diff options
| author |  letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2012-07-05 16:56:37 +0000 |
|---|---|---|
| committer | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2012-07-05 16:56:37 +0000 |
| commit | ffb64d16132dd80f72ecb619ef87e3eee1fa8bda (patch) | |
| tree | 5368562b42af1aeef7e19b4bd897c9fc5655769b /theories | |
| parent | a46ccd71539257bb55dcddd9ae8510856a5c9a16 (diff) | |
Kills the useless tactic annotations "in |- *"
Most of these heavyweight annotations were introduced a long time ago
by the automatic 7.x -> 8.0 translator
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15518 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories')
120 files changed, 4143 insertions, 4143 deletions
diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v index 67039072b..69dded848 100644 --- a/theories/Arith/Between.v +++ b/theories/Arith/Between.v @@ -74,7 +74,7 @@ Section Between. Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r. Proof. - red in |- *; auto with arith. + red; auto with arith. Qed. Hint Resolve in_int_intro: arith v62. @@ -149,7 +149,7 @@ Section Between. between k l -> (forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l. Proof. - induction 1; red in |- *; intros. + induction 1; red; intros. absurd (k < k); auto with arith. absurd (Q l); auto with arith. elim (exists_in_int k (S l)); auto with arith; intros l' inl' Ql'. diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v index dc4da448e..d304b99bd 100644 --- a/theories/Arith/Compare.v +++ b/theories/Arith/Compare.v @@ -39,7 +39,7 @@ Proof. lapply (lt_le_S m n); auto with arith. intro H'; lapply (le_lt_or_eq (S m) n); auto with arith. induction 1; auto with arith. - right; exists (n - S (S m)); simpl in |- *. + right; exists (n - S (S m)); simpl. rewrite (plus_comm m (n - S (S m))). rewrite (plus_n_Sm (n - S (S m)) m). rewrite (plus_n_Sm (n - S (S m)) (S m)). diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v index 1cb91f9a5..f6801da20 100644 --- a/theories/Arith/Compare_dec.v +++ b/theories/Arith/Compare_dec.v @@ -256,7 +256,7 @@ Lemma leb_correct : forall m n, m <= n -> leb m n = true. Proof. induction m as [| m IHm]. trivial. destruct n. intro H. elim (le_Sn_O _ H). - intros. simpl in |- *. apply IHm. apply le_S_n. assumption. + intros. simpl. apply IHm. apply le_S_n. assumption. Qed. Lemma leb_complete : forall m n, leb m n = true -> m <= n. diff --git a/theories/Arith/Div2.v b/theories/Arith/Div2.v index da1d9e989..68652b70c 100644 --- a/theories/Arith/Div2.v +++ b/theories/Arith/Div2.v @@ -43,7 +43,7 @@ Qed. Lemma lt_div2 : forall n, 0 < n -> div2 n < n. Proof. - intro n. pattern n in |- *. apply ind_0_1_SS. + intro n. pattern n. apply ind_0_1_SS. (* n = 0 *) inversion 1. (* n=1 *) @@ -99,12 +99,12 @@ Hint Unfold double: arith. Lemma double_S : forall n, double (S n) = S (S (double n)). Proof. - intro. unfold double in |- *. simpl in |- *. auto with arith. + intro. unfold double. simpl. auto with arith. Qed. Lemma double_plus : forall n (m:nat), double (n + m) = double n + double m. Proof. - intros m n. unfold double in |- *. + intros m n. unfold double. do 2 rewrite plus_assoc_reverse. rewrite (plus_permute n). reflexivity. Qed. @@ -115,7 +115,7 @@ Lemma even_odd_double : forall n, (even n <-> n = double (div2 n)) /\ (odd n <-> n = S (double (div2 n))). Proof. - intro n. pattern n in |- *. apply ind_0_1_SS. + intro n. pattern n. apply ind_0_1_SS. (* n = 0 *) split; split; auto with arith. intro H. inversion H. @@ -126,11 +126,11 @@ Proof. intros. destruct H as ((IH1,IH2),(IH3,IH4)). split; split. intro H. inversion H. inversion H1. - simpl in |- *. rewrite (double_S (div2 n0)). auto with arith. - simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. + simpl. rewrite (double_S (div2 n0)). auto with arith. + simpl. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. intro H. inversion H. inversion H1. - simpl in |- *. rewrite (double_S (div2 n0)). auto with arith. - simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. + simpl. rewrite (double_S (div2 n0)). auto with arith. + simpl. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith. Qed. (** Specializations *) diff --git a/theories/Arith/EqNat.v b/theories/Arith/EqNat.v index 1dc69f612..331990c54 100644 --- a/theories/Arith/EqNat.v +++ b/theories/Arith/EqNat.v @@ -23,7 +23,7 @@ Fixpoint eq_nat n m : Prop := end. Theorem eq_nat_refl : forall n, eq_nat n n. - induction n; simpl in |- *; auto. + induction n; simpl; auto. Qed. Hint Resolve eq_nat_refl: arith v62. @@ -35,7 +35,7 @@ Qed. Hint Immediate eq_eq_nat: arith v62. Lemma eq_nat_eq : forall n m, eq_nat n m -> n = m. - induction n; induction m; simpl in |- *; contradiction || auto with arith. + induction n; induction m; simpl; contradiction || auto with arith. Qed. Hint Immediate eq_nat_eq: arith v62. @@ -55,11 +55,11 @@ Proof. induction n. destruct m as [| n]. auto with arith. - intros; right; red in |- *; trivial with arith. + intros; right; red; trivial with arith. destruct m as [| n0]. - right; red in |- *; auto with arith. + right; red; auto with arith. intros. - simpl in |- *. + simpl. apply IHn. Defined. @@ -76,12 +76,12 @@ Fixpoint beq_nat n m : bool := Lemma beq_nat_refl : forall n, true = beq_nat n n. Proof. - intro x; induction x; simpl in |- *; auto. + intro x; induction x; simpl; auto. Qed. Definition beq_nat_eq : forall x y, true = beq_nat x y -> x = y. Proof. - double induction x y; simpl in |- *. + double induction x y; simpl. reflexivity. intros n H1 H2. discriminate H2. intros n H1 H2. discriminate H2. diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v index 513fd1108..6dd0272a8 100644 --- a/theories/Arith/Euclid.v +++ b/theories/Arith/Euclid.v @@ -19,16 +19,16 @@ Inductive diveucl a b : Set := Lemma eucl_dev : forall n, n > 0 -> forall m:nat, diveucl m n. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q r g e. - apply divex with (S q) r; simpl in |- *; auto with arith. + apply divex with (S q) r; simpl; auto with arith. elim plus_assoc. elim e; auto with arith. intros gtbn. - apply divex with 0 n; simpl in |- *; auto with arith. + apply divex with 0 n; simpl; auto with arith. Defined. Lemma quotient : @@ -36,17 +36,17 @@ Lemma quotient : n > 0 -> forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q Hq; exists (S q). elim Hq; intros r Hr. - exists r; simpl in |- *; elim Hr; intros. + exists r; simpl; elim Hr; intros. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists 0; exists n; simpl in |- *; auto with arith. + exists 0; exists n; simpl; auto with arith. Defined. Lemma modulo : @@ -54,15 +54,15 @@ Lemma modulo : n > 0 -> forall m:nat, {r : nat | exists q : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros r Hr; exists r. elim Hr; intros q Hq. - elim Hq; intros; exists (S q); simpl in |- *. + elim Hq; intros; exists (S q); simpl. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists n; exists 0; simpl in |- *; auto with arith. + exists n; exists 0; simpl; auto with arith. Defined. diff --git a/theories/Arith/Even.v b/theories/Arith/Even.v index 9a84a7f2e..23dc1d259 100644 --- a/theories/Arith/Even.v +++ b/theories/Arith/Even.v @@ -145,7 +145,7 @@ Lemma even_mult_aux : forall n m, (odd (n * m) <-> odd n /\ odd m) /\ (even (n * m) <-> even n \/ even m). Proof. - intros n; elim n; simpl in |- *; auto with arith. + intros n; elim n; simpl; auto with arith. intros m; split; split; auto with arith. intros H'; inversion H'. intros H'; elim H'; auto. diff --git a/theories/Arith/Factorial.v b/theories/Arith/Factorial.v index 82643cdcd..1432995e3 100644 --- a/theories/Arith/Factorial.v +++ b/theories/Arith/Factorial.v @@ -23,7 +23,7 @@ Arguments fact n%nat. Lemma lt_O_fact : forall n:nat, 0 < fact n. Proof. - simple induction n; unfold lt in |- *; simpl in |- *; auto with arith. + simple induction n; unfold lt; simpl; auto with arith. Qed. Lemma fact_neq_0 : forall n:nat, fact n <> 0. diff --git a/theories/Arith/Gt.v b/theories/Arith/Gt.v index f9bf0f2fd..04d44f9c9 100644 --- a/theories/Arith/Gt.v +++ b/theories/Arith/Gt.v @@ -47,7 +47,7 @@ Hint Immediate gt_S_n: arith v62. Theorem gt_S : forall n m, S n > m -> n > m \/ m = n. Proof. - intros n m H; unfold gt in |- *; apply le_lt_or_eq; auto with arith. + intros n m H; unfold gt; apply le_lt_or_eq; auto with arith. Qed. Lemma gt_pred : forall n m, m > S n -> pred m > n. @@ -110,23 +110,23 @@ Hint Resolve le_gt_S: arith v62. Theorem le_gt_trans : forall n m p, m <= n -> m > p -> n > p. Proof. - red in |- *; intros; apply lt_le_trans with m; auto with arith. + red; intros; apply lt_le_trans with m; auto with arith. Qed. Theorem gt_le_trans : forall n m p, n > m -> p <= m -> n > p. Proof. - red in |- *; intros; apply le_lt_trans with m; auto with arith. + red; intros; apply le_lt_trans with m; auto with arith. Qed. Lemma gt_trans : forall n m p, n > m -> m > p -> n > p. Proof. - red in |- *; intros n m p H1 H2. + red; intros n m p H1 H2. apply lt_trans with m; auto with arith. Qed. Theorem gt_trans_S : forall n m p, S n > m -> m > p -> n > p. Proof. - red in |- *; intros; apply lt_le_trans with m; auto with arith. + red; intros; apply lt_le_trans with m; auto with arith. Qed. Hint Resolve gt_trans_S le_gt_trans gt_le_trans: arith v62. @@ -142,7 +142,7 @@ Qed. Lemma plus_gt_reg_l : forall n m p, p + n > p + m -> n > m. Proof. - red in |- *; intros n m p H; apply plus_lt_reg_l with p; auto with arith. + red; intros n m p H; apply plus_lt_reg_l with p; auto with arith. Qed. Lemma plus_gt_compat_l : forall n m p, n > m -> p + n > p + m. diff --git a/theories/Arith/Le.v b/theories/Arith/Le.v index 717705a1c..35d200055 100644 --- a/theories/Arith/Le.v +++ b/theories/Arith/Le.v @@ -46,8 +46,8 @@ Qed. Theorem le_Sn_0 : forall n, ~ S n <= 0. Proof. - red in |- *; intros n H. - change (IsSucc 0) in |- *; elim H; simpl in |- *; auto with arith. + red; intros n H. + change (IsSucc 0); elim H; simpl; auto with arith. Qed. Hint Resolve le_0_n le_Sn_0: arith v62. diff --git a/theories/Arith/Lt.v b/theories/Arith/Lt.v index 1f7e6e018..940448202 100644 --- a/theories/Arith/Lt.v +++ b/theories/Arith/Lt.v @@ -51,7 +51,7 @@ Qed. Theorem lt_not_le : forall n m, n < m -> ~ m <= n. Proof. - red in |- *; intros n m Lt Le; exact (le_not_lt m n Le Lt). + red; intros n m Lt Le; exact (le_not_lt m n Le Lt). Qed. Hint Immediate le_not_lt lt_not_le: arith v62. @@ -107,12 +107,12 @@ Qed. Lemma lt_pred : forall n m, S n < m -> n < pred m. Proof. -induction 1; simpl in |- *; auto with arith. +induction 1; simpl; auto with arith. Qed. Hint Immediate lt_pred: arith v62. Lemma lt_pred_n_n : forall n, 0 < n -> pred n < n. -destruct 1; simpl in |- *; auto with arith. +destruct 1; simpl; auto with arith. Qed. Hint Resolve lt_pred_n_n: arith v62. @@ -159,7 +159,7 @@ Hint Immediate lt_le_weak: arith v62. Theorem le_or_lt : forall n m, n <= m \/ m < n. Proof. - intros n m; pattern n, m in |- *; apply nat_double_ind; auto with arith. + intros n m; pattern n, m; apply nat_double_ind; auto with arith. induction 1; auto with arith. Qed. diff --git a/theories/Arith/Minus.v b/theories/Arith/Minus.v index 7ec37a65e..85ac944cd 100644 --- a/theories/Arith/Minus.v +++ b/theories/Arith/Minus.v @@ -29,7 +29,7 @@ Implicit Types m n p : nat. Lemma minus_n_O : forall n, n = n - 0. