t2 b); intro.
exists (cons a (cons b nil)); exists (cons r1 nil);
- unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+ unfold adapted_couple_opt; unfold adapted_couple;
repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8;
- [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
- simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold ordered_Rlist; intros; simpl in H8; inversion H8;
+ [ simpl; assumption | elim (le_Sn_O _ H10) ].
+ simpl; unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
- simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro;
+ simpl; unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
intros; simpl in H8; inversion H8.
- unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *;
+ unfold constant_D_eq, open_interval; intros; simpl;
simpl in H9; rewrite H3 in H1; unfold adapted_couple in H1;
decompose [and] H1; apply (H16 0%nat).
- simpl in |- *; apply lt_O_Sn.
- unfold open_interval in |- *; simpl in |- *; rewrite H7; simpl in H13;
- rewrite H13; unfold Rmin in |- *; case (Rle_dec a b);
+ simpl; apply lt_O_Sn.
+ unfold open_interval; simpl; rewrite H7; simpl in H13;
+ rewrite H13; unfold Rmin; case (Rle_dec a b);
intro; [ assumption | elim n; assumption ].
elim (le_Sn_O _ H10).
intros; simpl in H8; elim (lt_n_O _ H8).
intros; simpl in H8; inversion H8;
- [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
+ [ simpl; assumption | elim (le_Sn_O _ H10) ].
assert (Hyp_min : Rmin t2 b = t2).
- unfold Rmin in |- *; case (Rle_dec t2 b); intro;
+ unfold Rmin; case (Rle_dec t2 b); intro;
[ reflexivity | elim n; assumption ].
unfold adapted_couple in H6; elim H6; clear H6; intros;
elim (RList_P20 _ (StepFun_P9 H6 H7)); intros s1 [s2 [s3 H9]];
@@ -379,141 +377,141 @@ Proof.
exists (cons t1 (cons s2 s3)); exists (cons r1 lf'); rewrite H3 in H1;
rewrite H9 in H6; unfold adapted_couple in H6, H1;
decompose [and] H1; decompose [and] H6; clear H1 H6;
- unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+ unfold adapted_couple_opt; unfold adapted_couple;
repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H1;
+ unfold ordered_Rlist; intros; simpl in H1;
induction i as [| i Hreci].
- simpl in |- *; apply Rle_trans with s1.
+ simpl; apply Rle_trans with s1.
replace s1 with t2.
apply (H12 0%nat).
- simpl in |- *; apply lt_O_Sn.
- simpl in H19; rewrite H19; symmetry in |- *; apply Hyp_min.
- apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
- change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *;
- apply (H16 (S i)); simpl in |- *; assumption.
- simpl in |- *; simpl in H14; rewrite H14; reflexivity.
- simpl in |- *; simpl in H18; rewrite H18; unfold Rmax in |- *;
+ simpl; apply lt_O_Sn.
+ simpl in H19; rewrite H19; symmetry ; apply Hyp_min.
+ apply (H16 0%nat); simpl; apply lt_O_Sn.
+ change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
+ apply (H16 (S i)); simpl; assumption.
+ simpl; simpl in H14; rewrite H14; reflexivity.
+ simpl; simpl in H18; rewrite H18; unfold Rmax;
case (Rle_dec a b); case (Rle_dec t2 b); intros; reflexivity || elim n;
assumption.
- simpl in |- *; simpl in H20; apply H20.
- intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+ simpl; simpl in H20; apply H20.
+ intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; simpl in H6; case (total_order_T x t2); intro.
+ simpl; simpl in H6; case (total_order_T x t2); intro.
elim s; intro.
apply (H17 0%nat);
- [ simpl in |- *; apply lt_O_Sn
- | unfold open_interval in |- *; simpl in |- *; elim H6; intros; split;
+ [ simpl; apply lt_O_Sn
+ | unfold open_interval; simpl; elim H6; intros; split;
assumption ].
rewrite b0; assumption.
rewrite H10; apply (H22 0%nat);
- [ simpl in |- *; apply lt_O_Sn
- | unfold open_interval in |- *; simpl in |- *; replace s1 with t2;
+ [ simpl; apply lt_O_Sn
+ | unfold open_interval; simpl; replace s1 with t2;
[ elim H6; intros; split; assumption
| simpl in H19; rewrite H19; rewrite Hyp_min; reflexivity ] ].
- simpl in |- *; simpl in H6; apply (H22 (S i));
- [ simpl in |- *; assumption
- | unfold open_interval in |- *; simpl in |- *; apply H6 ].
+ simpl; simpl in H6; apply (H22 (S i));
+ [ simpl; assumption
+ | unfold open_interval; simpl; apply H6 ].
intros; simpl in H1; rewrite H10;
change
(pos_Rl (cons r2 lf') i <> pos_Rl (cons r2 lf') (S i) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r2 lf') i)
- in |- *; rewrite <- H9; elim H8; intros; apply H6;
- simpl in |- *; apply H1.
+ ; rewrite <- H9; elim H8; intros; apply H6;
+ simpl; apply H1.
intros; induction i as [| i Hreci].
- simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
- apply (H12 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+ apply (H12 0%nat); simpl; apply lt_O_Sn.
rewrite <- Hyp_min; rewrite H6; simpl in H19; rewrite <- H19;
- apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
- elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl in |- *;
+ apply (H16 0%nat); simpl; apply lt_O_Sn.
+ elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl;
simpl in H1; apply H1.
exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
rewrite H3 in H1; unfold adapted_couple in H1, H6;
decompose [and] H6; decompose [and] H1; clear H6 H1;
- unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+ unfold adapted_couple_opt; unfold adapted_couple;
repeat split.
- rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1;
+ rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
induction i as [| i Hreci].
- simpl in |- *; replace s1 with t2.
- apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; replace s1 with t2.
+ apply (H16 0%nat); simpl; apply lt_O_Sn.
simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
change
(pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
- in |- *; apply (H12 i); simpl in |- *; apply lt_S_n;
+ ; apply (H12 i); simpl; apply lt_S_n;
assumption.
- simpl in |- *; simpl in H19; apply H19.
- rewrite H9; simpl in |- *; simpl in H13; rewrite H13; unfold Rmax in |- *;
+ simpl; simpl in H19; apply H19.
+ rewrite H9; simpl; simpl in H13; rewrite H13; unfold Rmax;
case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n;
assumption.
- rewrite H9; simpl in |- *; simpl in H15; rewrite H15; reflexivity.
- intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+ rewrite H9; simpl; simpl in H15; rewrite H15; reflexivity.
+ intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H22 0%nat).
- simpl in |- *; apply lt_O_Sn.
- unfold open_interval in |- *; simpl in |- *.
+ simpl; rewrite H9 in H6; simpl in H6; apply (H22 0%nat).
+ simpl; apply lt_O_Sn.
+ unfold open_interval; simpl.
replace t2 with s1.
assumption.
simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
- change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H17 i).
- simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
- rewrite H9 in H6; unfold open_interval in |- *; apply H6.
+ change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H17 i).
+ simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
+ rewrite H9 in H6; unfold open_interval; apply H6.
intros; simpl in H1; induction i as [| i Hreci].
- simpl in |- *; rewrite H9; right; simpl in |- *; replace s1 with t2.
+ simpl; rewrite H9; right; simpl; replace s1 with t2.
assumption.
simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
elim H8; intros; apply (H6 i).
- simpl in |- *; apply lt_S_n; apply H1.
+ simpl; apply lt_S_n; apply H1.
intros; rewrite H9; induction i as [| i Hreci].
- simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
- apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+ apply (H16 0%nat); simpl; apply lt_O_Sn.
rewrite <- Hyp_min; rewrite H6; simpl in H14; rewrite <- H14; right;
reflexivity.
elim H8; intros; rewrite <- H9; apply (H21 i); rewrite H9; rewrite H9 in H1;
- simpl in |- *; simpl in H1; apply lt_S_n; apply H1.
+ simpl; simpl in H1; apply lt_S_n; apply H1.
exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
rewrite H3 in H1; unfold adapted_couple in H1, H6;
decompose [and] H6; decompose [and] H1; clear H6 H1;
- unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+ unfold adapted_couple_opt; unfold adapted_couple;
repeat split.
- rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1;
+ rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
induction i as [| i Hreci].
- simpl in |- *; replace s1 with t2.
- apply (H15 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; replace s1 with t2.
+ apply (H15 0%nat); simpl; apply lt_O_Sn.
simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.
change
(pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
- in |- *; apply (H11 i); simpl in |- *; apply lt_S_n;
+ ; apply (H11 i); simpl; apply lt_S_n;
assumption.
- simpl in |- *; simpl in H18; apply H18.
- rewrite H9; simpl in |- *; simpl in H12; rewrite H12; unfold Rmax in |- *;
+ simpl; simpl in H18; apply H18.
+ rewrite H9; simpl; simpl in H12; rewrite H12; unfold Rmax;
case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n;
assumption.
- rewrite H9; simpl in |- *; simpl in H14; rewrite H14; reflexivity.
- intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+ rewrite H9; simpl; simpl in H14; rewrite H14; reflexivity.
+ intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H21 0%nat).
- simpl in |- *; apply lt_O_Sn.
- unfold open_interval in |- *; simpl in |- *; replace t2 with s1.
+ simpl; rewrite H9 in H6; simpl in H6; apply (H21 0%nat).
+ simpl; apply lt_O_Sn.
+ unfold open_interval; simpl; replace t2 with s1.
assumption.
simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.
- change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H16 i).
- simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
- rewrite H9 in H6; unfold open_interval in |- *; apply H6.
+ change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H16 i).
+ simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
+ rewrite H9 in H6; unfold open_interval; apply H6.
intros; simpl in H1; induction i as [| i Hreci].
- simpl in |- *; left; assumption.
+ simpl; left; assumption.
elim H8; intros; apply (H6 i).
- simpl in |- *; apply lt_S_n; apply H1.
+ simpl; apply lt_S_n; apply H1.
intros; rewrite H9; induction i as [| i Hreci].
- simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
- apply (H15 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+ apply (H15 0%nat); simpl; apply lt_O_Sn.
rewrite <- Hyp_min; rewrite H6; simpl in H13; rewrite <- H13; right;
reflexivity.
elim H8; intros; rewrite <- H9; apply (H20 i); rewrite H9; rewrite H9 in H1;
- simpl in |- *; simpl in H1; apply lt_S_n; apply H1.
+ simpl; simpl in H1; apply lt_S_n; apply H1.
rewrite H3 in H1; clear H4; unfold adapted_couple in H1; decompose [and] H1;
clear H1; clear H H7 H9; cut (Rmax a b = b);
[ intro; rewrite H in H5; rewrite <- H5; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ]
- | unfold Rmax in |- *; case (Rle_dec a b); intro;
+ [ assumption | simpl; right; left; reflexivity ]
+ | unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ] ].
Qed.
@@ -536,7 +534,7 @@ Proof.
simpl in H9; rewrite H9 in H16; cut (r1 <= Rmax a b).
intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H17 H16)).
rewrite <- H4; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ].
+ [ assumption | simpl; right; left; reflexivity ].
clear Hrecs3; induction lf2 as [| r5 lf2 Hreclf2].
simpl in H11; discriminate.
clear Hreclf2; assert (H17 : r3 = r4).
@@ -546,31 +544,31 @@ Proof.
simpl in H18; rewrite <- (H17 x).
rewrite <- (H18 x).
reflexivity.
- rewrite <- H12; unfold x in |- *; split.
+ rewrite <- H12; unfold x; split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
apply Rplus_lt_compat_l; assumption
| discrR ] ].
- unfold x in |- *; split.
+ unfold x; split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rlt_trans with s2;
[ apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
apply Rplus_lt_compat_l; assumption
@@ -578,8 +576,8 @@ Proof.
| assumption ].
assert (H18 : f s2 = r3).
apply (H8 0%nat);
- [ simpl in |- *; apply lt_O_Sn
- | unfold open_interval in |- *; simpl in |- *; split; assumption ].
+ [ simpl; apply lt_O_Sn
+ | unfold open_interval; simpl; split; assumption ].
assert (H19 : r3 = r5).
assert (H19 := H7 1%nat); simpl in H19;
assert (H20 := H19 (lt_n_S _ _ (lt_O_Sn _))); elim H20;
@@ -589,18 +587,18 @@ Proof.
rewrite <- (H22 (lt_O_Sn _) x).
rewrite <- (H23 (lt_n_S _ _ (lt_O_Sn _)) x).
reflexivity.
- unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split.
+ unfold open_interval; simpl; unfold x; split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
- unfold Rmin in |- *; case (Rle_dec r1 r0); intro;
+ unfold Rmin; case (Rle_dec r1 r0); intro;
assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
apply Rlt_le_trans with (r0 + Rmin r1 r0);
@@ -608,20 +606,20 @@ Proof.
assumption
| apply Rplus_le_compat_l; apply Rmin_r ]
| discrR ] ].
- unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split.
+ unfold open_interval; simpl; unfold x; split.
apply Rlt_trans with s2;
[ assumption
| apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
- unfold Rmin in |- *; case (Rle_dec r1 r0);
+ unfold Rmin; case (Rle_dec r1 r0);
intro; assumption
| discrR ] ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
apply Rlt_le_trans with (r1 + Rmin r1 r0);
@@ -638,20 +636,20 @@ Proof.
| elim H24; rewrite <- H17; assumption ].
elim H2; clear H2; intros; assert (H17 := H16 0%nat); simpl in H17;
elim (H17 (lt_O_Sn _)); assumption.
- rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl in |- *; apply lt_O_Sn.
+ rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl; apply lt_O_Sn.
Qed.
Lemma StepFun_P12 :
forall (a b:R) (f:R -> R) (l lf:Rlist),
adapted_couple_opt f a b l lf -> adapted_couple_opt f b a l lf.
Proof.
- unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; intros;
+ unfold adapted_couple_opt; unfold adapted_couple; intros;
decompose [and] H; clear H; repeat split; try assumption.
- rewrite H0; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ rewrite H0; unfold Rmin; case (Rle_dec a b); intro;
case (Rle_dec b a); intro; try reflexivity.
apply Rle_antisym; assumption.
apply Rle_antisym; auto with real.
- rewrite H3; unfold Rmax in |- *; case (Rle_dec a b); intro;
+ rewrite H3; unfold Rmax; case (Rle_dec a b); intro;
case (Rle_dec b a); intro; try reflexivity.
apply Rle_antisym; assumption.
apply Rle_antisym; auto with real.
@@ -691,10 +689,10 @@ Proof.
case (Req_dec a b); intro.
rewrite (StepFun_P8 H2 H4); rewrite (StepFun_P8 H H4); reflexivity.
assert (Hyp_min : Rmin a b = a).
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (Hyp_max : Rmax a b = b).
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
elim (RList_P20 _ (StepFun_P9 H H4)); intros s1 [s2 [s3 H5]]; rewrite H5 in H;
rewrite H5; induction lf1 as [| r3 lf1 Hreclf1].
@@ -718,34 +716,34 @@ Proof.
rewrite <- (H20 (lt_O_Sn _) x).
reflexivity.
assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro;
- [ idtac | elim H7; assumption ]; unfold x in |- *;
+ [ idtac | elim H7; assumption ]; unfold x;
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
apply Rplus_lt_compat_l; apply H
| discrR ] ].
rewrite <- H6; assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21;
- intro; [ idtac | elim H7; assumption ]; unfold x in |- *;
+ intro; [ idtac | elim H7; assumption ]; unfold x;
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
| discrR ] ].
apply Rlt_le_trans with r1;
[ apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
apply Rplus_lt_compat_l; apply H
@@ -754,64 +752,64 @@ Proof.
eapply StepFun_P13.
apply H4.
apply H2.
- unfold adapted_couple_opt in |- *; split.
+ unfold adapted_couple_opt; split.
apply H.
rewrite H5 in H3; apply H3.
assert (H8 : r1 <= s2).
eapply StepFun_P13.
apply H4.
apply H2.
- unfold adapted_couple_opt in |- *; split.
+ unfold adapted_couple_opt; split.
apply H.
rewrite H5 in H3; apply H3.
elim H7; intro.
- simpl in |- *; elim H8; intro.
+ simpl; elim H8; intro.
replace (r4 * (s2 - s1)) with (r3 * (r1 - r) + r3 * (s2 - r1));
[ idtac | rewrite H9; rewrite H6; ring ].
rewrite Rplus_assoc; apply Rplus_eq_compat_l;
change
(Int_SF lf1 (cons r1 r2) = Int_SF (cons r3 lf2) (cons r1 (cons s2 s3)))
- in |- *; apply H0 with r1 b.
+ ; apply H0 with r1 b.
unfold adapted_couple in H2; decompose [and] H2; clear H2;
replace b with (Rmax a b).
rewrite <- H12; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ].
+ [ assumption | simpl; right; left; reflexivity ].
eapply StepFun_P7.
apply H1.
apply H2.
- unfold adapted_couple_opt in |- *; split.
+ unfold adapted_couple_opt; split.
apply StepFun_P7 with a a r3.
apply H1.
unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
clear H H2; assert (H20 : r = a).
simpl in H13; rewrite H13; apply Hyp_min.
- unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci].
- simpl in |- *; rewrite <- H20; apply (H11 0%nat).
- simpl in |- *; apply lt_O_Sn.
+ unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H; induction i as [| i Hreci].
+ simpl; rewrite <- H20; apply (H11 0%nat).
+ simpl; apply lt_O_Sn.
induction i as [| i Hreci0].
- simpl in |- *; assumption.
- change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *;
- apply (H15 (S i)); simpl in |- *; apply lt_S_n; assumption.
- simpl in |- *; symmetry in |- *; apply Hyp_min.
+ simpl; assumption.
+ change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
+ apply (H15 (S i)); simpl; apply lt_S_n; assumption.
+ simpl; symmetry ; apply Hyp_min.
rewrite <- H17; reflexivity.
- simpl in H19; simpl in |- *; rewrite H19; reflexivity.
- intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+ simpl in H19; simpl; rewrite H19; reflexivity.
+ intros; simpl in H; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; apply (H16 0%nat).
- simpl in |- *; apply lt_O_Sn.
- simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *;
- simpl in |- *; apply H2.
+ simpl; apply (H16 0%nat).
+ simpl; apply lt_O_Sn.
+ simpl in H2; rewrite <- H20 in H2; unfold open_interval;
+ simpl; apply H2.
clear Hreci; induction i as [| i Hreci].
- simpl in |- *; simpl in H2; rewrite H9; apply (H21 0%nat).
- simpl in |- *; apply lt_O_Sn.
- unfold open_interval in |- *; simpl in |- *; elim H2; intros; split.
+ simpl; simpl in H2; rewrite H9; apply (H21 0%nat).
+ simpl; apply lt_O_Sn.
+ unfold open_interval; simpl; elim H2; intros; split.
apply Rle_lt_trans with r1; try assumption; rewrite <- H6; apply (H11 0%nat);
- simpl in |- *; apply lt_O_Sn.
+ simpl; apply lt_O_Sn.
assumption.
- clear Hreci; simpl in |- *; apply (H21 (S i)).
- simpl in |- *; apply lt_S_n; assumption.
- unfold open_interval in |- *; apply H2.
+ clear Hreci; simpl; apply (H21 (S i)).
+ simpl; apply lt_S_n; assumption.
+ unfold open_interval; apply H2.
elim H3; clear H3; intros; split.
rewrite H9;
change
@@ -819,64 +817,64 @@ Proof.
(i < pred (Rlength (cons r4 lf2)))%nat ->
pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/
f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r4 lf2) i)
- in |- *; rewrite <- H5; apply H3.
+ ; rewrite <- H5; apply H3.
rewrite H5 in H11; intros; simpl in H12; induction i as [| i Hreci].
- simpl in |- *; red in |- *; intro; rewrite H13 in H10;
+ simpl; red; intro; rewrite H13 in H10;
elim (Rlt_irrefl _ H10).
- clear Hreci; apply (H11 (S i)); simpl in |- *; apply H12.