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Hint Resolve minus_n_O: arith v62. @@ -37,21 +37,21 @@ Hint Resolve minus_n_O: arith v62. Lemma minus_Sn_m : forall n m, m <= n -> S (n - m) = S n - m. Proof. - intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern m, n; apply le_elim_rel; simpl; auto with arith. Qed. Hint Resolve minus_Sn_m: arith v62. Theorem pred_of_minus : forall n, pred n = n - 1. Proof. - intro x; induction x; simpl in |- *; auto with arith. + intro x; induction x; simpl; auto with arith. Qed. (** * Diagonal *) Lemma minus_diag : forall n, n - n = 0. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Lemma minus_diag_reverse : forall n, 0 = n - n. @@ -66,7 +66,7 @@ Notation minus_n_n := minus_diag_reverse. Lemma minus_plus_simpl_l_reverse : forall n m p, n - m = p + n - (p + m). Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve minus_plus_simpl_l_reverse: arith v62. @@ -74,7 +74,7 @@ Hint Resolve minus_plus_simpl_l_reverse: arith v62. Lemma plus_minus : forall n m p, n = m + p -> p = n - m. Proof. - intros n m p; pattern m, n in |- *; apply nat_double_ind; simpl in |- *; + intros n m p; pattern m, n; apply nat_double_ind; simpl; intros. replace (n0 - 0) with n0; auto with arith. absurd (0 = S (n0 + p)); auto with arith. @@ -83,20 +83,20 @@ Qed. Hint Immediate plus_minus: arith v62. Lemma minus_plus : forall n m, n + m - n = m. - symmetry in |- *; auto with arith. + symmetry ; auto with arith. Qed. Hint Resolve minus_plus: arith v62. Lemma le_plus_minus : forall n m, n <= m -> m = n + (m - n). Proof. - intros n m Le; pattern n, m in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern n, m; apply le_elim_rel; simpl; auto with arith. Qed. Hint Resolve le_plus_minus: arith v62. Lemma le_plus_minus_r : forall n m, n <= m -> n + (m - n) = m. Proof. - symmetry in |- *; auto with arith. + symmetry ; auto with arith. Qed. Hint Resolve le_plus_minus_r: arith v62. @@ -132,7 +132,7 @@ Qed. Lemma lt_minus : forall n m, m <= n -> 0 < m -> n - m < n. Proof. - intros n m Le; pattern m, n in |- *; apply le_elim_rel; simpl in |- *; + intros n m Le; pattern m, n; apply le_elim_rel; simpl; auto using le_minus with arith. intros; absurd (0 < 0); auto with arith. Qed. @@ -140,7 +140,7 @@ Hint Resolve lt_minus: arith v62. Lemma lt_O_minus_lt : forall n m, 0 < n - m -> m < n. Proof. - intros n m; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; + intros n m; pattern n, m; apply nat_double_ind; simpl; auto with arith. intros; absurd (0 < 0); trivial with arith. Qed. @@ -148,9 +148,9 @@ Hint Immediate lt_O_minus_lt: arith v62. Theorem not_le_minus_0 : forall n m, ~ m <= n -> n - m = 0. Proof. - intros y x; pattern y, x in |- *; apply nat_double_ind; - [ simpl in |- *; trivial with arith + intros y x; pattern y, x; apply nat_double_ind; + [ simpl; trivial with arith | intros n H; absurd (0 <= S n); [ assumption | apply le_O_n ] - | simpl in |- *; intros n m H1 H2; apply H1; unfold not in |- *; intros H3; + | simpl; intros n m H1 H2; apply H1; unfold not; intros H3; apply H2; apply le_n_S; assumption ]. Qed. diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v index 64b0d4dd3..0c44cfaf1 100644 --- a/theories/Arith/Mult.v +++ b/theories/Arith/Mult.v @@ -23,7 +23,7 @@ Implicit Types m n p : nat. Lemma mult_0_r : forall n, n * 0 = 0. Proof. - intro; symmetry in |- *; apply mult_n_O. + intro; symmetry ; apply mult_n_O. Qed. Lemma mult_0_l : forall n, 0 * n = 0. @@ -35,7 +35,7 @@ Qed. Lemma mult_1_l : forall n, 1 * n = n. Proof. - simpl in |- *; auto with arith. + simpl; auto with arith. Qed. Hint Resolve mult_1_l: arith v62. @@ -68,7 +68,7 @@ Hint Resolve mult_plus_distr_r: arith v62. Lemma mult_plus_distr_l : forall n m p, n * (m + p) = n * m + n * p. Proof. induction n. trivial. - intros. simpl in |- *. rewrite IHn. symmetry. apply plus_permute_2_in_4. + intros. simpl. rewrite IHn. symmetry. apply plus_permute_2_in_4. Qed. Lemma mult_minus_distr_r : forall n m p, (n - m) * p = n * p - m * p. @@ -137,13 +137,13 @@ Qed. Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n. Proof. - induction m; simpl in |- *; auto with arith. + induction m; simpl; auto with arith. Qed. Hint Resolve mult_O_le: arith v62. Lemma mult_le_compat_l : forall n m p, n <= m -> p * n <= p * m. Proof. - induction p as [| p IHp]; intros; simpl in |- *. + induction p as [| p IHp]; intros; simpl. apply le_n. auto using plus_le_compat. Qed. @@ -167,7 +167,7 @@ Proof. assumption. apply le_plus_l. (* m*p<=m0*q -> m*p<=(S m0)*q *) - simpl in |- *; apply le_trans with (m0 * q). + simpl; apply le_trans with (m0 * q). assumption. apply le_plus_r. Qed. @@ -232,7 +232,7 @@ Fixpoint mult_acc (s:nat) m n : nat := Lemma mult_acc_aux : forall n m p, m + n * p = mult_acc m p n. Proof. - induction n as [| p IHp]; simpl in |- *; auto. + induction n as [| p IHp]; simpl; auto. intros s m; rewrite <- plus_tail_plus; rewrite <- IHp. rewrite <- plus_assoc_reverse; apply f_equal2; auto. rewrite plus_comm; auto. @@ -242,7 +242,7 @@ Definition tail_mult n m := mult_acc 0 m n. Lemma mult_tail_mult : forall n m, n * m = tail_mult n m. Proof. - intros; unfold tail_mult in |- *; rewrite <- mult_acc_aux; auto. + intros; unfold tail_mult; rewrite <- mult_acc_aux; auto. Qed. (** [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus] @@ -250,4 +250,4 @@ Qed. Ltac tail_simpl := repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult; - simpl in |- *. + simpl. diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index e68ba9590..e121ce30d 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -29,7 +29,7 @@ Defined. Hint Resolve O_or_S eq_nat_dec: arith. Theorem dec_eq_nat : forall n m, decidable (n = m). - intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith. + intros x y; unfold decidable; elim (eq_nat_dec x y); auto with arith. Defined. Definition UIP_nat:= Eqdep_dec.UIP_dec eq_nat_dec. diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index c036d06e1..5428ada32 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -33,7 +33,7 @@ Definition plus_0_r n := eq_sym (plus_n_O n). Lemma plus_comm : forall n m, n + m = m + n. Proof. - intros n m; elim n; simpl in |- *; auto with arith. + intros n m; elim n; simpl; auto with arith. intros y H; elim (plus_n_Sm m y); auto with arith. Qed. Hint Immediate plus_comm: arith v62. @@ -45,7 +45,7 @@ Definition plus_Snm_nSm : forall n m, S n + m = n + S m:= Lemma plus_assoc : forall n m p, n + (m + p) = n + m + p. Proof. - intros n m p; elim n; simpl in |- *; auto with arith. + intros n m p; elim n; simpl; auto with arith. Qed. Hint Resolve plus_assoc: arith v62. @@ -64,42 +64,42 @@ Hint Resolve plus_assoc_reverse: arith v62. Lemma plus_reg_l : forall n m p, p + n = p + m -> n = m. Proof. - intros m p n; induction n; simpl in |- *; auto with arith. + intros m p n; induction n; simpl; auto with arith. Qed. Lemma plus_le_reg_l : forall n m p, p + n <= p + m -> n <= m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Lemma plus_lt_reg_l : forall n m p, p + n < p + m -> n < m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. (** * Compatibility with order *) Lemma plus_le_compat_l : forall n m p, n <= m -> p + n <= p + m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve plus_le_compat_l: arith v62. Lemma plus_le_compat_r : forall n m p, n <= m -> n + p <= m + p. Proof. - induction 1; simpl in |- *; auto with arith. + induction 1; simpl; auto with arith. Qed. Hint Resolve plus_le_compat_r: arith v62. Lemma le_plus_l : forall n m, n <= n + m. Proof. - induction n; simpl in |- *; auto with arith. + induction n; simpl; auto with arith. Qed. Hint Resolve le_plus_l: arith v62. Lemma le_plus_r : forall n m, m <= n + m. Proof. - intros n m; elim n; simpl in |- *; auto with arith. + intros n m; elim n; simpl; auto with arith. Qed. Hint Resolve le_plus_r: arith v62. @@ -117,7 +117,7 @@ Hint Immediate lt_plus_trans: arith v62. Lemma plus_lt_compat_l : forall n m p, n < m -> p + n < p + m. Proof. - induction p; simpl in |- *; auto with arith. + induction p; simpl; auto with arith. Qed. Hint Resolve plus_lt_compat_l: arith v62. @@ -131,18 +131,18 @@ Hint Resolve plus_lt_compat_r: arith v62. Lemma plus_le_compat : forall n m p q, n <= m -> p <= q -> n + p <= m + q. Proof. intros n m p q H H0. - elim H; simpl in |- *; auto with arith. + elim H; simpl; auto with arith. Qed. Lemma plus_le_lt_compat : forall n m p q, n <= m -> p < q -> n + p < m + q. Proof. - unfold lt in |- *. intros. change (S n + p <= m + q) in |- *. rewrite plus_Snm_nSm. + unfold lt. intros. change (S n + p <= m + q). rewrite plus_Snm_nSm. apply plus_le_compat; assumption. Qed. Lemma plus_lt_le_compat : forall n m p q, n < m -> p <= q -> n + p < m + q. Proof. - unfold lt in |- *. intros. change (S n + p <= m + q) in |- *. apply plus_le_compat; assumption. + unfold lt. intros. change (S n + p <= m + q). apply plus_le_compat; assumption. Qed. Lemma plus_lt_compat : forall n m p q, n < m -> p < q -> n + p < m + q. @@ -190,8 +190,8 @@ Fixpoint tail_plus n m : nat := end. Lemma plus_tail_plus : forall n m, n + m = tail_plus n m. -induction n as [| n IHn]; simpl in |- *; auto. -intro m; rewrite <- IHn; simpl in |- *; auto. +induction n as [| n IHn]; simpl; auto. +intro m; rewrite <- IHn; simpl; auto. Qed. (** * Discrimination *) diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index e264ccb5d..e579010ba 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -24,14 +24,14 @@ Definition gtof (a b:A) := f b > f a. Theorem well_founded_ltof : well_founded ltof. Proof. - red in |- *. + red. cut (forall n (a:A), f a < n -> Acc ltof a). intros H a; apply (H (S (f a))); auto with arith. induction n. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply Acc_intro. - unfold ltof in |- *; intros b ltfafb. + unfold ltof; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith. Defined. @@ -73,7 +73,7 @@ Proof. intros; absurd (f a < 0); auto with arith. intros a ltSma. apply F. - unfold ltof in |- *; intros b ltfafb. + unfold ltof; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith. Defined. @@ -108,7 +108,7 @@ Hypothesis H_compat : forall x y:A, R x y -> f x < f y. Theorem well_founded_lt_compat : well_founded R. Proof. - red in |- *. + red. cut (forall n (a:A), f a < n -> Acc R a). intros H a; apply (H (S (f a))); auto with arith. induction n. @@ -161,8 +161,8 @@ Lemma lt_wf_double_rec : (forall p q, p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m. Proof. - intros P Hrec p; pattern p in |- *; apply lt_wf_rec. - intros n H q; pattern q in |- *; apply lt_wf_rec; auto with arith. + intros P Hrec p; pattern p; apply lt_wf_rec. + intros n H q; pattern q; apply lt_wf_rec; auto with arith. Defined. Lemma lt_wf_double_ind : @@ -171,8 +171,8 @@ Lemma lt_wf_double_ind : (forall p (q:nat), p < n -> P p q) -> (forall p, p < m -> P n p) -> P n m) -> forall n m, P n m. Proof. - intros P Hrec p; pattern p in |- *; apply lt_wf_ind. - intros n H q; pattern q in |- *; apply lt_wf_ind; auto with arith. + intros P Hrec p; pattern p; apply lt_wf_ind. + intros n H q; pattern q; apply lt_wf_ind; auto with arith. Qed. Hint Resolve lt_wf: arith. @@ -190,7 +190,7 @@ Section LT_WF_REL. Remark acc_lt_rel : forall x:A, (exists n, F x n) -> Acc R x. Proof. intros x [n fxn]; generalize dependent x. - pattern n in |- *; apply lt_wf_ind; intros. + pattern n; apply lt_wf_ind; intros. constructor; intros. destruct (F_compat y x) as (x0,H1,H2); trivial. apply (H x0); auto. diff --git a/theories/Bool/BoolEq.v b/theories/Bool/BoolEq.v index d40e56bf6..dae282a17 100644 --- a/theories/Bool/BoolEq.v +++ b/theories/Bool/BoolEq.v @@ -52,12 +52,12 @@ Section Bool_eq_dec. Definition not_eq_false_beq : forall x y:A, x <> y -> false = beq x