+ clear Hreci; apply (H11 (S i)); simpl; apply H12.
rewrite H9; rewrite H10; rewrite H6; apply Rplus_eq_compat_l; rewrite <- H10;
apply H0 with r1 b.
unfold adapted_couple in H2; decompose [and] H2; clear H2;
replace b with (Rmax a b).
rewrite <- H12; apply RList_P7;
- [ assumption | simpl in |- *; right; left; reflexivity ].
+ [ assumption | simpl; right; left; reflexivity ].
eapply StepFun_P7.
apply H1.
apply H2.
- unfold adapted_couple_opt in |- *; split.
+ unfold adapted_couple_opt; split.
apply StepFun_P7 with a a r3.
apply H1.
unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
clear H H2; assert (H20 : r = a).
simpl in H13; rewrite H13; apply Hyp_min.
- unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci].
- simpl in |- *; rewrite <- H20; apply (H11 0%nat); simpl in |- *;
+ unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H; induction i as [| i Hreci].
+ simpl; rewrite <- H20; apply (H11 0%nat); simpl;
apply lt_O_Sn.
- rewrite H10; apply (H15 (S i)); simpl in |- *; assumption.
- simpl in |- *; symmetry in |- *; apply Hyp_min.
+ rewrite H10; apply (H15 (S i)); simpl; assumption.
+ simpl; symmetry ; apply Hyp_min.
rewrite <- H17; rewrite H10; reflexivity.
- simpl in H19; simpl in |- *; apply H19.
- intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+ simpl in H19; simpl; apply H19.
+ intros; simpl in H; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; apply (H16 0%nat).
- simpl in |- *; apply lt_O_Sn.
- simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *;
- simpl in |- *; apply H2.
- clear Hreci; simpl in |- *; apply (H21 (S i)).
- simpl in |- *; assumption.
- rewrite <- H10; unfold open_interval in |- *; apply H2.
+ simpl; apply (H16 0%nat).
+ simpl; apply lt_O_Sn.
+ simpl in H2; rewrite <- H20 in H2; unfold open_interval;
+ simpl; apply H2.
+ clear Hreci; simpl; apply (H21 (S i)).
+ simpl; assumption.
+ rewrite <- H10; unfold open_interval; apply H2.
elim H3; clear H3; intros; split.
rewrite H5 in H3; intros; apply (H3 (S i)).
- simpl in |- *; replace (Rlength lf2) with (S (pred (Rlength lf2))).
+ simpl; replace (Rlength lf2) with (S (pred (Rlength lf2))).
apply lt_n_S; apply H12.
- symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+ symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
intro; rewrite <- H13 in H12; elim (lt_n_O _ H12).
intros; simpl in H12; rewrite H10; rewrite H5 in H11; apply (H11 (S i));
- simpl in |- *; apply lt_n_S; apply H12.
- simpl in |- *; rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r;
+ simpl; apply lt_n_S; apply H12.
+ simpl; rewrite H9; unfold Rminus; rewrite Rplus_opp_r;
rewrite Rmult_0_r; rewrite Rplus_0_l;
change
(Int_SF lf1 (cons r1 r2) = Int_SF (cons r4 lf2) (cons s1 (cons s2 s3)))
- in |- *; eapply H0.
+ ; eapply H0.
apply H1.
- 2: rewrite H5 in H3; unfold adapted_couple_opt in |- *; split; assumption.
+ 2: rewrite H5 in H3; unfold adapted_couple_opt; split; assumption.
assert (H10 : r = a).
unfold adapted_couple in H2; decompose [and] H2; clear H2; simpl in H12;
rewrite H12; apply Hyp_min.
rewrite <- H9; rewrite H10; apply StepFun_P7 with a r r3;
[ apply H1
- | pattern a at 2 in |- *; rewrite <- H10; pattern r at 2 in |- *; rewrite H9;
+ | pattern a at 2; rewrite <- H10; pattern r at 2; rewrite H9;
apply H2 ].
Qed.
@@ -920,12 +918,12 @@ Qed.
Lemma StepFun_P18 :
forall a b c:R, RiemannInt_SF (mkStepFun (StepFun_P4 a b c)) = c * (b - a).
Proof.
- intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+ intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
replace
(Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
(subdivision (mkStepFun (StepFun_P4 a b c)))) with
(Int_SF (cons c nil) (cons a (cons b nil)));
- [ simpl in |- *; ring
+ [ simpl; ring
| apply StepFun_P17 with (fct_cte c) a b;
[ apply StepFun_P3; assumption
| apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].
@@ -933,7 +931,7 @@ Proof.
(Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
(subdivision (mkStepFun (StepFun_P4 a b c)))) with
(Int_SF (cons c nil) (cons b (cons a nil)));
- [ simpl in |- *; ring
+ [ simpl; ring
| apply StepFun_P17 with (fct_cte c) a b;
[ apply StepFun_P2; apply StepFun_P3; auto with real
| apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].
@@ -945,8 +943,8 @@ Lemma StepFun_P19 :
Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1.
Proof.
intros; induction l1 as [| r l1 Hrecl1];
- [ simpl in |- *; ring
- | induction l1 as [| r0 l1 Hrecl0]; simpl in |- *;
+ [ simpl; ring
+ | induction l1 as [| r0 l1 Hrecl0]; simpl;
[ ring | simpl in Hrecl1; rewrite Hrecl1; ring ] ].
Qed.
@@ -956,38 +954,38 @@ Lemma StepFun_P20 :
Proof.
intros l f H; induction l;
[ elim (lt_irrefl _ H)
- | simpl in |- *; rewrite RList_P18; rewrite RList_P14; reflexivity ].
+ | simpl; rewrite RList_P18; rewrite RList_P14; reflexivity ].
Qed.
Lemma StepFun_P21 :
forall (a b:R) (f:R -> R) (l:Rlist),
is_subdivision f a b l -> adapted_couple f a b l (FF l f).
Proof.
- intros; unfold adapted_couple in |- *; unfold is_subdivision in X;
+ intros; unfold adapted_couple; unfold is_subdivision in X;
unfold adapted_couple in X; elim X; clear X; intros;
decompose [and] p; clear p; repeat split; try assumption.
apply StepFun_P20; rewrite H2; apply lt_O_Sn.
intros; assert (H5 := H4 _ H3); unfold constant_D_eq, open_interval in H5;
- unfold constant_D_eq, open_interval in |- *; intros;
+ unfold constant_D_eq, open_interval; intros;
induction l as [| r l Hrecl].
discriminate.
- unfold FF in |- *; rewrite RList_P12.
- simpl in |- *;
- change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i))) in |- *;
+ unfold FF; rewrite RList_P12.
+ simpl;
+ change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i)));
rewrite RList_P13; try assumption; rewrite (H5 x0 H6);
rewrite H5.
reflexivity.
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; elim H6;
intros; apply Rlt_trans with x0; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
rewrite (Rplus_comm (pos_Rl (cons r l) i));
@@ -1003,22 +1001,22 @@ Lemma StepFun_P22 :
is_subdivision f a b lf ->
is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg).
Proof.
- unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0;
+ unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (Hyp_max : Rmax a b = b).
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
apply existT with (FF (cons_ORlist lf lg) f); unfold adapted_couple in p, p0;
decompose [and] p; decompose [and] p0; clear p p0;
rewrite Hyp_min in H6; rewrite Hyp_min in H1; rewrite Hyp_max in H0;
- rewrite Hyp_max in H5; unfold adapted_couple in |- *;
+ rewrite Hyp_max in H5; unfold adapted_couple;
repeat split.
apply RList_P2; assumption.
- rewrite Hyp_min; symmetry in |- *; apply Rle_antisym.
+ rewrite Hyp_min; symmetry ; apply Rle_antisym.
induction lf as [| r lf Hreclf].
- simpl in |- *; right; symmetry in |- *; assumption.
+ simpl; right; symmetry ; assumption.
assert
(H10 :
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).
@@ -1026,7 +1024,7 @@ Proof.
(RList_P3 (cons_ORlist (cons r lf) lg)
(pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
apply H10; exists 0%nat; split;
- [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ].
+ [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].
elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
intros H12 _; assert (H13 := H12 H10); elim H13; intro.
elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
@@ -1039,16 +1037,16 @@ Proof.
clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
intros; apply H17; [ assumption | apply le_O_n | assumption ].
induction lf as [| r lf Hreclf].
- simpl in |- *; right; assumption.
+ simpl; right; assumption.
assert (H8 : In a (cons_ORlist (cons r lf) lg)).
elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
elim (RList_P3 (cons r lf) a); intros; apply H12;
exists 0%nat; split;
- [ symmetry in |- *; assumption | simpl in |- *; apply lt_O_Sn ].
+ [ symmetry ; assumption | simpl; apply lt_O_Sn ].
apply RList_P5; [ apply RList_P2; assumption | assumption ].
rewrite Hyp_max; apply Rle_antisym.
induction lf as [| r lf Hreclf].
- simpl in |- *; right; assumption.
+ simpl; right; assumption.
assert
(H8 :
In
@@ -1061,7 +1059,7 @@ Proof.
(pred (Rlength (cons_ORlist (cons r lf) lg)))));
intros _ H10; apply H10;
exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
- split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ].
+ split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].
elim
(RList_P9 (cons r lf) lg
(pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1076,8 +1074,8 @@ Proof.
elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
elim (RList_P6 (cons r lf)); intros; apply H17;
[ assumption
- | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption
- | simpl in |- *; apply lt_n_Sn ].
+ | simpl; simpl in H14; apply lt_n_Sm_le; assumption
+ | simpl; apply lt_n_Sn ].
elim
(RList_P3 lg
(pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1085,23 +1083,23 @@ Proof.
intros H13 _; assert (H14 := H13 H12); elim H14; intros;
elim H15; clear H15; intros.
rewrite H15; assert (H17 : Rlength lg = S (pred (Rlength lg))).
- apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+ apply S_pred with 0%nat; apply neq_O_lt; red; intro;
rewrite <- H17 in H16; elim (lt_n_O _ H16).
rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
[ assumption
| rewrite H17 in H16; apply lt_n_Sm_le; assumption
| apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].
induction lf as [| r lf Hreclf].
- simpl in |- *; right; symmetry in |- *; assumption.
+ simpl; right; symmetry ; assumption.
assert (H8 : In b (cons_ORlist (cons r lf) lg)).
elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
elim (RList_P3 (cons r lf) b); intros; apply H12;
exists (pred (Rlength (cons r lf))); split;
- [ symmetry in |- *; assumption | simpl in |- *; apply lt_n_Sn ].