y. Proof. intros x y H. - symmetry in |- *. + symmetry . apply not_true_is_false. intro. apply H. apply beq_eq. - symmetry in |- *. + symmetry . assumption. Defined. diff --git a/theories/FSets/FMapAVL.v b/theories/FSets/FMapAVL.v index bed5ce742..980cfeac6 100644 --- a/theories/FSets/FMapAVL.v +++ b/theories/FSets/FMapAVL.v @@ -603,12 +603,12 @@ Qed. Lemma lt_leaf : forall x, lt_tree x (Leaf elt). Proof. - unfold lt_tree in |- *; intros; intuition_in. + unfold lt_tree; intros; intuition_in. Qed. Lemma gt_leaf : forall x, gt_tree x (Leaf elt). Proof. - unfold gt_tree in |- *; intros; intuition_in. + unfold gt_tree; intros; intuition_in. Qed. Lemma lt_tree_node : forall x y l r e h, @@ -1388,8 +1388,8 @@ Lemma fold_equiv_aux : L.fold f (elements_aux acc s) a = L.fold f acc (fold f s a). Proof. simple induction s. - simpl in |- *; intuition. - simpl in |- *; intros. + simpl; intuition. + simpl; intros. rewrite H. simpl. apply H0. @@ -1399,11 +1399,11 @@ Lemma fold_equiv : forall (A : Type) (s : t elt) (f : key -> elt -> A -> A) (a : A), fold f s a = fold' f s a. Proof. - unfold fold', elements in |- *. - simple induction s; simpl in |- *; auto; intros. + unfold fold', elements. + simple induction s; simpl; auto; intros. rewrite fold_equiv_aux. rewrite H0. - simpl in |- *; auto. + simpl; auto. Qed. Lemma fold_1 : diff --git a/theories/FSets/FSetBridge.v b/theories/FSets/FSetBridge.v index f0a588379..1ac544e1f 100644 --- a/theories/FSets/FSetBridge.v +++ b/theories/FSets/FSetBridge.v @@ -44,7 +44,7 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. Definition add : forall (x : elt) (s : t), {s' : t | Add x s s'}. Proof. intros; exists (add x s); auto. - unfold Add in |- *; intuition. + unfold Add; intuition. elim (E.eq_dec x y); auto. intros; right. eapply add_3; eauto. @@ -131,7 +131,7 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x}), compat_P E.eq P -> compat_bool E.eq (fdec Pdec). Proof. - unfold compat_P, compat_bool, Proper, respectful, fdec in |- *; intros. + unfold compat_P, compat_bool, Proper, respectful, fdec; intros. generalize (E.eq_sym H0); case (Pdec x); case (Pdec y); firstorder. Qed. @@ -147,11 +147,11 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. intuition. eauto with set. generalize (filter_2 H0 H1). - unfold fdec in |- *. + unfold fdec. case (Pdec x); intuition. inversion H2. apply filter_3; auto. - unfold fdec in |- *; simpl in |- *. + unfold fdec; simpl. case (Pdec x); intuition. Qed. @@ -162,17 +162,17 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. intros. generalize (for_all_1 (s:=s) (f:=fdec Pdec)) (for_all_2 (s:=s) (f:=fdec Pdec)). - case (for_all (fdec Pdec) s); unfold For_all in |- *; [ left | right ]; + case (for_all (fdec Pdec) s); unfold For_all; [ left | right ]; intros. assert (compat_bool E.eq (fdec Pdec)); auto. generalize (H0 H3 Logic.eq_refl _ H2). - unfold fdec in |- *. + unfold fdec. case (Pdec x); intuition. inversion H4. intuition. absurd (false = true); [ auto with bool | apply H; auto ]. intro. - unfold fdec in |- *. + unfold fdec. case (Pdec x); intuition. Qed. @@ -183,19 +183,19 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. intros. generalize (exists_1 (s:=s) (f:=fdec Pdec)) (exists_2 (s:=s) (f:=fdec Pdec)). - case (exists_ (fdec Pdec) s); unfold Exists in |- *; [ left | right ]; + case (exists_ (fdec Pdec) s); unfold Exists; [ left | right ]; intros. elim H0; auto; intros. exists x; intuition. generalize H4. - unfold fdec in |- *. + unfold fdec. case (Pdec x); intuition. inversion H2. intuition. elim H2; intros. absurd (false = true); [ auto with bool | apply H; auto ]. exists x; intuition. - unfold fdec in |- *. + unfold fdec. case (Pdec x); intuition. Qed. @@ -212,26 +212,26 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. exists (partition (fdec Pdec) s). generalize (partition_1 s (f:=fdec Pdec)) (partition_2 s (f:=fdec Pdec)). case (partition (fdec Pdec) s). - intros s1 s2; simpl in |- *. + intros s1 s2; simpl. intros; assert (compat_bool E.eq (fdec Pdec)); auto. intros; assert (compat_bool E.eq (fun x => negb (fdec Pdec x))). - generalize H2; unfold compat_bool, Proper, respectful in |- *; intuition; + generalize H2; unfold compat_bool, Proper, respectful; intuition; apply (f_equal negb); auto. intuition. - generalize H4; unfold For_all, Equal in |- *; intuition. + generalize H4; unfold For_all, Equal; intuition. elim (H0 x); intros. assert (fdec Pdec x = true). eapply filter_2; eauto with set. - generalize H8; unfold fdec in |- *; case (Pdec x); intuition. + generalize H8; unfold fdec; case (Pdec x); intuition. inversion H9. - generalize H; unfold For_all, Equal in |- *; intuition. + generalize H; unfold For_all, Equal; intuition. elim (H0 x); intros. cut ((fun x => negb (fdec Pdec x)) x = true). - unfold fdec in |- *; case (Pdec x); intuition. - change ((fun x => negb (fdec Pdec x)) x = true) in |- *. + unfold fdec; case (Pdec x); intuition. + change ((fun x => negb (fdec Pdec x)) x = true). apply (filter_2 (s:=s) (x:=x)); auto. set (b := fdec Pdec x) in *; generalize (Logic.eq_refl b); - pattern b at -1 in |- *; case b; unfold b in |- *; + pattern b at -1; case b; unfold b; [ left | right ]. elim (H4 x); intros _ B; apply B; auto with set. elim (H x); intros _ B; apply B; auto with set. @@ -308,7 +308,7 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. intros; generalize (min_elt_1 (s:=s)) (min_elt_2 (s:=s)) (min_elt_3 (s:=s)). case (min_elt s); [ left | right ]; auto. - exists e; unfold For_all in |- *; eauto. + exists e; unfold For_all; eauto. Qed. Definition max_elt : @@ -318,7 +318,7 @@ Module DepOfNodep (Import M: S) <: Sdep with Module E := M.E. intros; generalize (max_elt_1 (s:=s)) (max_elt_2 (s:=s)) (max_elt_3 (s:=s)). case (max_elt s); [ left | right ]; auto. - exists e; unfold For_all in |- *; eauto. + exists e; unfold For_all; eauto. Qed. Definition elt := elt. @@ -360,7 +360,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma empty_1 : Empty empty. Proof. - unfold empty in |- *; case M.empty; auto. + unfold empty; case M.empty; auto. Qed. Definition is_empty (s : t) : bool := @@ -368,12 +368,12 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma is_empty_1 : forall s : t, Empty s -> is_empty s = true. Proof. - intros; unfold is_empty in |- *; case (M.is_empty s); auto. + intros; unfold is_empty; case (M.is_empty s); auto. Qed. Lemma is_empty_2 : forall s : t, is_empty s = true -> Empty s. Proof. - intro s; unfold is_empty in |- *; case (M.is_empty s); auto. + intro s; unfold is_empty; case (M.is_empty s); auto. intros; discriminate H. Qed. @@ -382,12 +382,12 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma mem_1 : forall (s : t) (x : elt), In x s -> mem x s = true. Proof. - intros; unfold mem in |- *; case (M.mem x s); auto. + intros; unfold mem; case (M.mem x s); auto. Qed. Lemma mem_2 : forall (s : t) (x : elt), mem x s = true -> In x s. Proof. - intros s x; unfold mem in |- *; case (M.mem x s); auto. + intros s x; unfold mem; case (M.mem x s); auto. intros; discriminate H. Qed. @@ -398,12 +398,12 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma equal_1 : forall s s' : t, Equal s s' -> equal s s' = true. Proof. - intros; unfold equal in |- *; case M.equal; intuition. + intros; unfold equal; case M.equal; intuition. Qed. Lemma equal_2 : forall s s' : t, equal s s' = true -> Equal s s'. Proof. - intros s s'; unfold equal in |- *; case (M.equal s s'); intuition; + intros s s'; unfold equal; case (M.equal s s'); intuition; inversion H. Qed. @@ -412,12 +412,12 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma subset_1 : forall s s' : t, Subset s s' -> subset s s' = true. Proof. - intros; unfold subset in |- *; case M.subset; intuition. + intros; unfold subset; case M.subset; intuition. Qed. Lemma subset_2 : forall s s' : t, subset s s' = true -> Subset s s'. Proof. - intros s s'; unfold subset in |- *; case (M.subset s s'); intuition; + intros s s'; unfold subset; case (M.subset s s'); intuition; inversion H. Qed. @@ -429,14 +429,14 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma choose_1 : forall (s : t) (x : elt), choose s = Some x -> In x s. Proof. - intros s x; unfold choose in |- *; case (M.choose s). + intros s x; unfold choose; case (M.choose s). simple destruct s0; intros; injection H; intros; subst; auto. intros; discriminate H. Qed. Lemma choose_2 : forall s : t, choose s = None -> Empty s. Proof. - intro s; unfold choose in |- *; case (M.choose s); auto. + intro s; unfold choose; case (M.choose s); auto. simple destruct s0; intros; discriminate H. Qed. @@ -453,17 +453,17 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma elements_1 : forall (s : t) (x : elt), In x s -> InA E.eq x (elements s). Proof. - intros; unfold elements in |- *; case (M.elements s); firstorder. + intros; unfold elements; case (M.elements s); firstorder. Qed. Lemma elements_2 : forall (s : t) (x : elt), InA E.eq x (elements s) -> In x s. Proof. - intros s x; unfold elements in |- *; case (M.elements s); firstorder. + intros s x; unfold elements; case (M.elements s); firstorder. Qed. Lemma elements_3 : forall s : t, sort E.lt (elements s). Proof. - intros; unfold elements in |- *; case (M.elements s); firstorder. + intros; unfold elements; case (M.elements s); firstorder. Qed. Hint Resolve elements_3. @@ -478,7 +478,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma min_elt_1 : forall (s : t) (x : elt), min_elt s = Some x -> In x s. Proof. - intros s x; unfold min_elt in |- *; case (M.min_elt s). + intros s x; unfold min_elt; case (M.min_elt s). simple destruct s0; intros; injection H; intros; subst; intuition. intros; discriminate H. Qed. @@ -486,15 +486,15 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma min_elt_2 : forall (s : t) (x y : elt), min_elt s = Some x -> In y s -> ~ E.lt y x. Proof. - intros s x y; unfold min_elt in |- *; case (M.min_elt s). - unfold For_all in |- *; simple destruct s0; intros; injection H; intros; + intros s x y; unfold min_elt; case (M.min_elt s). + unfold For_all; simple destruct s0; intros; injection H; intros; subst; firstorder. intros; discriminate H. Qed. Lemma min_elt_3 : forall s : t, min_elt s = None -> Empty s. Proof. - intros s; unfold min_elt in |- *; case (M.min_elt s); auto. + intros s; unfold min_elt; case (M.min_elt s); auto. simple destruct s0; intros; discriminate H. Qed. @@ -506,7 +506,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma max_elt_1 : forall (s : t) (x : elt), max_elt s = Some x -> In x s. Proof. - intros s x; unfold max_elt in |- *; case (M.max_elt s). + intros s x; unfold max_elt; case (M.max_elt s). simple destruct s0; intros; injection H; intros; subst; intuition. intros; discriminate H. Qed. @@ -514,15 +514,15 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma max_elt_2 : forall (s : t) (x y : elt), max_elt s = Some x -> In y s -> ~ E.lt x y. Proof. - intros s x y; unfold max_elt in |- *; case (M.max_elt s). - unfold For_all in |- *; simple destruct s0; intros; injection H; intros; + intros s x y; unfold max_elt; case (M.max_elt s). + unfold For_all; simple destruct s0; intros; injection H; intros; subst; firstorder. intros; discriminate H. Qed. Lemma max_elt_3 : forall s : t, max_elt s = None -> Empty s. Proof. - intros s; unfold max_elt in |- *; case (M.max_elt s); auto. + intros s; unfold max_elt; case (M.max_elt s); auto. simple destruct s0; intros; discriminate H. Qed. @@ -530,20 +530,20 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma add_1 : forall (s : t) (x y : elt), E.eq x y -> In y (add x s). Proof. - intros; unfold add in |- *; case (M.add x s); unfold Add in |- *; + intros; unfold add; case (M.add x s); unfold Add; firstorder. Qed. Lemma add_2 : forall (s : t) (x y : elt), In y s -> In y (add x s). Proof. - intros; unfold add in |- *; case (M.add x s); unfold Add in |- *; + intros; unfold add; case (M.add x s); unfold Add; firstorder. Qed. Lemma add_3 : forall (s : t) (x y : elt), ~ E.eq x y -> In y (add x s) -> In y s. Proof. - intros s x y; unfold add in |- *; case (M.add x s); unfold Add in |- *; + intros s x y; unfold add; case (M.add x s); unfold Add; firstorder. Qed. @@ -551,30 +551,30 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma remove_1 : forall (s : t) (x y : elt), E.eq x y -> ~ In y (remove x s). Proof. - intros; unfold remove in |- *; case (M.remove x s); firstorder. + intros; unfold remove; case (M.remove x s); firstorder. Qed. Lemma remove_2 : forall (s : t) (x y : elt), ~ E.eq x y -> In y s -> In y (remove x s). Proof. - intros; unfold remove in |- *; case (M.remove x s); firstorder. + intros; unfold remove; case (M.remove x s); firstorder. Qed. Lemma remove_3 : forall (s : t) (x y : elt), In y (remove x s) -> In y s. Proof. - intros s x y; unfold remove in |- *; case (M.remove x s); firstorder. + intros s x y; unfold remove; case (M.remove x s); firstorder. Qed. Definition singleton (x : elt) : t := let (s, _) := singleton x in s. Lemma singleton_1 : forall x y : elt, In y (singleton x) -> E.eq x y. Proof. - intros x y; unfold singleton in |- *; case (M.singleton x); firstorder. + intros x y; unfold singleton; case (M.singleton x); firstorder. Qed. Lemma singleton_2 : forall x y : elt, E.eq x y -> In y (singleton x). Proof. - intros x y; unfold singleton in |- *; case (M.singleton x); firstorder. + intros x y; unfold singleton; case (M.singleton x); firstorder. Qed. Definition union (s s' : t) : t := let (s'', _) := union s s' in s''. @@ -582,60 +582,60 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. Lemma union_1 : forall (s s' : t) (x : elt), In x (union s s') -> In x s \/ In x s'. Proof. - intros s s' x; unfold union in |- *; case (M.union s s'); firstorder. + intros s s' x; unfold union; case (M.union s s'); firstorder. Qed. Lemma union_2 : forall (s s' : t) (x : elt), In x s -> In x (union s s'). Proof. - intros s s' x; unfold union in |- *; case (M.union s s'); firstorder. + intros s s' x; unfold union; case (M.union s s'); firstorder. Qed. Lemma union_3 : forall (s s' : t) (x : elt), In x s' -> In x (union s s'). Proof. - intros s s' x; unfold union in |- *; case (M.union s s'); firstorder. + intros s s' x; unfold union; case (M.union s s'); firstorder. Qed. Definition inter (s s' : t) : t := let (s'', _) := inter s s' in s''. Lemma inter_1 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s. Proof. - intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder. + intros s s' x; unfold inter; case (M.inter s s'); firstorder. Qed. Lemma inter_2 : forall (s s' : t) (x : elt), In x (inter s s') -> In x s'. Proof. - intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder. + intros s s' x; unfold inter; case (M.inter s s'); firstorder. Qed. Lemma inter_3 : forall (s s' : t) (x : elt), In x s -> In x s' -> In x (inter s s'). Proof. - intros s s' x; unfold inter in |- *; case (M.inter s s'); firstorder. + intros s s' x; unfold inter; case (M.inter s s'); firstorder. Qed. Definition diff (s s' : t) : t := let (s'', _) := diff s s' in s''. Lemma diff_1 : forall (s s' : t) (x : elt), In x (diff s s') -> In x s. Proof. - intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder. + intros s s' x; unfold diff; case (M.diff s s'); firstorder. Qed. Lemma diff_2 : forall (s s' : t) (x : elt), In x (diff s s') -> ~ In x s'. Proof. - intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder. + intros s s' x; unfold diff; case (M.diff s s'); firstorder. Qed. Lemma diff_3 : forall (s s' : t) (x : elt), In x s -> ~ In x s' -> In x (diff s s'). Proof. - intros s s' x; unfold diff in |- *; case (M.diff s s'); firstorder. + intros s s' x; unfold diff; case (M.diff s s'); firstorder. Qed. Definition cardinal (s : t) : nat := let (f, _) := cardinal s in f. Lemma cardinal_1 : forall s, cardinal s = length (elements s). Proof. - intros; unfold cardinal in |- *; case (M.cardinal s); unfold elements in *; + intros; unfold cardinal; case (M.cardinal s); unfold elements in *; destruct (M.elements s); auto. Qed. @@ -646,7 +646,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (A : Type) (i : A) (f : elt -> A -> A), fold f s i = fold_left (fun a e => f e a) (elements s) i. Proof. - intros; unfold fold in |- *; case (M.fold f s i); unfold elements in *; + intros; unfold fold; case (M.fold f s i); unfold elements in *; destruct (M.elements s); auto. Qed. @@ -673,7 +673,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (x : elt) (f : elt -> bool), compat_bool E.eq f -> In x (filter f s) -> In x s. Proof. - intros s x f; unfold filter in |- *; case M.filter; intuition. + intros s x f; unfold filter; case M.filter; intuition. generalize (i (compat_P_aux H)); firstorder. Qed. @@ -681,7 +681,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (x : elt) (f : elt -> bool), compat_bool E.eq f -> In x (filter f s) -> f x = true. Proof. - intros s x f; unfold filter in |- *; case M.filter; intuition. + intros s x f; unfold filter; case M.filter; intuition. generalize (i (compat_P_aux H)); firstorder. Qed. @@ -689,7 +689,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (x : elt) (f : elt -> bool), compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s). Proof. - intros s x f; unfold filter in |- *; case M.filter; intuition. + intros s x f; unfold filter; case M.filter; intuition. generalize (i (compat_P_aux H)); firstorder. Qed. @@ -703,7 +703,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. compat_bool E.eq f -> For_all (fun x => f x = true) s -> for_all f s = true. Proof. - intros s f; unfold for_all in |- *; case M.for_all; intuition; elim n; + intros s f; unfold for_all; case M.for_all; intuition; elim n; auto. Qed. @@ -712,7 +712,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. compat_bool E.eq f -> for_all f s = true -> For_all (fun x => f x = true) s. Proof. - intros s f; unfold for_all in |- *; case M.for_all; intuition; + intros s f; unfold for_all; case M.for_all; intuition; inversion H0. Qed. @@ -725,7 +725,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (f : elt -> bool), compat_bool E.eq f -> Exists (fun x => f x = true) s -> exists_ f s = true. Proof. - intros s f; unfold exists_ in |- *; case M.exists_; intuition; elim n; + intros s f; unfold exists_; case M.exists_; intuition; elim n; auto. Qed. @@ -733,7 +733,7 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (f : elt -> bool), compat_bool E.eq f -> exists_ f s = true -> Exists (fun x => f x = true) s. Proof. - intros s f; unfold exists_ in |- *; case M.exists_; intuition; + intros s f; unfold exists_; case M.exists_; intuition; inversion H0. Qed. @@ -745,10 +745,10 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (f : elt -> bool), compat_bool E.eq f -> Equal (fst (partition f s)) (filter f s). Proof. - intros s f; unfold partition in |- *; case M.partition. + intros s f; unfold partition; case M.partition. intro p; case p; clear p; intros s1 s2 H C. generalize (H (compat_P_aux C)); clear H; intro H. - simpl in |- *; unfold Equal in |- *; intuition. + simpl; unfold Equal; intuition. apply filter_3; firstorder. elim (H2 a); intros. assert (In a s). @@ -763,13 +763,13 @@ Module NodepOfDep (M: Sdep) <: S with Module E := M.E. forall (s : t) (f : elt -> bool), compat_bool E.eq f -> Equal (snd (partition f s)) (filter (fun x => negb (f x)) s). Proof. - intros s f; unfold partition in |- *; case M.partition. + intros s f; unfold partition; case M.partition. intro p; case p; clear p; intros s1 s2 H C. generalize (H (compat_P_aux C)); clear H; intro H. assert (D : compat_bool E.eq (fun x => negb (f x))). generalize C; unfold compat_bool, Proper, respectful; intros; apply (f_equal negb); auto. - simpl in |- *; unfold Equal in |- *; intuition. + simpl; unfold Equal; intuition. apply filter_3; firstorder. elim (H2 a); intros. assert (In a s). diff --git a/theories/Init/Datatypes.v b/theories/Init/Datatypes.v index 5828fc3f8..296612593 100644 --- a/theories/Init/Datatypes.v +++ b/theories/Init/Datatypes.v @@ -72,7 +72,7 @@ Hint Resolve andb_prop: bool. Lemma andb_true_intro : forall b1 b2:bool, b1 = true /\ b2 = true -> andb b1 b2 = true. Proof. - destruct b1; destruct b2; simpl in |- *; intros [? ?]; assumption. + destruct b1; destruct b2; simpl; intros [? ?]; assumption. Qed. Hint Resolve andb_true_intro: bool. @@ -203,7 +203,7 @@ Lemma injective_projections : forall (A B:Type) (p1 p2:A * B), fst p1 = fst p2 -> snd p1 = snd p2 -> p1 = p2. Proof. - destruct p1; destruct p2; simpl in |- *; intros Hfst Hsnd. + destruct p1; destruct p2; simpl; intros Hfst Hsnd. rewrite Hfst; rewrite Hsnd; reflexivity. Qed. diff --git a/theories/Init/Logic.v b/theories/Init/Logic.v index 1dc998cdf..50a715988 100644 --- a/theories/Init/Logic.v +++ b/theories/Init/Logic.v @@ -261,12 +261,12 @@ Section universal_quantification. Theorem inst : forall x:A, all (fun x => P x) -> P x. Proof. - unfold all in |- *; auto. + unfold all; auto. Qed. Theorem gen : forall (B:Prop) (f:forall y:A, B -> P y), B -> all P. Proof. - red in |- *; auto. + red; auto. Qed. End universal_quantification. @@ -305,7 +305,7 @@ Section Logic_lemmas. Theorem absurd : forall A C:Prop, A -> ~ A -> C. Proof. - unfold not in |- *; intros A C h1 h2. + unfold not; intros A C h1 h2. destruct (h2 h1). Qed. @@ -334,7 +334,7 @@ Section Logic_lemmas. Theorem not_eq_sym : x <> y -> y <> x. Proof. - red in |- *; intros h1 h2; apply h1; destruct h2; trivial. + red; intros h1 h2; apply h1; destruct h2; trivial. Qed. End equality. diff --git a/theories/Init/Logic_Type.v b/theories/Init/Logic_Type.v index c244ac24f..e501ac527 100644 --- a/theories/Init/Logic_Type.v +++ b/theories/Init/Logic_Type.v @@ -44,7 +44,7 @@ Section identity_is_a_congruence. Lemma not_identity_sym : notT (identity x y) -> notT (identity y x). Proof. - red in |- *; intros H H'; apply H; destruct H'; trivial. + red; intros H H'; apply H; destruct H'; trivial. Qed. End identity_is_a_congruence. diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v index cbb960ceb..4db490e28 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.v @@ -54,7 +54,7 @@ Hint Immediate eq_add_S: core. Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m. Proof. - red in |- *; auto. + red; auto. Qed. Hint Resolve not_eq_S: core. @@ -93,7 +93,7 @@ Hint Resolve (f_equal2 (A1:=nat) (A2:=nat)): core. Lemma plus_n_O : forall n:nat, n = n + 0. Proof. - induction n; simpl in |- *; auto. + induction n; simpl; auto. Qed. Hint Resolve plus_n_O: core. @@ -104,7 +104,7 @@ Qed. Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m. Proof. - intros n m; induction n; simpl in |- *; auto. + intros n m; induction n; simpl; auto. Qed. Hint Resolve plus_n_Sm: core. diff --git a/theories/Init/Wf.v b/theories/Init/Wf.v index 2bb7eae94..df1a9df67 100644 --- a/theories/Init/Wf.v +++ b/theories/Init/Wf.v @@ -103,7 +103,7 @@ Section Well_founded. Lemma Fix_eq : forall x:A, Fix x = F (fun (y:A) (p:R y x) => Fix y). Proof. - intro x; unfold Fix in |- *. + intro x; unfold Fix. rewrite <- Fix_F_eq. apply F_ext; intros. apply Fix_F_inv. diff --git a/theories/Lists/ListSet.v b/theories/Lists/ListSet.v index d67baf57f..26bfa6d35 100644 --- a/theories/Lists/ListSet.v +++ b/theories/Lists/ListSet.v @@ -85,15 +85,15 @@ Section first_definitions. Lemma set_In_dec : forall (a:A) (x:set), {set_In a x} + {~ set_In a x}. Proof. - unfold set_In in |- *. + unfold set_In. (*** Realizer set_mem. Program_all. ***) simple induction x. auto. intros a0 x0 Ha0. case (Aeq_dec a a0); intro eq. - rewrite eq; simpl in |- *; auto with datatypes. + rewrite eq; simpl; auto with datatypes. elim Ha0. auto with datatypes. - right; simpl in |- *; unfold not in |- *; intros [Hc1| Hc2]; + right; simpl; unfold not; intros [Hc1| Hc2]; auto with datatypes. Qed. @@ -102,7 +102,7 @@ Section first_definitions. (set_In a x -> P y) -> P z -> P (if set_mem a x then y else z). Proof. - simple induction x; simpl in |- *; intros. + simple induction x; simpl; intros. assumption. elim (Aeq_dec a a0); auto with datatypes. Qed. @@ -113,11 +113,11 @@ Section first_definitions. (~ set_In a x -> P z) -> P (if set_mem a x then y else z). Proof. - simple induction x; simpl in |- *; intros. - apply H0; red in |- *; trivial. + simple induction x; simpl; intros. + apply H0; red; trivial. case (Aeq_dec a a0); auto with datatypes. intro; apply H; intros; auto. - apply H1; red in |- *; intro. + apply H1; red; intro. case H3; auto. Qed. @@ -125,7 +125,7 @@ Section first_definitions. Lemma set_mem_correct1 : forall (a:A) (x:set), set_mem a x = true -> set_In a x. Proof. - simple induction x; simpl in |- *. + simple induction x; simpl. discriminate. intros a0 l; elim (Aeq_dec a a0); auto with datatypes. Qed. @@ -133,7 +133,7 @@ Section first_definitions. Lemma set_mem_correct2 : forall (a:A) (x:set), set_In a x -> set_mem a x = true. Proof. - simple induction x; simpl in |- *. + simple induction x; simpl. intro Ha; elim Ha. intros a0 l; elim (Aeq_dec a a0); auto with datatypes. intros H1 H2 [H3| H4]. @@ -144,17 +144,17 @@ Section first_definitions. Lemma set_mem_complete1 : forall (a:A) (x:set), set_mem a x = false -> ~ set_In a x. Proof. - simple induction x; simpl in |- *. + simple induction x; simpl. tauto. intros a0 l; elim (Aeq_dec a a0). intros; discriminate H0. - unfold not in |- *; intros; elim H1; auto with datatypes. + unfold not; intros; elim H1; auto with datatypes. Qed. Lemma set_mem_complete2 : forall (a:A) (x:set), ~ set_In a x -> set_mem a x = false. Proof. - simple induction x; simpl in |- *. + simple induction x; simpl. tauto. intros a0 l; elim (Aeq_dec a a0). intros; elim H0; auto with datatypes. @@ -165,7 +165,7 @@ Section first_definitions. forall (a b:A) (x:set), set_In a x -> set_In a (set_add b x). Proof. - unfold set_In in |- *; simple induction x; simpl in |- *. + unfold set_In; simple induction x; simpl. auto with datatypes. intros a0 l H [Ha0a| Hal]. elim (Aeq_dec b a0); left; assumption. @@ -176,11 +176,11 @@ Section first_definitions. forall (a b:A) (x:set), a = b -> set_In a (set_add b x). Proof. - unfold set_In in |- *; simple induction x; simpl in |- *. + unfold set_In; simple induction x; simpl. auto with datatypes. intros a0 l H Hab. elim (Aeq_dec b a0); - [ rewrite Hab; intro Hba0; rewrite Hba0; simpl in |- *; + [ rewrite Hab; intro Hba0; rewrite Hba0; simpl; auto with datatypes | auto with datatypes ]. Qed. @@ -198,13 +198,13 @@ Section first_definitions. forall (a b:A) (x:set), set_In a (set_add b x) -> a = b \/ set_In a x. Proof. - unfold set_In in |- *. + unfold set_In. simple induction x. - simpl in |- *; intros [H1| H2]; auto with datatypes. - simpl in |- *; do 3 intro. + simpl; intros [H1| H2]; auto with datatypes. + simpl; do 3 intro. elim (Aeq_dec b a0). - simpl in |- *; tauto. - simpl in |- *; intros; elim H0. + simpl; tauto. + simpl; intros; elim H0. trivial with datatypes. tauto. tauto. @@ -220,7 +220,7 @@ Section first_definitions. Lemma set_add_not_empty : forall (a:A) (x:set), set_add a x <> empty_set. Proof. - simple induction x; simpl in |- *. + simple induction x; simpl. discriminate. intros; elim (Aeq_dec a a0); intros; discriminate. Qed. @@ -229,13 +229,13 @@ Section first_definitions. Lemma set_union_intro1 : forall (a:A) (x y:set), set_In a x -> set_In a (set_union x y). Proof. - simple induction y; simpl in |- *; auto with datatypes. + simple induction y; simpl; auto with datatypes. Qed. Lemma set_union_intro2 : forall (a:A) (x y:set), set_In a y -> set_In a (set_union x y). Proof. - simple induction y; simpl in |- *. + simple induction y; simpl. tauto. intros; elim H0; auto with datatypes. Qed. @@ -253,7 +253,7 @@ Section first_definitions. forall (a:A) (x y:set), set_In a (set_union x y) -> set_In a x \/ set_In a y. Proof. - simple induction y; simpl in |- *. + simple induction y; simpl. auto with datatypes. intros. generalize (set_add_elim _ _ _ H0). @@ -280,11 +280,11 @@ Section first_definitions. Proof. simple induction x. auto with datatypes. - simpl in |- *; intros a0 l Hrec y [Ha0a| Hal] Hy. - simpl in |- *; rewrite Ha0a. + simpl; intros a0 l Hrec y [Ha0a| Hal] Hy. + simpl; rewrite Ha0a. generalize (set_mem_correct1 a y). generalize (set_mem_complete1 a y). - elim (set_mem a y); simpl in |- *; intros. + elim (set_mem a y); simpl; intros. auto with datatypes. absurd (set_In a y); auto with datatypes. elim (set_mem a0 y); [ right; auto with datatypes | auto with datatypes ]. @@ -295,9 +295,9 @@ Section first_definitions. Proof. simple induction x. auto with datatypes. - simpl in |- *; intros a0 l Hrec y. + simpl; intros a0 l Hrec y. generalize (set_mem_correct1 a0 y). - elim (set_mem a0 y); simpl in |- *; intros. + elim (set_mem a0 y); simpl; intros. elim H0; eauto with datatypes. eauto with datatypes. Qed. @@ -306,10 +306,10 @@ Section first_definitions. forall (a:A) (x y:set), set_In a (set_inter x y) -> set_In a y. Proof. simple induction x. - simpl in |- *; tauto. - simpl in |- *; intros a0 l Hrec y. + simpl; tauto. + simpl; intros a0 l Hrec y. generalize (set_mem_correct1 a0 y). - elim (set_mem a0 y); simpl in |- *; intros. + elim (set_mem a0 y); simpl; intros. elim H0; [ intro Hr; rewrite <- Hr; eauto with datatypes | eauto with datatypes ]. eauto with datatypes. @@ -329,8 +329,8 @@ Section first_definitions. set_In a x -> ~ set_In a y -> set_In a (set_diff x y). Proof. simple induction x. - simpl in |- *; tauto. - simpl in |- *; intros a0 l Hrec y [Ha0a| Hal] Hay. + simpl; tauto. + simpl; intros a0 l Hrec y [Ha0a| Hal] Hay. rewrite Ha0a; generalize (set_mem_complete2 _ _ Hay). elim (set_mem a y); [ intro Habs; discriminate Habs | auto with datatypes ]. @@ -341,8 +341,8 @@ Section first_definitions. forall (a:A) (x y:set), set_In a (set_diff x y) -> set_In a x. Proof. simple induction x. - simpl in |- *; tauto. - simpl in |- *; intros a0 l Hrec y; elim (set_mem a0 y). + simpl; tauto. + simpl; intros a0 l Hrec y; elim (set_mem a0 y). eauto with datatypes. intro; generalize (set_add_elim _ _ _ H). intros [H1| H2]; eauto with datatypes. @@ -350,7 +350,7 @@ Section first_definitions. Lemma set_diff_elim2 : forall (a:A) (x y:set), set_In a (set_diff x y) -> ~ set_In a y. - intros a x y; elim x; simpl in |- *. + intros a x y; elim x; simpl. intros; contradiction. intros a0 l Hrec. apply set_mem_ind2; auto. @@ -359,7 +359,7 @@ Section first_definitions. Qed. Lemma set_diff_trivial : forall (a:A) (x:set), ~ set_In a (set_diff x x). - red in |- *; intros a x H. + red; intros a x H. apply (set_diff_elim2 _ _ _ H). apply (set_diff_elim1 _ _ _ H). Qed. diff --git a/theories/Lists/Streams.v b/theories/Lists/Streams.v index 7a6f38fc2..85ecf97e2 100644 --- a/theories/Lists/Streams.v +++ b/theories/Lists/Streams.v @@ -49,21 +49,21 @@ Qed. Lemma tl_nth_tl : forall (n:nat) (s:Stream), tl (Str_nth_tl n s) = Str_nth_tl n (tl s). Proof. - simple induction n; simpl in |- *; auto. + simple induction n; simpl; auto. Qed. Hint Resolve tl_nth_tl: datatypes v62. Lemma Str_nth_tl_plus : forall (n m:nat) (s:Stream), Str_nth_tl n (Str_nth_tl m s) = Str_nth_tl (n + m) s. -simple induction n; simpl in |- *; intros; auto with datatypes. +simple induction n; simpl; intros; auto with datatypes. rewrite <- H. rewrite tl_nth_tl; trivial with datatypes. Qed. Lemma Str_nth_plus : forall (n m:nat) (s:Stream), Str_nth n (Str_nth_tl m s) = Str_nth (n + m) s. -intros; unfold Str_nth in |- *; rewrite Str_nth_tl_plus; +intros; unfold Str_nth; rewrite Str_nth_tl_plus; trivial with datatypes. Qed. @@ -89,7 +89,7 @@ Qed. Theorem sym_EqSt : forall s1 s2:Stream, EqSt s1 s2 -> EqSt s2 s1. coinduction Eq_sym. -case H; intros; symmetry in |- *; assumption. +case H; intros; symmetry ; assumption. case H; intros; assumption. Qed. @@ -110,10 +110,10 @@ Qed. Theorem eqst_ntheq : forall (n:nat) (s1 s2:Stream), EqSt s1 s2 -> Str_nth n s1 = Str_nth n s2. -unfold Str_nth in |- *; simple induction n. +unfold Str_nth; simple induction n. intros s1 s2 H; case H; trivial with datatypes. intros m hypind. -simpl in |- *. +simpl. intros s1 s2 H. apply hypind. case H; trivial with datatypes. diff --git a/theories/Logic/Berardi.v b/theories/Logic/Berardi.v index 2b3886874..58e339b4c 100644 --- a/theories/Logic/Berardi.v +++ b/theories/Logic/Berardi.v @@ -45,7 +45,7 @@ Lemma AC_IF : (B -> Q e1) -> (~ B -> Q e2) -> Q (IFProp B e1 e2). Proof. intros P B e1 e2 Q p1 p2. -unfold IFProp in |- *. +unfold IFProp. case (EM B); assumption. Qed. @@ -76,7 +76,7 @@ Record retract_cond : Prop := Lemma AC : forall r:retract_cond, retract -> forall a:A, j2 r (i2 r a) = a. Proof. intros r. -case r; simpl in |- *. +case r; simpl. trivial. Qed. @@ -113,7 +113,7 @@ Lemma retract_pow_U_U : retract (pow U) U. Proof. exists g f. intro a. -unfold f, g in |- *; simpl in |- *. +unfold f, g; simpl. apply AC. exists (fun x:pow U => x) (fun x:pow U => x). trivial. @@ -130,8 +130,8 @@ Definition R : U := g (fun u:U => Not_b (u U u)). Lemma not_has_fixpoint : R R = Not_b (R R). Proof. -unfold R at 1 in |- *. -unfold g in |- *. +unfold R at 1. +unfold g. rewrite AC with (r := L1 U U) (a := fun u:U => Not_b (u U u)). trivial. exists (fun x:pow U => x) (fun x:pow U => x); trivial. @@ -141,7 +141,7 @@ Qed. Theorem classical_proof_irrelevence : T = F. Proof. generalize not_has_fixpoint. -unfold Not_b in |- *. +unfold Not_b. apply AC_IF. intros is_true is_false. elim is_true; elim is_false; trivial. diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index 8b11f09b9..b93b7688a 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -345,7 +345,7 @@ Lemma rel_choice_and_proof_irrel_imp_guarded_rel_choice : RelationalChoice -> ProofIrrelevance -> GuardedRelationalChoice. Proof. intros rel_choice proof_irrel. - red in |- *; intros A B P R H. + red; intros A B P R H. destruct (rel_choice _ _ (fun (x:sigT P) (y:B) => R (projT1 x) y)) as (R',(HR'R,H0)). intros (x,HPx). destruct (H x HPx) as (y,HRxy). @@ -581,7 +581,7 @@ Lemma classical_denumerable_description_imp_fun_choice : (forall x y, decidable (R x y)) -> FunctionalChoice_on_rel R. Proof. intros A Descr. - red in |- *; intros R Rdec H. + red; intros R Rdec H. set (R':= fun x y => R x y /\ forall y', R x y' -> y <= y'). destruct (Descr R') as (f,Hf). intro x. diff --git a/theories/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v index 2e1e99e8a..07ed643b7 100644 --- a/theories/Logic/ClassicalFacts.v +++ b/theories/Logic/ClassicalFacts.v @@ -148,7 +148,7 @@ Proof. case (prop_ext_retract_A_A_imp_A Ext A a); intros g1 g2 g1_o_g2. exists (fun f => (fun x:A => f (g1 x x)) (g2 (fun x => f (g1 x x)))). intro f. - pattern (g1 (g2 (fun x:A => f (g1 x x)))) at 1 in |- *. + pattern (g1 (g2 (fun x:A => f (g1 x x)))) at 1. rewrite (g1_o_g2 (fun x:A => f (g1 x x))). reflexivity. Qed. @@ -192,12 +192,12 @@ Section Proof_irrelevance_gen. case (ext_prop_fixpoint Ext bool true); intros G Gfix. set (neg := fun b:bool => bool_elim bool false true b). generalize (eq_refl (G neg)). - pattern (G neg) at 1 in |- *. + pattern (G neg) at 1. apply Ind with (b := G neg); intro Heq. rewrite (bool_elim_redl bool false true). - change (true = neg true) in |- *; rewrite Heq; apply Gfix. + change (true = neg true); rewrite Heq; apply Gfix. rewrite (bool_elim_redr bool false true). - change (neg false = false) in |- *; rewrite Heq; symmetry in |- *; + change (neg false = false); rewrite Heq; symmetry ; apply Gfix. Qed. @@ -207,9 +207,9 @@ Section Proof_irrelevance_gen. intros Ext Ind A a1 a2. set (f := fun b:bool => bool_elim A a1 a2 b). rewrite (bool_elim_redl A a1 a2). - change (f true = a2) in |- *. + change (f true = a2). rewrite (bool_elim_redr A a1 a2). - change (f true = f false) in |- *. + change (f true = f false). rewrite (aux Ext Ind). reflexivity. Qed. @@ -344,8 +344,8 @@ Section Proof_irrelevance_EM_CC. Lemma p2p1 : forall A:Prop, A -> b2p (p2b A). Proof. - unfold p2b in |- *; intro A; apply or_dep_elim with (b := em A); - unfold b2p in |- *; intros. + unfold p2b; intro A; apply or_dep_elim with (b := em A); + unfold b2p; intros. apply (or_elim_redl A (~ A) B (fun _ => b1) (fun _ => b2)). destruct (b H). Qed. @@ -353,8 +353,8 @@ Section Proof_irrelevance_EM_CC. Lemma p2p2 : b1 <> b2 -> forall A:Prop, b2p (p2b A) -> A. Proof. intro not_eq_b1_b2. - unfold p2b in |- *; intro A; apply or_dep_elim with (b := em A); - unfold b2p in |- *; intros. + unfold p2b; intro A; apply or_dep_elim with (b := em A); + unfold b2p; intros. assumption. destruct not_eq_b1_b2. rewrite <- (or_elim_redr A (~ A) B (fun _ => b1) (fun _ => b2)) in H. diff --git a/theories/Logic/Classical_Pred_Type.v b/theories/Logic/Classical_Pred_Type.v index 9d57fe88b..8da364a38 100644 --- a/theories/Logic/Classical_Pred_Type.v +++ b/theories/Logic/Classical_Pred_Type.v @@ -42,7 +42,7 @@ Qed. Lemma not_ex_all_not : forall P:U -> Prop, ~ (exists n : U, P n) -> forall n:U, ~ P n. Proof. (* Intuitionistic *) -unfold not in |- *; intros P notex n abs. +unfold not; intros P notex n abs. apply notex. exists n; trivial. Qed. @@ -52,20 +52,20 @@ Lemma not_ex_not_all : Proof. intros P H n. apply NNPP. -red in |- *; intro K; apply H; exists n; trivial. +red; intro K; apply H; exists n; trivial. Qed. Lemma ex_not_not_all : forall P:U -> Prop, (exists n : U, ~ P n) -> ~ (forall n:U, P n). Proof. (* Intuitionistic *) -unfold not in |- *; intros P exnot allP. +unfold not; intros P exnot allP. elim exnot; auto. Qed. Lemma all_not_not_ex : forall P:U -> Prop, (forall n:U, ~ P n) -> ~ (exists n : U, P n). Proof. (* Intuitionistic *) -unfold not in |- *; intros P allnot exP; elim exP; intros n p. +unfold not; intros P allnot exP; elim exP; intros n p. apply allnot with n; auto. Qed. diff --git a/theories/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v index 5d7764e7e..c48165c61 100644 --- a/theories/Logic/Classical_Prop.v +++ b/theories/Logic/Classical_Prop.v @@ -20,7 +20,7 @@ Axiom classic : forall P:Prop, P \/ ~ P. Lemma NNPP : forall p:Prop, ~ ~ p -> p. Proof. -unfold not in |- *; intros; elim (classic p); auto. +unfold not; intros; elim (classic p); auto. intro NP; elim (H NP). Qed. @@ -35,7 +35,7 @@ Qed. Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P. Proof. -intros; apply NNPP; red in |- *. +intros; apply NNPP; red. intro; apply H; intro; absurd P; trivial. Qed. @@ -68,7 +68,7 @@ Qed. Lemma or_not_and : forall P Q:Prop, ~ P \/ ~ Q -> ~ (P /\ Q). Proof. -simple induction 1; red in |- *; simple induction 2; auto. +simple induction 1; red; simple induction 2; auto. Qed. Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q. diff --git a/theories/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v index e8e8b94ce..b5e7b2c41 100644 --- a/theories/Logic/Diaconescu.v +++ b/theories/Logic/Diaconescu.v @@ -61,7 +61,7 @@ Variable pred_extensionality : PredicateExtensionality. Lemma prop_ext : forall A B:Prop, (A <-> B) -> A = B. Proof. intros A B H. - change ((fun _ => A) true = (fun _ => B) true) in |- *. + change ((fun _ => A) true = (fun _ => B) true). rewrite pred_extensionality with (P := fun _:bool => A) (Q := fun _:bool => B). reflexivity. @@ -134,8 +134,8 @@ right. intro HP. assert (Hequiv : forall b:bool, class_of_true b <-> class_of_false b). intro b; split. -unfold class_of_false in |- *; right; assumption. -unfold class_of_true in |- *; right; assumption. +unfold class_of_false; right; assumption. +unfold class_of_true; right; assumption. assert (Heq : class_of_true = class_of_false). apply pred_extensionality with (1 := Hequiv). apply diff_true_false. diff --git a/theories/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v index 2ed5d428c..ba43600fc 100644 --- a/theories/Logic/Eqdep_dec.v +++ b/theories/Logic/Eqdep_dec.v @@ -61,7 +61,7 @@ Section EqdepDec. Let nu_constant : forall (y:A) (u v:x = y), nu u = nu v. intros. - unfold nu in |- *. + unfold nu. case (eq_dec x y); intros. reflexivity. @@ -75,7 +75,7 @@ Section EqdepDec. Remark nu_left_inv : forall (y:A) (u:x = y), nu_inv (nu u) = u. Proof. intros. - case u; unfold nu_inv in |- *. + case u; unfold nu_inv. apply trans_sym_eq. Qed. @@ -115,7 +115,7 @@ Section EqdepDec. Proof. intros. cut (proj (ex_intro P x y) y = proj (ex_intro P x y') y). - simpl in |- *. + simpl. case (eq_dec x x). intro e. elim e using K_dec; trivial. diff --git a/theories/Logic/Hurkens.v b/theories/Logic/Hurkens.v index bb03c6664..cf2c8a16b 100644 --- a/theories/Logic/Hurkens.v +++ b/theories/Logic/Hurkens.v @@ -46,7 +46,7 @@ Lemma Omega : forall i:U -> bool, induct i -> b2p (i WF). Proof. intros i y. apply y. -unfold le, WF, induct in |- *. +unfold le, WF, induct. apply p2p2. intros x H0. apply y. @@ -55,7 +55,7 @@ Qed. Lemma lemma1 : induct (fun u => p2b (I u)). Proof. -unfold induct in |- *. +unfold induct. intros x p. apply (p2p2 (I x)). intro q. diff --git a/theories/NArith/Ndist.v b/theories/NArith/Ndist.v index 7097159c7..490bf745a 100644 --- a/theories/NArith/Ndist.v +++ b/theories/NArith/Ndist.v @@ -33,7 +33,7 @@ Definition Nplength (a:N) := Lemma Nplength_infty : forall a:N, Nplength a = infty -> a = N0. Proof. simple induction a; trivial. - unfold Nplength in |- *; intros; discriminate H. + unfold Nplength; intros; discriminate H. Qed. Lemma Nplength_zeros : @@ -46,9 +46,9 @@ Proof. intros. simpl in H1. discriminate H1. simple induction k. trivial. generalize H0. case n. intros. inversion H3. - intros. simpl in |- *. unfold N.testbit_nat in H. apply (H n0). simpl in H1. inversion H1. reflexivity. + intros. simpl. unfold N.testbit_nat in H. apply (H n0). simpl in H1. inversion H1. reflexivity. exact (lt_S_n n1 n0 H3). - simpl in |- *. intros n H. inversion H. intros. inversion H0. + simpl. intros n H. inversion H. intros. inversion H0. Qed. Lemma Nplength_one : @@ -56,7 +56,7 @@ Lemma Nplength_one : Proof. simple induction a. intros. inversion H. simple induction p. intros. simpl in H0. inversion H0. reflexivity. - intros. simpl in H0. inversion H0. simpl in |- *. unfold N.testbit_nat in H. apply H. reflexivity. + intros. simpl in H0. inversion H0. simpl. unfold N.testbit_nat in H. apply H. reflexivity. intros. simpl in H. inversion H. reflexivity. Qed. @@ -70,9 +70,9 @@ Proof. intros. absurd (N.testbit_nat (Npos (xI p0)) 0 = false). trivial with bool. auto with bool arith. intros. generalize H0 H1. case n. intros. simpl in H3. discriminate H3. - intros. simpl in |- *. unfold Nplength in H. + intros. simpl. unfold Nplength in H. cut (ni (Pplength p0) = ni n0). intro. inversion H4. reflexivity. - apply H. intros. change (N.testbit_nat (Npos (xO p0)) (S k) = false) in |- *. apply H2. apply lt_n_S. exact H4. + apply H. intros. change (N.testbit_nat (Npos (xO p0)) (S k) = false). apply H2. apply lt_n_S. exact H4. exact H3. intro. case n. trivial. intros. simpl in H0. discriminate H0. @@ -90,10 +90,10 @@ Definition ni_min (d d':natinf) := Lemma ni_min_idemp : forall d:natinf, ni_min d d = d. Proof. simple induction d; trivial. - unfold ni_min in |- *. + unfold ni_min. simple induction n; trivial. intros. - simpl in |- *. + simpl. inversion H. rewrite H1. rewrite H1. @@ -105,7 +105,7 @@ Proof. simple induction d. simple induction d'; trivial. simple induction d'; trivial. elim n. simple induction n0; trivial. intros. elim n1; trivial. intros. unfold ni_min in H. cut (min n0 n2 = min n2 n0). - intro. unfold ni_min in |- *. simpl in |- *. rewrite H1. reflexivity. + intro. unfold ni_min. simpl. rewrite H1. reflexivity. cut (ni (min n0 n2) = ni (min n2 n0)). intros. inversion H1; trivial. exact (H n2). @@ -116,11 +116,11 @@ Lemma ni_min_assoc : Proof. simple induction d; trivial. simple induction d'; trivial. simple induction d''; trivial. - unfold ni_min in |- *. intro. cut (min (min n n0) n1 = min n (min n0 n1)). + unfold ni_min. intro. cut (min (min n n0) n1 = min n (min n0 n1)). intro. rewrite H. reflexivity. generalize n0 n1. elim n; trivial. simple induction n3; trivial. simple induction n5; trivial. - intros. simpl in |- *. auto. + intros. simpl. auto. Qed. Lemma ni_min_O_l : forall d:natinf, ni_min (ni 0) d = ni 0. @@ -152,42 +152,42 @@ Qed. Lemma ni_le_antisym : forall d d':natinf, ni_le d d' -> ni_le d' d -> d = d'. Proof. - unfold ni_le in |- *. intros d d'. rewrite ni_min_comm. intro H. rewrite H. trivial. + unfold ni_le. intros d d'. rewrite ni_min_comm. intro H. rewrite H. trivial. Qed. Lemma ni_le_trans : forall d d' d'':natinf, ni_le d d' -> ni_le d' d'' -> ni_le d d''. Proof. - unfold ni_le in |- *. intros. rewrite <- H. rewrite ni_min_assoc. rewrite H0. reflexivity. + unfold ni_le. intros. rewrite <- H. rewrite ni_min_assoc. rewrite H0. reflexivity. Qed. Lemma ni_le_min_1 : forall d d':natinf, ni_le (ni_min d d') d. Proof. - unfold ni_le in |- *. intros. rewrite (ni_min_comm d d'). rewrite ni_min_assoc. + unfold ni_le. intros. rewrite (ni_min_comm d d'). rewrite ni_min_assoc. rewrite ni_min_idemp. reflexivity. Qed. Lemma ni_le_min_2 : forall d d':natinf, ni_le (ni_min d d') d'. Proof. - unfold ni_le in |- *. intros. rewrite ni_min_assoc. rewrite ni_min_idemp. reflexivity. + unfold ni_le. intros. rewrite ni_min_assoc. rewrite ni_min_idemp. reflexivity. Qed. Lemma ni_min_case : forall d d':natinf, ni_min d d' = d \/ ni_min d d' = d'. Proof. simple induction d. intro. right. exact (ni_min_inf_l d'). simple induction d'. left. exact (ni_min_inf_r (ni n)). - unfold ni_min in |- *. cut (forall n0:nat, min n n0 = n \/ min n n0 = n0). + unfold ni_min. cut (forall n0:nat, min n n0 = n \/ min n n0 = n0). intros. case (H n0). intro. left. rewrite H0. reflexivity. intro. right. rewrite H0. reflexivity. elim n. intro. left. reflexivity. simple induction n1. right. reflexivity. - intros. case (H n2). intro. left. simpl in |- *. rewrite H1. reflexivity. - intro. right. simpl in |- *. rewrite H1. reflexivity. + intros. case (H n2). intro. left. simpl. rewrite H1. reflexivity. + intro. right. simpl. rewrite H1. reflexivity. Qed. Lemma ni_le_total : forall d d':natinf, ni_le d d' \/ ni_le d' d. Proof. - unfold ni_le in |- *. intros. rewrite (ni_min_comm d' d). apply ni_min_case. + unfold ni_le. intros. rewrite (ni_min_comm d' d). apply ni_min_case. Qed. Lemma ni_le_min_induc : @@ -201,7 +201,7 @@ Proof. apply