+ [ symmetry ; assumption | simpl; apply lt_n_Sn ].
apply RList_P7; [ apply RList_P2; assumption | assumption ].
- apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl in |- *;
+ apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl;
apply lt_O_Sn.
- intros; unfold constant_D_eq, open_interval in |- *; intros;
+ intros; unfold constant_D_eq, open_interval; intros;
cut
(exists l : R,
constant_D_eq f
@@ -1111,10 +1109,10 @@ Proof.
assert
(Hyp_cons :
exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).
- apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
+ apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
- unfold FF in |- *; rewrite RList_P12.
- change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *;
+ unfold FF; rewrite RList_P12.
+ change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
rewrite <- Hyp_cons; rewrite RList_P13.
assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.
unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
@@ -1126,13 +1124,13 @@ Proof.
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
@@ -1151,7 +1149,7 @@ Proof.
apply le_O_n.
apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
[ assumption
- | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+ | apply lt_pred_n_n; apply neq_O_lt; red; intro;
rewrite <- H13 in H8; elim (lt_n_O _ H8) ].
assumption.
assumption.
@@ -1162,7 +1160,7 @@ Proof.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.
apply RList_P2; assumption.
apply lt_n_Sm_le; apply lt_n_S; assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
elim (lt_n_O _ H8).
rewrite H0; assumption.
set
@@ -1170,24 +1168,24 @@ Proof.
fun j:nat =>
pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lf)%nat);
assert (H12 : Nbound I).
- unfold Nbound in |- *; exists (Rlength lf); intros; unfold I in H12; elim H12;
+ unfold Nbound; exists (Rlength lf); intros; unfold I in H12; elim H12;
intros; apply lt_le_weak; assumption.
assert (H13 : exists n : nat, I n).
- exists 0%nat; unfold I in |- *; split.
+ exists 0%nat; unfold I; split.
apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).
- right; symmetry in |- *.
+ right; symmetry .
apply RList_P15; try assumption; rewrite H1; assumption.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13.
apply RList_P2; assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength (cons_ORlist lf lg))).
assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H15 in H8;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H15 in H8;
elim (lt_n_O _ H8).
- apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H5;
+ apply neq_O_lt; red; intro; rewrite <- H13 in H5;
rewrite <- H6 in H11; rewrite <- H5 in H11; elim (Rlt_irrefl _ H11).
assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
- exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval in |- *;
+ exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval;
intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (Rlength lf))%nat).
elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
apply lt_S_n; replace (S (pred (Rlength lf))) with (Rlength lf).
@@ -1205,11 +1203,11 @@ Proof.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.
apply RList_P2; assumption.
apply lt_n_Sm_le; apply lt_n_S; assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
elim (lt_n_O _ H8).
right; apply RList_P16; try assumption; rewrite H0; assumption.
rewrite <- H20; reflexivity.
- apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+ apply S_pred with 0%nat; apply neq_O_lt; red; intro;
rewrite <- H19 in H18; elim (lt_n_O _ H18).
assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
rewrite (H18 x1).
@@ -1221,11 +1219,11 @@ Proof.
assert (H22 : (S x0 < Rlength lf)%nat).
replace (Rlength lf) with (S (pred (Rlength lf)));
[ apply lt_n_S; assumption
- | symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+ | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
intro; rewrite <- H22 in H21; elim (lt_n_O _ H21) ].
elim (Rle_dec (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro.
assert (H23 : (S x0 <= x0)%nat).
- apply H20; unfold I in |- *; split; assumption.
+ apply H20; unfold I; split; assumption.
elim (le_Sn_n _ H23).
assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)).
auto with real.
@@ -1255,22 +1253,22 @@ Lemma StepFun_P24 :
is_subdivision f a b lf ->
is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg).
Proof.
- unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0;
+ unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (Hyp_max : Rmax a b = b).
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
apply existT with (FF (cons_ORlist lf lg) g); unfold adapted_couple in p, p0;
decompose [and] p; decompose [and] p0; clear p p0;
rewrite Hyp_min in H1; rewrite Hyp_min in H6; rewrite Hyp_max in H0;
- rewrite Hyp_max in H5; unfold adapted_couple in |- *;
+ rewrite Hyp_max in H5; unfold adapted_couple;
repeat split.
apply RList_P2; assumption.
- rewrite Hyp_min; symmetry in |- *; apply Rle_antisym.
+ rewrite Hyp_min; symmetry ; apply Rle_antisym.
induction lf as [| r lf Hreclf].
- simpl in |- *; right; symmetry in |- *; assumption.
+ simpl; right; symmetry ; assumption.
assert
(H10 :
In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).
@@ -1278,7 +1276,7 @@ Proof.
(RList_P3 (cons_ORlist (cons r lf) lg)
(pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
apply H10; exists 0%nat; split;
- [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ].
+ [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].
elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
intros H12 _; assert (H13 := H12 H10); elim H13; intro.
elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
@@ -1291,16 +1289,16 @@ Proof.
clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
intros; apply H17; [ assumption | apply le_O_n | assumption ].
induction lf as [| r lf Hreclf].
- simpl in |- *; right; assumption.
+ simpl; right; assumption.
assert (H8 : In a (cons_ORlist (cons r lf) lg)).
elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
elim (RList_P3 (cons r lf) a); intros; apply H12;
exists 0%nat; split;
- [ symmetry in |- *; assumption | simpl in |- *; apply lt_O_Sn ].
+ [ symmetry ; assumption | simpl; apply lt_O_Sn ].
apply RList_P5; [ apply RList_P2; assumption | assumption ].
rewrite Hyp_max; apply Rle_antisym.
induction lf as [| r lf Hreclf].
- simpl in |- *; right; assumption.
+ simpl; right; assumption.
assert
(H8 :
In
@@ -1313,7 +1311,7 @@ Proof.
(pred (Rlength (cons_ORlist (cons r lf) lg)))));
intros _ H10; apply H10;
exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
- split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ].
+ split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].
elim
(RList_P9 (cons r lf) lg
(pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1327,8 +1325,8 @@ Proof.
elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
elim (RList_P6 (cons r lf)); intros; apply H17;
[ assumption
- | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption
- | simpl in |- *; apply lt_n_Sn ].
+ | simpl; simpl in H14; apply lt_n_Sm_le; assumption
+ | simpl; apply lt_n_Sn ].
elim
(RList_P3 lg
(pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1336,23 +1334,23 @@ Proof.
intros H13 _; assert (H14 := H13 H12); elim H14; intros;
elim H15; clear H15; intros; rewrite H15;
assert (H17 : Rlength lg = S (pred (Rlength lg))).
- apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+ apply S_pred with 0%nat; apply neq_O_lt; red; intro;
rewrite <- H17 in H16; elim (lt_n_O _ H16).
rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
[ assumption
| rewrite H17 in H16; apply lt_n_Sm_le; assumption
| apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].
induction lf as [| r lf Hreclf].
- simpl in |- *; right; symmetry in |- *; assumption.
+ simpl; right; symmetry ; assumption.
assert (H8 : In b (cons_ORlist (cons r lf) lg)).
elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
elim (RList_P3 (cons r lf) b); intros; apply H12;
exists (pred (Rlength (cons r lf))); split;
- [ symmetry in |- *; assumption | simpl in |- *; apply lt_n_Sn ].
+ [ symmetry ; assumption | simpl; apply lt_n_Sn ].
apply RList_P7; [ apply RList_P2; assumption | assumption ].
- apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl in |- *;
+ apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl;
apply lt_O_Sn.
- unfold constant_D_eq, open_interval in |- *; intros;
+ unfold constant_D_eq, open_interval; intros;
cut
(exists l : R,
constant_D_eq g
@@ -1362,10 +1360,10 @@ Proof.
assert
(Hyp_cons :
exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).
- apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
+ apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
- unfold FF in |- *; rewrite RList_P12.
- change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *;
+ unfold FF; rewrite RList_P12.
+ change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
rewrite <- Hyp_cons; rewrite RList_P13.
assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.
unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
@@ -1377,13 +1375,13 @@ Proof.
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
@@ -1402,7 +1400,7 @@ Proof.
apply le_O_n.
apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
[ assumption
- | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+ | apply lt_pred_n_n; apply neq_O_lt; red; intro;
rewrite <- H13 in H8; elim (lt_n_O _ H8) ].
rewrite H1; assumption.
apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).
@@ -1411,7 +1409,7 @@ Proof.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.
apply RList_P2; assumption.
apply lt_n_Sm_le; apply lt_n_S; assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
elim (lt_n_O _ H8).
rewrite H0; assumption.
set
@@ -1419,24 +1417,24 @@ Proof.
fun j:nat =>
pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lg)%nat);
assert (H12 : Nbound I).
- unfold Nbound in |- *; exists (Rlength lg); intros; unfold I in H12; elim H12;
+ unfold Nbound; exists (Rlength lg); intros; unfold I in H12; elim H12;
intros; apply lt_le_weak; assumption.
assert (H13 : exists n : nat, I n).
- exists 0%nat; unfold I in |- *; split.
+ exists 0%nat; unfold I; split.
apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).
- right; symmetry in |- *; rewrite H1; rewrite <- H6; apply RList_P15;
+ right; symmetry ; rewrite H1; rewrite <- H6; apply RList_P15;
try assumption; rewrite H1; assumption.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13;
[ apply RList_P2; assumption
| apply le_O_n
| apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
[ assumption
- | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+ | apply lt_pred_n_n; apply neq_O_lt; red; intro;
rewrite <- H15 in H8; elim (lt_n_O _ H8) ] ].
- apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H0;
+ apply neq_O_lt; red; intro; rewrite <- H13 in H0;
rewrite <- H1 in H11; rewrite <- H0 in H11; elim (Rlt_irrefl _ H11).
assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
- exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval in |- *;
+ exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval;
intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (Rlength lg))%nat).
elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
apply lt_S_n; replace (S (pred (Rlength lg))) with (Rlength lg).
@@ -1454,12 +1452,12 @@ Proof.
elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.
apply RList_P2; assumption.
apply lt_n_Sm_le; apply lt_n_S; assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
elim (lt_n_O _ H8).
right; rewrite H0; rewrite <- H5; apply RList_P16; try assumption.
rewrite H0; assumption.
rewrite <- H20; reflexivity.
- apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+ apply S_pred with 0%nat; apply neq_O_lt; red; intro;
rewrite <- H19 in H18; elim (lt_n_O _ H18).
assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
rewrite (H18 x1).
@@ -1471,11 +1469,11 @@ Proof.
assert (H22 : (S x0 < Rlength lg)%nat).
replace (Rlength lg) with (S (pred (Rlength lg))).
apply lt_n_S; assumption.
- symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+ symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
intro; rewrite <- H22 in H21; elim (lt_n_O _ H21).
elim (Rle_dec (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro.
assert (H23 : (S x0 <= x0)%nat);
- [ apply H20; unfold I in |- *; split; assumption | elim (le_Sn_n _ H23) ].
+ [ apply H20; unfold I; split; assumption | elim (le_Sn_n _ H23) ].
assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)).
auto with real.
clear b0; apply RList_P17; try assumption;
@@ -1511,35 +1509,35 @@ Proof.
intros i H8 x1 H10; unfold open_interval in H10, H9, H4;
rewrite (H9 _ H8 _ H10); rewrite (H4 _ H8 _ H10);
assert (H11 : l1 <> nil).
- red in |- *; intro H11; rewrite H11 in H8; elim (lt_n_O _ H8).
+ red; intro H11; rewrite H11 in H8; elim (lt_n_O _ H8).
destruct (RList_P19 _ H11) as (r,(r0,H12));
- rewrite H12; unfold FF in |- *;
+ rewrite H12; unfold FF;
change
(pos_Rl x0 i + l * pos_Rl x i =
pos_Rl
(app_Rlist (mid_Rlist (cons r r0) r) (fun x2:R => f x2 + l * g x2))
- (S i)) in |- *; rewrite RList_P12.
+ (S i)); rewrite RList_P12.
rewrite RList_P13.
rewrite <- H12; rewrite (H9 _ H8); try rewrite (H4 _ H8);
reflexivity ||
(elim H10; clear H10; intros; split;
[ apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
apply Rlt_trans with x1; assumption
| discrR ] ]
| apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double;
rewrite (Rplus_comm (pos_Rl l1 i)); apply Rplus_lt_compat_l;
apply Rlt_trans with x1; assumption
| discrR ] ] ]).
rewrite <- H12; assumption.
- rewrite RList_P14; simpl in |- *; rewrite H12 in H8; simpl in H8;
+ rewrite RList_P14; simpl; rewrite H12 in H8; simpl in H8;
apply lt_n_S; apply H8.
Qed.
@@ -1558,7 +1556,7 @@ Qed.
Lemma StepFun_P28 :
forall (a b l:R) (f g:StepFun a b), IsStepFun (fun x:R => f x + l * g x) a b.
Proof.
- intros a b l f g; unfold IsStepFun in |- *; assert (H := pre f);
+ intros a b l f g; unfold IsStepFun; assert (H := pre f);
assert (H0 := pre g); unfold IsStepFun in H, H0; elim H;
elim H0; intros; apply existT with (cons_ORlist x0 x);
apply StepFun_P27; assumption.
@@ -1567,7 +1565,7 @@ Qed.
Lemma StepFun_P29 :
forall (a b:R) (f:StepFun a b), is_subdivision f a b (subdivision f).
Proof.
- intros a b f; unfold is_subdivision in |- *;
+ intros a b f; unfold is_subdivision;
apply existT with (subdivision_val f); apply StepFun_P1.
Qed.
@@ -1576,7 +1574,7 @@ Lemma StepFun_P30 :
RiemannInt_SF (mkStepFun (StepFun_P28 l f g)) =
RiemannInt_SF f + l * RiemannInt_SF g.
Proof.
- intros a b l f g; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
+ intros a b l f g; unfold RiemannInt_SF; case (Rle_dec a b);
(intro;
replace
(Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g)))
@@ -1613,10 +1611,10 @@ Lemma StepFun_P31 :
adapted_couple f a b l lf ->
adapted_couple (fun x:R => Rabs (f x)) a b l (app_Rlist lf Rabs).
Proof.
- unfold adapted_couple in |- *; intros; decompose [and] H; clear H;
+ unfold adapted_couple; intros; decompose [and] H; clear H;
repeat split; try assumption.
- symmetry in |- *; rewrite H3; rewrite RList_P18; reflexivity.
- intros; unfold constant_D_eq, open_interval in |- *;
+ symmetry ; rewrite H3; rewrite RList_P18; reflexivity.
+ intros; unfold constant_D_eq, open_interval;
unfold constant_D_eq, open_interval in H5; intros;
rewrite (H5 _ H _ H4); rewrite RList_P12;
[ reflexivity | rewrite H3 in H; simpl in H; apply H ].
@@ -1625,8 +1623,8 @@ Qed.
Lemma StepFun_P32 :
forall (a b:R) (f:StepFun a b), IsStepFun (fun x:R => Rabs (f x)) a b.
Proof.
- intros a b f; unfold IsStepFun in |- *; apply existT with (subdivision f);
- unfold is_subdivision in |- *;
+ intros a b f; unfold IsStepFun; apply existT with (subdivision f);
+ unfold is_subdivision;
apply existT with (app_Rlist (subdivision_val f) Rabs);
apply StepFun_P31; apply StepFun_P1.
Qed.
@@ -1636,8 +1634,8 @@ Lemma StepFun_P33 :
ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1.
Proof.
simple induction l2; intros.
- simpl in |- *; rewrite Rabs_R0; right; reflexivity.
- simpl in |- *; induction l1 as [| r1 l1 Hrecl1].
+ simpl; rewrite Rabs_R0; right; reflexivity.
+ simpl; induction l1 as [| r1 l1 Hrecl1].
rewrite Rabs_R0; right; reflexivity.
induction l1 as [| r2 l1 Hrecl0].
rewrite Rabs_R0; right; reflexivity.
@@ -1645,7 +1643,7 @@ Proof.
apply Rabs_triang.
rewrite Rabs_mult; rewrite (Rabs_right (r2 - r1));
[ apply Rplus_le_compat_l; apply H; apply RList_P4 with r1; assumption
- | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *;
+ | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
apply lt_O_Sn ].
Qed.
@@ -1654,7 +1652,7 @@ Lemma StepFun_P34 :
a <= b ->
Rabs (RiemannInt_SF f) <= RiemannInt_SF (mkStepFun (StepFun_P32 f)).
Proof.
- intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+ intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
replace
(Int_SF (subdivision_val (mkStepFun (StepFun_P32 f)))
(subdivision (mkStepFun (StepFun_P32 f)))) with
@@ -1678,18 +1676,18 @@ Lemma StepFun_P35 :
Proof.
simple induction l; intros.
right; reflexivity.
- simpl in |- *; induction r0 as [| r0 r1 Hrecr0].
+ simpl; induction r0 as [| r0 r1 Hrecr0].
right; reflexivity.
- simpl in |- *; apply Rplus_le_compat.
+ simpl; apply Rplus_le_compat.
case (Req_dec r r0); intro.
rewrite H4; right; ring.
do 2 rewrite <- (Rmult_comm (r0 - r)); apply Rmult_le_compat_l.
- apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *;
+ apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
apply lt_O_Sn.
apply H3; split.
apply Rmult_lt_reg_l with 2.
prove_sup0.
- unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
assert (H5 : r = a).
apply H1.
@@ -1702,7 +1700,7 @@ Proof.
discrR.
apply Rmult_lt_reg_l with 2.
prove_sup0.
- unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l; rewrite double; assert (H5 : r0 <= b).
replace b with
@@ -1710,9 +1708,9 @@ Proof.
replace r0 with (pos_Rl (cons r (cons r0 r1)) 1).
elim (RList_P6 (cons r (cons r0 r1))); intros; apply H5.
assumption.
- simpl in |- *; apply le_n_S.
+ simpl; apply le_n_S.
apply le_O_n.
- simpl in |- *; apply lt_n_Sn.
+ simpl; apply lt_n_Sn.
reflexivity.
apply Rle_lt_trans with (r + b).
apply Rplus_le_compat_l; assumption.
@@ -1732,7 +1730,7 @@ Proof.
intros; apply H3; elim H4; intros; split; try assumption.
apply Rle_lt_trans with r0; try assumption.
rewrite <- H1.
- simpl in |- *; apply (H0 0%nat); simpl in |- *; apply lt_O_Sn.
+ simpl; apply (H0 0%nat); simpl; apply lt_O_Sn.
Qed.
Lemma StepFun_P36 :
@@ -1743,16 +1741,16 @@ Lemma StepFun_P36 :
(forall x:R, a < x < b -> f x <= g x) ->
RiemannInt_SF f <= RiemannInt_SF g.
Proof.
- intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+ intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l).
replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l).
unfold is_subdivision in X; elim X; clear X; intros;
unfold adapted_couple in p; decompose [and] p; clear p;
assert (H5 : Rmin a b = a);
- [ unfold Rmin in |- *; case (Rle_dec a b); intro;
+ [ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ]
| assert (H7 : Rmax a b = b);
- [ unfold Rmax in |- *; case (Rle_dec a b); intro;
+ [ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ]
| rewrite H5 in H3; rewrite H7 in H2; eapply StepFun_P35 with a b;
assumption ] ].
@@ -1811,27 +1809,27 @@ Proof.
assert (H7 : r1 <= b).
rewrite <- H4; apply RList_P7; [ assumption | left; reflexivity ].
assert (H8 : IsStepFun g' a b).
- unfold IsStepFun in |- *; assert (H8 := pre g); unfold IsStepFun in H8;
+ unfold IsStepFun; assert (H8 := pre g); unfold IsStepFun in H8;
elim H8; intros lg H9; unfold is_subdivision in H9;
elim H9; clear H9; intros lg2 H9; split with (cons a lg);
- unfold is_subdivision in |- *; split with (cons (f a) lg2);
+ unfold is_subdivision; split with (cons (f a) lg2);
unfold adapted_couple in H9; decompose [and] H9; clear H9;
- unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H9;
+ unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H9;
induction i as [| i Hreci].