ni_le_antisym. apply H1. apply ni_le_refl. exact H2. exact H. - intro. rewrite H2. apply ni_le_antisym. apply H1. unfold ni_le in |- *. rewrite ni_min_comm. exact H2. + intro. rewrite H2. apply ni_le_antisym. apply H1. unfold ni_le. rewrite ni_min_comm. exact H2. apply ni_le_refl. exact H0. Qed. @@ -209,15 +209,15 @@ Qed. Lemma le_ni_le : forall m n:nat, m <= n -> ni_le (ni m) (ni n). Proof. cut (forall m n:nat, m <= n -> min m n = m). - intros. unfold ni_le, ni_min in |- *. rewrite (H m n H0). reflexivity. + intros. unfold ni_le, ni_min. rewrite (H m n H0). reflexivity. simple induction m. trivial. simple induction n0. intro. inversion H0. - intros. simpl in |- *. rewrite (H n1 (le_S_n n n1 H1)). reflexivity. + intros. simpl. rewrite (H n1 (le_S_n n n1 H1)). reflexivity. Qed. Lemma ni_le_le : forall m n:nat, ni_le (ni m) (ni n) -> m <= n. Proof. - unfold ni_le in |- *. unfold ni_min in |- *. intros. inversion H. apply le_min_r. + unfold ni_le. unfold ni_min. intros. inversion H. apply le_min_r. Qed. Lemma Nplength_lb : @@ -225,7 +225,7 @@ Lemma Nplength_lb : (forall k:nat, k < n -> N.testbit_nat a k = false) -> ni_le (ni n) (Nplength a). Proof. simple induction a. intros. exact (ni_min_inf_r (ni n)). - intros. unfold Nplength in |- *. apply le_ni_le. case (le_or_lt n (Pplength p)). trivial. + intros. unfold Nplength. apply le_ni_le. case (le_or_lt n (Pplength p)). trivial. intro. absurd (N.testbit_nat (Npos p) (Pplength p) = false). rewrite (Nplength_one (Npos p) (Pplength p) @@ -238,7 +238,7 @@ Lemma Nplength_ub : forall (a:N) (n:nat), N.testbit_nat a n = true -> ni_le (Nplength a) (ni n). Proof. simple induction a. intros. discriminate H. - intros. unfold Nplength in |- *. apply le_ni_le. case (le_or_lt (Pplength p) n). trivial. + intros. unfold Nplength. apply le_ni_le. case (le_or_lt (Pplength p) n). trivial. intro. absurd (N.testbit_nat (Npos p) n = true). rewrite (Nplength_zeros (Npos p) (Pplength p) @@ -262,7 +262,7 @@ Definition Npdist (a a':N) := Nplength (N.lxor a a'). Lemma Npdist_eq_1 : forall a:N, Npdist a a = infty. Proof. - intros. unfold Npdist in |- *. rewrite N.lxor_nilpotent. reflexivity. + intros. unfold Npdist. rewrite N.lxor_nilpotent. reflexivity. Qed. Lemma Npdist_eq_2 : forall a a':N, Npdist a a' = infty -> a = a'. @@ -274,7 +274,7 @@ Qed. Lemma Npdist_comm : forall a a':N, Npdist a a' = Npdist a' a. Proof. - unfold Npdist in |- *. intros. rewrite N.lxor_comm. reflexivity. + unfold Npdist. intros. rewrite N.lxor_comm. reflexivity. Qed. (** $d$ is an ultrametric distance, that is, not only $d(a,a')\leq @@ -296,8 +296,8 @@ Lemma Nplength_ultra_1 : Proof. simple induction a. intros. unfold ni_le in H. unfold Nplength at 1 3 in H. rewrite (ni_min_inf_l (Nplength a')) in H. - rewrite (Nplength_infty a' H). simpl in |- *. apply ni_le_refl. - intros. unfold Nplength at 1 in |- *. apply Nplength_lb. intros. + rewrite (Nplength_infty a' H). simpl. apply ni_le_refl. + intros. unfold Nplength at 1. apply Nplength_lb. intros. cut (forall a'':N, N.lxor (Npos p) a' = a'' -> N.testbit_nat a'' k = false). intros. apply H1. reflexivity. intro a''. case a''. intro. reflexivity. @@ -329,7 +329,7 @@ Lemma Npdist_ultra : forall a a' a'':N, ni_le (ni_min (Npdist a a'') (Npdist a'' a')) (Npdist a a'). Proof. - intros. unfold Npdist in |- *. cut (N.lxor (N.lxor a a'') (N.lxor a'' a') = N.lxor a a'). + intros. unfold Npdist. cut (N.lxor (N.lxor a a'') (N.lxor a'' a') = N.lxor a a'). intro. rewrite <- H. apply Nplength_ultra. rewrite N.lxor_assoc. rewrite <- (N.lxor_assoc a'' a'' a'). rewrite N.lxor_nilpotent. rewrite N.lxor_0_l. reflexivity. diff --git a/theories/Program/Tactics.v b/theories/Program/Tactics.v index 9694e3fd1..54011fee8 100644 --- a/theories/Program/Tactics.v +++ b/theories/Program/Tactics.v @@ -310,7 +310,7 @@ Ltac refine_hyp c := possibly using [program_simplify] to use standard goal-cleaning tactics. *) Ltac program_simplify := -simpl in |- *; intros ; destruct_all_rec_calls ; repeat (destruct_conjs; simpl proj1_sig in * ); +simpl; intros ; destruct_all_rec_calls ; repeat (destruct_conjs; simpl proj1_sig in * ); subst*; autoinjections ; try discriminates ; try (solve [ red ; intros ; destruct_conjs ; autoinjections ; discriminates ]). diff --git a/theories/Program/Wf.v b/theories/Program/Wf.v index 8ef1eb4e6..6494ed87c 100644 --- a/theories/Program/Wf.v +++ b/theories/Program/Wf.v @@ -52,7 +52,7 @@ Section Well_founded. Lemma Fix_eq : forall x:A, Fix_sub x = F_sub x (fun y:{ y:A | R y x} => Fix_sub (proj1_sig y)). Proof. - intro x; unfold Fix_sub in |- *. + intro x; unfold Fix_sub. rewrite <- (Fix_F_eq ). apply F_ext; intros. apply Fix_F_inv. diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index 05a27cc43..c20c3ba7f 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -488,7 +488,7 @@ Definition Qc_eq_bool (x y : Qc) := Lemma Qc_eq_bool_correct : forall x y : Qc, Qc_eq_bool x y = true -> x=y. Proof. - intros x y; unfold Qc_eq_bool in |- *; case (Qc_eq_dec x y); simpl in |- *; auto. + intros x y; unfold Qc_eq_bool; case (Qc_eq_dec x y); simpl; auto. intros _ H; inversion H. Qed. diff --git a/theories/QArith/Qreals.v b/theories/QArith/Qreals.v index 24f6d7204..3730bcd7f 100644 --- a/theories/QArith/Qreals.v +++ b/theories/QArith/Qreals.v @@ -21,7 +21,7 @@ Hint Resolve IZR_nz Rmult_integral_contrapositive. Lemma eqR_Qeq : forall x y : Q, Q2R x = Q2R y -> x==y. Proof. -unfold Qeq, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qeq, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. apply eq_IZR. do 2 rewrite mult_IZR. @@ -36,24 +36,24 @@ Qed. Lemma Qeq_eqR : forall x y : Q, x==y -> Q2R x = Q2R y. Proof. -unfold Qeq, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qeq, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. set (X1 := IZR x1) in *; assert (X2nz := IZR_nz x2); set (X2 := IZR (Zpos x2)) in *. set (Y1 := IZR y1) in *; assert (Y2nz := IZR_nz y2); set (Y2 := IZR (Zpos y2)) in *. assert ((X1 * Y2)%R = (Y1 * X2)%R). - unfold X1, X2, Y1, Y2 in |- *; do 2 rewrite <- mult_IZR. + unfold X1, X2, Y1, Y2; do 2 rewrite <- mult_IZR. apply IZR_eq; auto. clear H. field_simplify_eq; auto. ring_simplify X1 Y2 (Y2 * X1)%R. -rewrite H0 in |- *; ring. +rewrite H0; ring. Qed. Lemma Rle_Qle : forall x y : Q, (Q2R x <= Q2R y)%R -> x<=y. Proof. -unfold Qle, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qle, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. apply le_IZR. do 2 rewrite mult_IZR. @@ -65,37 +65,37 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos. -unfold X2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_le; +unfold X2; replace 0%R with (IZR 0); auto; apply IZR_le; auto with zarith. -unfold Y2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_le; +unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_le; auto with zarith. Qed. Lemma Qle_Rle : forall x y : Q, x<=y -> (Q2R x <= Q2R y)%R. Proof. -unfold Qle, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qle, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. set (X1 := IZR x1) in *; assert (X2nz := IZR_nz x2); set (X2 := IZR (Zpos x2)) in *. set (Y1 := IZR y1) in *; assert (Y2nz := IZR_nz y2); set (Y2 := IZR (Zpos y2)) in *. assert (X1 * Y2 <= Y1 * X2)%R. - unfold X1, X2, Y1, Y2 in |- *; do 2 rewrite <- mult_IZR. + unfold X1, X2, Y1, Y2; do 2 rewrite <- mult_IZR. apply IZR_le; auto. clear H. replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos; apply Rlt_le; apply Rinv_0_lt_compat. -unfold X2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. -unfold Y2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. Qed. Lemma Rlt_Qlt : forall x y : Q, (Q2R x < Q2R y)%R -> x<y. Proof. -unfold Qlt, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qlt, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. apply lt_IZR. do 2 rewrite mult_IZR. @@ -107,38 +107,38 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat. -unfold X2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. -unfold Y2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. Qed. Lemma Qlt_Rlt : forall x y : Q, x<y -> (Q2R x < Q2R y)%R. Proof. -unfold Qlt, Q2R in |- *; intros (x1, x2) (y1, y2); unfold Qnum, Qden in |- *; +unfold Qlt, Q2R; intros (x1, x2) (y1, y2); unfold Qnum, Qden; intros. set (X1 := IZR x1) in *; assert (X2nz := IZR_nz x2); set (X2 := IZR (Zpos x2)) in *. set (Y1 := IZR y1) in *; assert (Y2nz := IZR_nz y2); set (Y2 := IZR (Zpos y2)) in *. assert (X1 * Y2 < Y1 * X2)%R. - unfold X1, X2, Y1, Y2 in |- *; do 2 rewrite <- mult_IZR. + unfold X1, X2, Y1, Y2; do 2 rewrite <- mult_IZR. apply IZR_lt; auto. clear H. replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat; apply Rinv_0_lt_compat. -unfold X2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. -unfold Y2 in |- *; replace 0%R with (IZR 0); auto; apply IZR_lt; red in |- *; +unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; auto with zarith. Qed. Lemma Q2R_plus : forall x y : Q, Q2R (x+y) = (Q2R x + Q2R y)%R. Proof. -unfold Qplus, Qeq, Q2R in |- *; intros (x1, x2) (y1, y2); - unfold Qden, Qnum in |- *. +unfold Qplus, Qeq, Q2R; intros (x1, x2) (y1, y2); + unfold Qden, Qnum. simpl_mult. rewrite plus_IZR. do 3 rewrite mult_IZR. @@ -147,8 +147,8 @@ Qed. Lemma Q2R_mult : forall x y : Q, Q2R (x*y) = (Q2R x * Q2R y)%R. Proof. -unfold Qmult, Qeq, Q2R in |- *; intros (x1, x2) (y1, y2); - unfold Qden, Qnum in |- *. +unfold Qmult, Qeq, Q2R; intros (x1, x2) (y1, y2); + unfold Qden, Qnum. simpl_mult. do 2 rewrite mult_IZR. field; auto. @@ -156,24 +156,24 @@ Qed. Lemma Q2R_opp : forall x : Q, Q2R (- x) = (- Q2R x)%R. Proof. -unfold Qopp, Qeq, Q2R in |- *; intros (x1, x2); unfold Qden, Qnum in |- *. +unfold Qopp, Qeq, Q2R; intros (x1, x2); unfold Qden, Qnum. rewrite Ropp_Ropp_IZR. field; auto. Qed. Lemma Q2R_minus : forall x y : Q, Q2R (x-y) = (Q2R x - Q2R y)%R. -unfold Qminus in |- *; intros; rewrite Q2R_plus; rewrite Q2R_opp; auto. +unfold Qminus; intros; rewrite Q2R_plus; rewrite Q2R_opp; auto. Qed. Lemma Q2R_inv : forall x : Q, ~ x==0 -> Q2R (/x) = (/ Q2R x)%R. Proof. -unfold Qinv, Q2R, Qeq in |- *; intros (x1, x2); unfold Qden, Qnum in |- *. +unfold Qinv, Q2R, Qeq; intros (x1, x2); unfold Qden, Qnum. case x1. -simpl in |- *; intros; elim H; trivial. +simpl; intros; elim H; trivial. intros; field; auto. intros; - change (IZR (Zneg x2)) with (- IZR (' x2))%R in |- *; - change (IZR (Zneg p)) with (- IZR (' p))%R in |- *; + change (IZR (Zneg x2)) with (- IZR (' x2))%R; + change (IZR (Zneg p)) with (- IZR (' p))%R; field; (*auto 8 with real.