- simpl in |- *; rewrite H12; replace (Rmin r1 b) with r1.
- simpl in H0; rewrite <- H0; apply (H 0%nat); simpl in |- *; apply lt_O_Sn.
- unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+ simpl; rewrite H12; replace (Rmin r1 b) with r1.
+ simpl in H0; rewrite <- H0; apply (H 0%nat); simpl; apply lt_O_Sn.
+ unfold Rmin; case (Rle_dec r1 b); intro;
[ reflexivity | elim n; assumption ].
apply (H10 i); apply lt_S_n.
replace (S (pred (Rlength lg))) with (Rlength lg).
apply H9.
apply S_pred with 0%nat; apply neq_O_lt; intro; rewrite <- H14 in H9;
elim (lt_n_O _ H9).
- simpl in |- *; assert (H14 : a <= b).
+ simpl; assert (H14 : a <= b).
rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
[ assumption | left; reflexivity ].
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (H14 : a <= b).
rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
@@ -1840,30 +1838,30 @@ Proof.
rewrite <- H11; induction lg as [| r0 lg Hreclg].
simpl in H13; discriminate.
reflexivity.
- unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec r1 b); intros;
+ unfold Rmax; case (Rle_dec a b); case (Rle_dec r1 b); intros;
reflexivity || elim n; assumption.
- simpl in |- *; rewrite H13; reflexivity.
+ simpl; rewrite H13; reflexivity.
intros; simpl in H9; induction i as [| i Hreci].
- unfold constant_D_eq, open_interval in |- *; simpl in |- *; intros;
+ unfold constant_D_eq, open_interval; simpl; intros;
assert (H16 : Rmin r1 b = r1).
- unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+ unfold Rmin; case (Rle_dec r1 b); intro;
[ reflexivity | elim n; assumption ].
rewrite H16 in H12; rewrite H12 in H14; elim H14; clear H14; intros _ H14;
- unfold g' in |- *; case (Rle_dec r1 x); intro r3.
+ unfold g'; case (Rle_dec r1 x); intro r3.
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H14)).
reflexivity.
change
(constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
- (pos_Rl lg2 i)) in |- *; clear Hreci; assert (H16 := H15 i);
+ (pos_Rl lg2 i)); clear Hreci; assert (H16 := H15 i);
assert (H17 : (i < pred (Rlength lg))%nat).
apply lt_S_n.
replace (S (pred (Rlength lg))) with (Rlength lg).
assumption.
- apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+ apply S_pred with 0%nat; apply neq_O_lt; red; intro;
rewrite <- H14 in H9; elim (lt_n_O _ H9).
assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
- unfold constant_D_eq, open_interval in |- *; intros;
- assert (H19 := H18 _ H14); rewrite <- H19; unfold g' in |- *;
+ unfold constant_D_eq, open_interval; intros;
+ assert (H19 := H18 _ H14); rewrite <- H19; unfold g';
case (Rle_dec r1 x); intro.
reflexivity.
elim n; replace r1 with (Rmin r1 b).
@@ -1874,17 +1872,17 @@ Proof.
elim (RList_P3 lg (pos_Rl lg i)); intros; apply H21; exists i; split.
reflexivity.
apply lt_trans with (pred (Rlength lg)); try assumption.
- apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H22 in H17;
+ apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H22 in H17;
elim (lt_n_O _ H17).
- unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+ unfold Rmin; case (Rle_dec r1 b); intro;
[ reflexivity | elim n0; assumption ].
exists (mkStepFun H8); split.
- simpl in |- *; unfold g' in |- *; case (Rle_dec r1 b); intro.
+ simpl; unfold g'; case (Rle_dec r1 b); intro.
assumption.
elim n; assumption.
intros; simpl in H9; induction i as [| i Hreci].
- unfold constant_D_eq, co_interval in |- *; simpl in |- *; intros; simpl in H0;
- rewrite H0; elim H10; clear H10; intros; unfold g' in |- *;
+ unfold constant_D_eq, co_interval; simpl; intros; simpl in H0;
+ rewrite H0; elim H10; clear H10; intros; unfold g';
case (Rle_dec r1 x); intro r3.
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H11)).
reflexivity.
@@ -1892,21 +1890,21 @@ Proof.
change
(constant_D_eq (mkStepFun H8)
(co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
- (f (pos_Rl (cons r1 l) i))) in |- *; assert (H10 := H6 i);
+ (f (pos_Rl (cons r1 l) i))); assert (H10 := H6 i);
assert (H11 : (i < pred (Rlength (cons r1 l)))%nat).
- simpl in |- *; apply lt_S_n; assumption.
+ simpl; apply lt_S_n; assumption.
assert (H12 := H10 H11); unfold constant_D_eq, co_interval in H12;
- unfold constant_D_eq, co_interval in |- *; intros;
- rewrite <- (H12 _ H13); simpl in |- *; unfold g' in |- *;
+ unfold constant_D_eq, co_interval; intros;
+ rewrite <- (H12 _ H13); simpl; unfold g';
case (Rle_dec r1 x); intro.
reflexivity.
elim n; elim H13; clear H13; intros;
apply Rle_trans with (pos_Rl (cons r1 l) i); try assumption;
- change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i) in |- *;
+ change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i);
elim (RList_P6 (cons r1 l)); intros; apply H15;
[ assumption
| apply le_O_n
- | simpl in |- *; apply lt_trans with (Rlength l);
+ | simpl; apply lt_trans with (Rlength l);
[ apply lt_S_n; assumption | apply lt_n_Sn ] ].
Qed.
@@ -1914,7 +1912,7 @@ Lemma StepFun_P39 :
forall (a b:R) (f:StepFun a b),
RiemannInt_SF f = - RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))).
Proof.
- intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); case (Rle_dec b a);
+ intros; unfold RiemannInt_SF; case (Rle_dec a b); case (Rle_dec b a);
intros.
assert (H : adapted_couple f a b (subdivision f) (subdivision_val f));
[ apply StepFun_P1
@@ -1927,16 +1925,16 @@ Proof.
| assert (H1 : a = b);
[ apply Rle_antisym; assumption
| rewrite (StepFun_P8 H H1); assert (H2 : b = a);
- [ symmetry in |- *; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ].
+ [ symmetry ; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ].
rewrite Ropp_involutive; eapply StepFun_P17;
[ apply StepFun_P1
| apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
- elim H; intros; unfold is_subdivision in |- *;
+ elim H; intros; unfold is_subdivision;
elim p; intros; apply p0 ].
apply Ropp_eq_compat; eapply StepFun_P17;
[ apply StepFun_P1
| apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
- elim H; intros; unfold is_subdivision in |- *;
+ elim H; intros; unfold is_subdivision;
elim p; intros; apply p0 ].
assert (H : a < b);
[ auto with real
@@ -1953,34 +1951,34 @@ Lemma StepFun_P40 :
adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f).
Proof.
intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; unfold adapted_couple in H1, H2;
- unfold adapted_couple in |- *; decompose [and] H1;
+ unfold adapted_couple; decompose [and] H1;
decompose [and] H2; clear H1 H2; repeat split.
apply RList_P25; try assumption.
- rewrite H10; rewrite H4; unfold Rmin, Rmax in |- *; case (Rle_dec a b);
+ rewrite H10; rewrite H4; unfold Rmin, Rmax; case (Rle_dec a b);
case (Rle_dec b c); intros;
(right; reflexivity) || (elim n; left; assumption).
rewrite RList_P22.
- rewrite H5; unfold Rmin, Rmax in |- *; case (Rle_dec a b); case (Rle_dec a c);
+ rewrite H5; unfold Rmin, Rmax; case (Rle_dec a b); case (Rle_dec a c);
intros;
[ reflexivity
| elim n; apply Rle_trans with b; left; assumption
| elim n; left; assumption
| elim n0; left; assumption ].
- red in |- *; intro; rewrite H1 in H6; discriminate.
+ red; intro; rewrite H1 in H6; discriminate.
rewrite RList_P24.
- rewrite H9; unfold Rmin, Rmax in |- *; case (Rle_dec b c); case (Rle_dec a c);
+ rewrite H9; unfold Rmin, Rmax; case (Rle_dec b c); case (Rle_dec a c);
intros;
[ reflexivity
| elim n; apply Rle_trans with b; left; assumption
| elim n; left; assumption
| elim n0; left; assumption ].
- red in |- *; intro; rewrite H1 in H11; discriminate.
+ red; intro; rewrite H1 in H11; discriminate.
apply StepFun_P20.
- rewrite RList_P23; apply neq_O_lt; red in |- *; intro.
+ rewrite RList_P23; apply neq_O_lt; red; intro.
assert (H2 : (Rlength l1 + Rlength l2)%nat = 0%nat).
- symmetry in |- *; apply H1.
+ symmetry ; apply H1.
elim (plus_is_O _ _ H2); intros; rewrite H12 in H6; discriminate.
- unfold constant_D_eq, open_interval in |- *; intros;
+ unfold constant_D_eq, open_interval; intros;
elim (le_or_lt (S (S i)) (Rlength l1)); intro.
assert (H14 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i).
apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; apply le_S_n;
@@ -1993,28 +1991,28 @@ Proof.
elim (RList_P20 _ H16); intros r1 [r2 [r3 H17]]; rewrite H17;
change
(f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
- in |- *; rewrite RList_P12.
+ ; rewrite RList_P12.
induction i as [| i Hreci].
- simpl in |- *; assert (H18 := H8 0%nat);
+ simpl; assert (H18 := H8 0%nat);
unfold constant_D_eq, open_interval in H18;
assert (H19 : (0 < pred (Rlength l1))%nat).
- rewrite H17; simpl in |- *; apply lt_O_Sn.
+ rewrite H17; simpl; apply lt_O_Sn.
assert (H20 := H18 H19); repeat rewrite H20.
reflexivity.
assert (H21 : r1 <= r2).
rewrite H17 in H3; apply (H3 0%nat).
- simpl in |- *; apply lt_O_Sn.
+ simpl; apply lt_O_Sn.
elim H21; intro.
split.
- rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2;
+ rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
- rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2;
+ rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite (Rplus_comm r1); rewrite double;
apply Rplus_lt_compat_l; assumption
@@ -2043,13 +2041,13 @@ Proof.
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l1 (S i)));
rewrite double; apply Rplus_lt_compat_l; assumption
@@ -2057,21 +2055,21 @@ Proof.
elim H2; intros; rewrite H22 in H23;
elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)).
assumption.
- simpl in |- *; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption.
+ simpl; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption.
rewrite RList_P14; rewrite H17 in H1; simpl in H1; apply H1.
inversion H12.
assert (H16 : pos_Rl (cons_Rlist l1 l2) (S i) = b).
rewrite RList_P29.
- rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin in |- *;
+ rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin;
case (Rle_dec b c); intro; [ reflexivity | elim n; left; assumption ].
rewrite H15; apply le_n.
induction l1 as [| r l1 Hrecl1].
simpl in H15; discriminate.
- clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption.
+ clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.
assert (H17 : pos_Rl (cons_Rlist l1 l2) i = b).
rewrite RList_P26.
replace i with (pred (Rlength l1));
- [ rewrite H4; unfold Rmax in |- *; case (Rle_dec a b); intro;
+ [ rewrite H4; unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; left; assumption ]
| rewrite H15; reflexivity ].
rewrite H15; apply lt_n_Sn.
@@ -2089,22 +2087,22 @@ Proof.
apply le_S_n; apply le_trans with (S i); [ assumption | apply le_n_Sn ].
induction l1 as [| r l1 Hrecl1].
simpl in H6; discriminate.
- clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption.
- symmetry in |- *; apply minus_Sn_m; apply le_S_n; assumption.
+ clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.
+ symmetry ; apply minus_Sn_m; apply le_S_n; assumption.
assert (H18 : (2 <= Rlength l1)%nat).
clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17;
induction l1 as [| r l1 Hrecl1].
discriminate.
clear Hrecl1; induction l1 as [| r0 l1 Hrecl1].
simpl in H5; simpl in H4; assert (H0 : Rmin a b < Rmax a b).
- unfold Rmin, Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmin, Rmax; case (Rle_dec a b); intro;
[ assumption | elim n; left; assumption ].
rewrite <- H5 in H0; rewrite <- H4 in H0; elim (Rlt_irrefl _ H0).
- clear Hrecl1; simpl in |- *; repeat apply le_n_S; apply le_O_n.
+ clear Hrecl1; simpl; repeat apply le_n_S; apply le_O_n.
elim (RList_P20 _ H18); intros r1 [r2 [r3 H19]]; rewrite H19;
change
(f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
- in |- *; rewrite RList_P12.
+ ; rewrite RList_P12.
induction i as [| i Hreci].
assert (H20 := le_S_n _ _ H15); assert (H21 := le_trans _ _ _ H18 H20);
elim (le_Sn_O _ H21).
@@ -2122,7 +2120,7 @@ Proof.
assert (H21 : (S i - Rlength l1 < pred (Rlength l2))%nat).
apply lt_pred; rewrite minus_Sn_m.
apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.
- rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *;
+ rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
rewrite RList_P23 in H1; apply lt_n_S; assumption.
apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].
apply le_S_n; assumption.
@@ -2134,7 +2132,7 @@ Proof.
apply H7; apply lt_pred.
rewrite minus_Sn_m.
apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.
- rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *;
+ rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
rewrite RList_P23 in H1; apply lt_n_S; assumption.
apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].
apply le_S_n; assumption.
@@ -2142,13 +2140,13 @@ Proof.
split.
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
apply Rmult_lt_reg_l with 2;
[ prove_sup0
- | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+ | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - Rlength l1)));
rewrite double; apply Rplus_lt_compat_l; assumption
@@ -2159,14 +2157,14 @@ Proof.
rewrite H17 in H26; simpl in H24; rewrite H24 in H25;
elim (Rlt_irrefl _ (Rlt_trans _ _ _ H25 H26)).
assert (H23 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S i - Rlength l1)).
- rewrite H19; simpl in |- *; simpl in H16; apply H16.
+ rewrite H19; simpl; simpl in H16; apply H16.
assert
(H24 :
pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l2 (S (S i - Rlength l1))).
- rewrite H19; simpl in |- *; simpl in H17; apply H17.
+ rewrite H19; simpl; simpl in H17; apply H17.
rewrite <- H23; rewrite <- H24; assumption.
- simpl in |- *; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption.
- rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl in |- *; apply H1.
+ simpl; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption.
+ rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl; apply H1.
Qed.
Lemma StepFun_P41 :
@@ -2191,11 +2189,11 @@ Lemma StepFun_P42 :
Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2.
Proof.
intros l1 l2 f; induction l1 as [| r l1 IHl1]; intros H;
- [ simpl in |- *; ring
+ [ simpl; ring
| destruct l1 as [| r0 r1];
- [ simpl in H; simpl in |- *; destruct l2 as [| r0 r1];
- [ simpl in |- *; ring | simpl in |- *; simpl in H; rewrite H; ring ]
- | simpl in |- *; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1;
+ [ simpl in H; simpl; destruct l2 as [| r0 r1];
+ [ simpl; ring | simpl; simpl in H; rewrite H; ring ]
+ | simpl; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1;
rewrite <- H; reflexivity ] ].
Qed.
@@ -2231,27 +2229,27 @@ Proof.
(Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)).
replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
- symmetry in |- *; apply StepFun_P42.
+ symmetry ; apply StepFun_P42.
unfold adapted_couple in H1, H2; decompose [and] H1; decompose [and] H2;
- clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+ clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
case (Rle_dec a b); case (Rle_dec b c); intros; reflexivity || elim n;
assumption.
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2;
+ [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2;
assumption
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+ [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
| assumption ].
eapply StepFun_P17; [ apply (StepFun_P40 H H0 H1 H2) | apply H3 ].
replace (Int_SF lf2 l2) with 0.
rewrite Rplus_0_r; eapply StepFun_P17;
[ apply H1 | rewrite <- H0 in H3; apply H3 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H2 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H2 | assumption ].
replace (Int_SF lf1 l1) with 0.
rewrite Rplus_0_l; eapply StepFun_P17;
[ apply H2 | rewrite H in H3; apply H3 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H1 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].
elim n; apply Rle_trans with b; assumption.
apply Rplus_eq_reg_l with (Int_SF lf2 l2);
replace (Int_SF lf2 l2 + (Int_SF lf1 l1 + - Int_SF lf2 l2)) with
@@ -2266,24 +2264,24 @@ Proof.
replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
apply StepFun_P42.
unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
- clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin in |- *;
+ clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin;
case (Rle_dec a c); case (Rle_dec b c); intros;
[ elim n; assumption
| reflexivity
| elim n0; assumption
| elim n1; assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+ [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+ [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
| assumption ].
eapply StepFun_P17;
[ apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | apply H1 ].
replace (Int_SF lf3 l3) with 0.
rewrite Rplus_0_r; eapply StepFun_P17;
[ apply H1 | apply StepFun_P2; rewrite <- H0 in H2; apply H2 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H3 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].
replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).
ring.
elim r; intro.
@@ -2291,19 +2289,19 @@ Proof.
(Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)).
replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
- symmetry in |- *; apply StepFun_P42.
+ symmetry ; apply StepFun_P42.
unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
- clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin in |- *;
+ clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin;
case (Rle_dec a c); case (Rle_dec a b); intros;
[ elim n; assumption
| elim n1; assumption
| reflexivity
| elim n1; assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+ [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+ [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
| assumption ].
eapply StepFun_P17.
assert (H0 : c < a).
@@ -2313,7 +2311,7 @@ Proof.
replace (Int_SF lf1 l1) with 0.
rewrite Rplus_0_r; eapply StepFun_P17;
[ apply H3 | rewrite <- H in H2; apply H2 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H1 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].
assert (H : b < a).
auto with real.
replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).
@@ -2323,19 +2321,19 @@ Proof.
(Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)).
replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
- symmetry in |- *; apply StepFun_P42.
+ symmetry ; apply StepFun_P42.
unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
- clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+ clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin;
case (Rle_dec a c); case (Rle_dec a b); intros;
[ elim n; assumption
| reflexivity
| elim n0; assumption
| elim n1; assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+ [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+ [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
| assumption ].
eapply StepFun_P17.
apply (StepFun_P40 H H0 (StepFun_P2 H1) H3).
@@ -2343,7 +2341,7 @@ Proof.
replace (Int_SF lf3 l3) with 0.
rewrite Rplus_0_r; eapply StepFun_P17;
[ apply H1 | rewrite <- H0 in H2; apply StepFun_P2; apply H2 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H3 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].
assert (H : c < a).
auto with real.
replace (Int_SF lf1 l1) with (Int_SF lf2 l2 + Int_SF lf3 l3).
@@ -2353,19 +2351,19 @@ Proof.
(Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)).
replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
- symmetry in |- *; apply StepFun_P42.
+ symmetry ; apply StepFun_P42.
unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
- clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+ clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
case (Rle_dec a c); case (Rle_dec b c); intros;
[ elim n; assumption
| elim n1; assumption
| reflexivity
| elim n1; assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+ [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+ [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
| assumption ].
eapply StepFun_P17.
apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)).
@@ -2373,7 +2371,7 @@ Proof.
replace (Int_SF lf2 l2) with 0.
rewrite Rplus_0_l; eapply StepFun_P17;
[ apply H3 | rewrite H0 in H1; apply H1 ].
- symmetry in |- *; eapply StepFun_P8; [ apply H2 | assumption ].
+ symmetry ; eapply StepFun_P8; [ apply H2 | assumption ].
elim n; apply Rle_trans with a; try assumption.
auto with real.
assert (H : c < b).
@@ -2386,56 +2384,56 @@ Proof.
(Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)).
replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
- symmetry in |- *; apply StepFun_P42.
+ symmetry ; apply StepFun_P42.
unfold adapted_couple in H2, H1; decompose [and] H2; decompose [and] H1;
- clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+ clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
case (Rle_dec a b); case (Rle_dec b c); intros;
[ elim n1; assumption
| elim n1; assumption
| elim n0; assumption
| reflexivity ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+ [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
| assumption ].
eapply StepFun_P17;
- [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+ [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
| assumption ].
eapply StepFun_P17.
apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)).
apply StepFun_P2; apply H3.
- unfold RiemannInt_SF in |- *; case (Rle_dec a c); intro.
+ unfold RiemannInt_SF; case (Rle_dec a c); intro.
eapply StepFun_P17.
apply H3.
change
(adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
- (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1.
+ (subdivision_val (mkStepFun pr3))); apply StepFun_P1.
apply Ropp_eq_compat; eapply StepFun_P17.
apply H3.
change
(adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
- (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1.
- unfold RiemannInt_SF in |- *; case (Rle_dec b c); intro.
+ (subdivision_val (mkStepFun pr3))); apply StepFun_P1.
+ unfold RiemannInt_SF; case (Rle_dec b c); intro.
eapply StepFun_P17.
apply H2.
change
(adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
- (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1.
+ (subdivision_val (mkStepFun pr2))); apply StepFun_P1.
apply Ropp_eq_compat; eapply StepFun_P17.
apply H2.
change
(adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
- (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1.
- unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+ (subdivision_val (mkStepFun pr2))); apply StepFun_P1.
+ unfold RiemannInt_SF; case (Rle_dec a b); intro.
eapply StepFun_P17.
apply H1.
change
(adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
- (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1.
+ (subdivision_val (mkStepFun pr1))); apply StepFun_P1.
apply Ropp_eq_compat; eapply StepFun_P17.
apply H1.
change
(adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
- (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1.
+ (subdivision_val (mkStepFun pr1))); apply StepFun_P1.
Qed.
Lemma StepFun_P44 :
@@ -2451,7 +2449,7 @@ Proof.
adapted_couple f a b l1 lf1 ->
a <= c <= b ->
{ l:Rlist & { l0:Rlist & adapted_couple f a c l l0 } }).
- intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X.
+ intro X; unfold IsStepFun; unfold is_subdivision; eapply X.
apply H2.
split; assumption.
clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.
@@ -2463,11 +2461,11 @@ Proof.
simpl in H2; assert (H7 : a <= b).
elim H0; intros; apply Rle_trans with c; assumption.
replace a with (Rmin a b).
- pattern b at 2 in |- *; replace b with (Rmax a b).
+ pattern b at 2; replace b with (Rmax a b).
rewrite <- H2; rewrite H3; reflexivity.
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
split with (cons r nil); split with lf1; assert (H2 : c = b).
rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.
@@ -2481,22 +2479,22 @@ Proof.
split with (cons r (cons c nil)); split with (cons r3 nil);
unfold adapted_couple in H; decompose [and] H; clear H;
assert (H6 : r = a).
- simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity
| elim n; elim H0; intros; apply Rle_trans with c; assumption ].
- elim H0; clear H0; intros; unfold adapted_couple in |- *; repeat split.
- rewrite H6; unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8;
- [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
- simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro;
+ elim H0; clear H0; intros; unfold adapted_couple; repeat split.
+ rewrite H6; unfold ordered_Rlist; intros; simpl in H8; inversion H8;
+ [ simpl; assumption | elim (le_Sn_O _ H10) ].
+ simpl; unfold Rmin; case (Rle_dec a c); intro;
[ assumption | elim n; assumption ].
- simpl in |- *; unfold Rmax in |- *; case (Rle_dec a c); intro;
+ simpl; unfold Rmax; case (Rle_dec a c); intro;
[ reflexivity | elim n; assumption ].
- unfold constant_D_eq, open_interval in |- *; intros; simpl in H8;
+ unfold constant_D_eq, open_interval; intros; simpl in H8;
inversion H8.
- simpl in |- *; assert (H10 := H7 0%nat);
+ simpl; assert (H10 := H7 0%nat);
assert (H12 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).
- simpl in |- *; apply lt_O_Sn.
- apply (H10 H12); unfold open_interval in |- *; simpl in |- *;
+ simpl; apply lt_O_Sn.
+ apply (H10 H12); unfold open_interval; simpl;
rewrite H11 in H9; simpl in H9; elim H9; clear H9;
intros; split; try assumption.
apply Rlt_le_trans with c; assumption.
@@ -2510,42 +2508,42 @@ Proof.
assert (H14 : a <= b).
elim H0; intros; apply Rle_trans with c; assumption.
assert (H16 : r = a).
- simpl in H7; rewrite H7; unfold Rmin in |- *; case (Rle_dec a b); intro;
+ simpl in H7; rewrite H7; unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
induction l1' as [| r4 l1' Hrecl1'].
simpl in H13; discriminate.
- clear Hrecl1'; unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci].
- simpl in |- *; replace r4 with r1.
+ clear Hrecl1'; unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H; induction i as [| i Hreci].
+ simpl; replace r4 with r1.
apply (H5 0%nat).
- simpl in |- *; apply lt_O_Sn.
- simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro;
+ simpl; apply lt_O_Sn.
+ simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c); intro;
[ reflexivity | elim n; left; assumption ].
- apply (H9 i); simpl in |- *; apply lt_S_n; assumption.
- simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro;
+ apply (H9 i); simpl; apply lt_S_n; assumption.
+ simpl; unfold Rmin; case (Rle_dec a c); intro;
[ assumption | elim n; elim H0; intros; assumption ].
replace (Rmax a c) with (Rmax r1 c).
rewrite <- H11; reflexivity.
- unfold Rmax in |- *; case (Rle_dec r1 c); case (Rle_dec a c); intros;
+ unfold Rmax; case (Rle_dec r1 c); case (Rle_dec a c); intros;
[ reflexivity
| elim n; elim H0; intros; assumption
| elim n; left; assumption
| elim n0; left; assumption ].
- simpl in |- *; simpl in H13; rewrite H13; reflexivity.
- intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+ simpl; simpl in H13; rewrite H13; reflexivity.
+ intros; simpl in H; unfold constant_D_eq, open_interval; intros;
induction i as [| i Hreci].
- simpl in |- *; assert (H17 := H10 0%nat);
+ simpl; assert (H17 := H10 0%nat);
assert (H18 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).
- simpl in |- *; apply lt_O_Sn.
- apply (H17 H18); unfold open_interval in |- *; simpl in |- *; simpl in H4;
+ simpl; apply lt_O_Sn.
+ apply (H17 H18); unfold open_interval; simpl; simpl in H4;
elim H4; clear H4; intros; split; try assumption;
replace r1 with r4.
assumption.
- simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro;
+ simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c); intro;
[ reflexivity | elim n; left; assumption ].
- clear Hreci; simpl in |- *; apply H15.
- simpl in |- *; apply lt_S_n; assumption.
- unfold open_interval in |- *; apply H4.
+ clear Hreci; simpl; apply H15.
+ simpl; apply lt_S_n; assumption.
+ unfold open_interval; apply H4.
split.
left; assumption.
elim H0; intros; assumption.
@@ -2567,7 +2565,7 @@ Proof.
adapted_couple f a b l1 lf1 ->
a <= c <= b ->
{ l:Rlist & { l0:Rlist & adapted_couple f c b l l0 } }).
- intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X;
+ intro X; unfold IsStepFun; unfold is_subdivision; eapply X;
[ apply H2 | split; assumption ].
clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.
intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
@@ -2578,11 +2576,11 @@ Proof.
simpl in H2; assert (H7 : a <= b).
elim H0; intros; apply Rle_trans with c; assumption.
replace a with (Rmin a b).
- pattern b at 2 in |- *; replace b with (Rmax a b).
+ pattern b at 2; replace b with (Rmax a b).
rewrite <- H2; rewrite H3; reflexivity.
- unfold Rmax in |- *; case (Rle_dec a b); intro;
+ unfold Rmax; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
- unfold Rmin in |- *; case (Rle_dec a b); intro;
+ unfold Rmin; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
split with (cons r nil); split with lf1; assert (H2 : c = b).
rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.
@@ -2595,32 +2593,32 @@ Proof.
elim H1; intro.
split with (cons c (cons r1 r2)); split with (cons r3 lf1);
unfold adapted_couple in H; decompose [and] H; clear H;
- unfold adapted_couple in |- *; repeat split.
- unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci].
- simpl in |- *; assumption.
- clear Hreci; apply (H2 (S i)); simpl in |- *; assumption.
- simpl in |- *; unfold Rmin in |- *; case (Rle_dec c b); intro;
+ unfold adapted_couple; repeat split.
+ unfold ordered_Rlist; intros; simpl in H; induction i as [| i Hreci].
+ simpl; assumption.
+ clear Hreci; apply (H2 (S i)); simpl; assumption.
+ simpl; unfold Rmin; case (Rle_dec c b); intro;
[ reflexivity | elim n; elim H0; intros; assumption ].
replace (Rmax c b) with (Rmax a b).
rewrite <- H3; reflexivity.
- unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec c b); intros;
+ unfold Rmax; case (Rle_dec a b); case (Rle_dec c b); intros;
[ reflexivity
| elim n; elim H0; intros; assumption
| elim n; elim H0; intros; apply Rle_trans with c; assumption
| elim n0; elim H0; intros; apply Rle_trans with c; assumption ].
- simpl in |- *; simpl in H5; apply H5.
+ simpl; simpl in H5; apply H5.
intros; simpl in H; induction i as [| i Hreci].
- unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *;
+ unfold constant_D_eq, open_interval; intros; simpl;
apply (H7 0%nat).
- simpl in |- *; apply lt_O_Sn.
- unfold open_interval in |- *; simpl in |- *; simpl in H6; elim H6; clear H6;
+ simpl; apply lt_O_Sn.
+ unfold open_interval; simpl; simpl in H6; elim H6; clear H6;
intros; split; try assumption; apply Rle_lt_trans with c;
try assumption; replace r with a.
elim H0; intros; assumption.
- simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intros;
+ simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b); intros;
[ reflexivity
| elim n; elim H0; intros; apply Rle_trans with c; assumption ].
- clear Hreci; apply (H7 (S i)); simpl in |- *; assumption.
+ clear Hreci; apply (H7 (S i)); simpl; assumption.
cut (adapted_couple f r1 b (cons r1 r2) lf1).
cut (r1 <= c <= b).
intros; elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with l1';