*) repeat split; auto; auto with real. Qed. @@ -181,7 +181,7 @@ Qed. Lemma Q2R_div : forall x y : Q, ~ y==0 -> Q2R (x/y) = (Q2R x / Q2R y)%R. Proof. -unfold Qdiv, Rdiv in |- *. +unfold Qdiv, Rdiv. intros; rewrite Q2R_mult. rewrite Q2R_inv; auto. Qed. @@ -205,7 +205,7 @@ Qed. Let ex2 : forall x y : Q, ~ y==0 -> (x/y)*y == x. intros; QField. intro; apply H; apply eqR_Qeq. -rewrite H0; unfold Q2R in |- *; simpl in |- *; field; auto with real. +rewrite H0; unfold Q2R; simpl; field; auto with real. Qed. End LegacyQField. diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v index 5e91accfe..9f78e0b66 100644 --- a/theories/Reals/Alembert.v +++ b/theories/Reals/Alembert.v @@ -31,23 +31,23 @@ Proof. { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }). intro X; apply X. apply completeness. - unfold Un_cv in H0; unfold bound in |- *; cut (0 < / 2); + unfold Un_cv in H0; unfold bound; cut (0 < / 2); [ intro | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H0 (/ 2) H1); intros. exists (sum_f_R0 An x + 2 * An (S x)). - unfold is_upper_bound in |- *; intros; unfold EUn in H3; elim H3; intros. + unfold is_upper_bound; intros; unfold EUn in H3; elim H3; intros. rewrite H4; assert (H5 := lt_eq_lt_dec x1 x). elim H5; intros. elim a; intro. replace (sum_f_R0 An x) with (sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)). - pattern (sum_f_R0 An x1) at 1 in |- *; rewrite <- Rplus_0_r; + pattern (sum_f_R0 An x1) at 1; rewrite <- Rplus_0_r; rewrite Rplus_assoc; apply Rplus_le_compat_l. left; apply Rplus_lt_0_compat. apply tech1; intros; apply H. apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. - symmetry in |- *; apply tech2; assumption. - rewrite b; pattern (sum_f_R0 An x) at 1 in |- *; rewrite <- Rplus_0_r; + symmetry ; apply tech2; assumption. + rewrite b; pattern (sum_f_R0 An x) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. replace (sum_f_R0 An x1) with @@ -64,14 +64,14 @@ Proof. left; apply H. rewrite tech3. replace (1 - / 2) with (/ 2). - unfold Rdiv in |- *; rewrite Rinv_involutive. - pattern 2 at 3 in |- *; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2); + unfold Rdiv; rewrite Rinv_involutive. + pattern 2 at 3; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2); apply Rmult_le_compat_l. left; prove_sup0. left; apply Rplus_lt_reg_r with ((/ 2) ^ S (x1 - S x)). replace ((/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x))) with 1; [ idtac | ring ]. - rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. apply pow_lt; apply Rinv_0_lt_compat; prove_sup0. discrR. @@ -80,14 +80,14 @@ Proof. ring. discrR. discrR. - pattern 1 at 3 in |- *; replace 1 with (/ 1); + pattern 1 at 3; replace 1 with (/ 1); [ apply tech7; discrR | apply Rinv_1 ]. replace (An (S x)) with (An (S x + 0)%nat). apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)). left; apply Rinv_0_lt_compat; prove_sup0. intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n). intro; replace (S x + S i)%nat with (S (S x + i)). - apply H6; unfold ge in |- *; apply tech8. + apply H6; unfold ge; apply tech8. apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. intros; unfold R_dist in H2; apply Rmult_lt_reg_l with (/ An n). apply Rinv_0_lt_compat; apply H. @@ -96,20 +96,20 @@ Proof. rewrite Rmult_1_r; replace (An (S n) * / An n) with (Rabs (Rabs (An (S n) / An n) - 0)). apply H2; assumption. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; rewrite Rabs_right. - unfold Rdiv in |- *; reflexivity. - left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + unfold Rdiv; reflexivity. + left; unfold Rdiv; change (0 < An (S n) * / An n); apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ]. - red in |- *; intro; assert (H8 := H n); rewrite H7 in H8; + red; intro; assert (H8 := H n); rewrite H7 in H8; elim (Rlt_irrefl _ H8). replace (S x + 0)%nat with (S x); [ reflexivity | ring ]. - symmetry in |- *; apply tech2; assumption. - exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. + symmetry ; apply tech2; assumption. + exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity. intro X; elim X; intros. exists x; apply Un_cv_crit_lub; - [ unfold Un_growing in |- *; intro; rewrite tech5; - pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + [ unfold Un_growing; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H | apply p ]. Defined. @@ -131,14 +131,14 @@ Proof. assert (H6 := Alembert_C1 Wn H2 H4). elim H5; intros. elim H6; intros. - exists (x - x0); unfold Un_cv in |- *; unfold Un_cv in p; + exists (x - x0); unfold Un_cv; unfold Un_cv in p; unfold Un_cv in p0; intros; cut (0 < eps / 2). intro; elim (p (eps / 2) H8); clear p; intros. elim (p0 (eps / 2) H8); clear p0; intros. set (N := max x1 x2). exists N; intros; replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n). - unfold R_dist in |- *; + unfold R_dist; replace (sum_f_R0 Vn n - sum_f_R0 Wn n - (x - x0)) with (sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)); [ idtac | ring ]; apply Rle_lt_trans with @@ -146,29 +146,29 @@ Proof. apply Rabs_triang. rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2). apply Rplus_lt_compat. - unfold R_dist in H9; apply H9; unfold ge in |- *; apply le_trans with N; - [ unfold N in |- *; apply le_max_l | assumption ]. - unfold R_dist in H10; apply H10; unfold ge in |- *; apply le_trans with N; - [ unfold N in |- *; apply le_max_r | assumption ]. - right; symmetry in |- *; apply double_var. - symmetry in |- *; apply tech11; intro; unfold Vn, Wn in |- *; - unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2)); + unfold R_dist in H9; apply H9; unfold ge; apply le_trans with N; + [ unfold N; apply le_max_l | assumption ]. + unfold R_dist in H10; apply H10; unfold ge; apply le_trans with N; + [ unfold N; apply le_max_r | assumption ]. + right; symmetry ; apply double_var. + symmetry ; apply tech11; intro; unfold Vn, Wn; + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_eq_reg_l with 2. rewrite Rmult_minus_distr_l; repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. ring. discrR. discrR. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. cut (forall n:nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)). intro; cut (forall n:nat, / Wn n <= 2 * / Rabs (An n)). intro; cut (forall n:nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)). - intro; unfold Un_cv in |- *; intros; unfold Un_cv in H0; cut (0 < eps / 3). + intro; unfold Un_cv; intros; unfold Un_cv in H0; cut (0 < eps / 3). intro; elim (H0 (eps / 3) H8); intros. exists x; intros. assert (H11 := H9 n H10). - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H11; unfold Rminus in H11; rewrite Ropp_0 in H11; rewrite Rplus_0_r in H11; rewrite Rabs_Rabsolu in H11; rewrite Rabs_right. @@ -179,13 +179,13 @@ Proof. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H11; exact H11. - left; change (0 < Wn (S n) / Wn n) in |- *; unfold Rdiv in |- *; + left; change (0 < Wn (S n) / Wn n); unfold Rdiv; apply Rmult_lt_0_compat. apply H2. apply Rinv_0_lt_compat; apply H2. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + intro; unfold Rdiv; rewrite Rabs_mult; rewrite <- Rmult_assoc; replace 3 with (2 * (3 * / 2)); [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; apply Rle_trans with (Wn (S n) * 2 * / Rabs (An n)). @@ -218,32 +218,32 @@ Proof. rewrite Rmult_1_l; elim (H4 n); intros; assumption. discrR. apply Rabs_no_R0; apply H. - red in |- *; intro; assert (H6 := H2 n); rewrite H5 in H6; + red; intro; assert (H6 := H2 n); rewrite H5 in H6; elim (Rlt_irrefl _ H6). intro; split. - unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + unfold Wn; unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; prove_sup0. - pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; - unfold Rminus in |- *; rewrite Rplus_assoc; apply Rplus_le_compat_l. + pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; rewrite double; + unfold Rminus; rewrite Rplus_assoc; apply Rplus_le_compat_l. apply Rplus_le_reg_l with (An n). rewrite Rplus_0_r; rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; apply RRle_abs. - unfold Wn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + unfold Wn; unfold Rdiv; repeat rewrite <- (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; prove_sup0. - unfold Rminus in |- *; rewrite double; + unfold Rminus; rewrite double; replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l. rewrite <- Rabs_Ropp; apply RRle_abs. cut (forall n:nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)). intro; cut (forall n:nat, / Vn n <= 2 * / Rabs (An n)). intro; cut (forall n:nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)). - intro; unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / 3). + intro; unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / 3). intro; elim (H0 (eps / 3) H7); intros. exists x; intros. assert (H10 := H8 n H9). - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H10; unfold Rminus in H10; rewrite Ropp_0 in H10; rewrite Rplus_0_r in H10; rewrite Rabs_Rabsolu in H10; rewrite Rabs_right. @@ -254,13 +254,13 @@ Proof. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H10; exact H10. - left; change (0 < Vn (S n) / Vn n) in |- *; unfold Rdiv in |- *; + left; change (0 < Vn (S n) / Vn n); unfold Rdiv; apply Rmult_lt_0_compat. apply H1. apply Rinv_0_lt_compat; apply H1. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + intro; unfold Rdiv; rewrite Rabs_mult; rewrite <- Rmult_assoc; replace 3 with (2 * (3 * / 2)); [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; apply Rle_trans with (Vn (S n) * 2 * / Rabs (An n)). @@ -293,44 +293,44 @@ Proof. rewrite Rmult_1_l; elim (H3 n); intros; assumption. discrR. apply Rabs_no_R0; apply H. - red in |- *; intro; assert (H5 := H1 n); rewrite H4 in H5; + red; intro; assert (H5 := H1 n); rewrite H4 in H5; elim (Rlt_irrefl _ H5). intro; split. - unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + unfold Vn; unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; prove_sup0. - pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; + pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; rewrite double; rewrite Rplus_assoc; apply Rplus_le_compat_l. apply Rplus_le_reg_l with (- An n); rewrite Rplus_0_r; rewrite <- (Rplus_comm (An n)); rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite <- Rabs_Ropp; apply RRle_abs. - unfold Vn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + unfold Vn; unfold Rdiv; repeat rewrite <- (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; prove_sup0. - unfold Rminus in |- *; rewrite double; + unfold Rminus; rewrite double; replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l; apply RRle_abs. - intro; unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + intro; unfold Wn; unfold Rdiv; rewrite <- (Rmult_0_r (/ 2)); rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. apply Rinv_0_lt_compat; prove_sup0. - apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus in |- *; + apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rle_lt_trans with (Rabs (An n)). apply RRle_abs. - rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. - intro; unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + intro; unfold Vn; unfold Rdiv; rewrite <- (Rmult_0_r (/ 2)); rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. apply Rinv_0_lt_compat; prove_sup0. - apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus in |- *; + apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm (- An n)); rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r; apply Rle_lt_trans with (Rabs (An n)). rewrite <- Rabs_Ropp; apply RRle_abs. - rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. Defined. @@ -347,11 +347,11 @@ Proof. intro; assert (H4 := Alembert_C2 Bn H2 H3). elim H4; intros. exists x0; unfold Bn in p; apply tech12; assumption. - unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x). + unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x). intro; elim (H1 (eps / Rabs x) H4); intros. - exists x0; intros; unfold R_dist in |- *; unfold Rminus in |- *; + exists x0; intros; unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; - unfold Bn in |- *; + unfold Bn; replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs x). apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. @@ -360,22 +360,22 @@ Proof. rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H5; replace (Rabs (An (S n) / An n)) with (R_dist (Rabs (An (S n) * / An n)) 0). apply H5; assumption. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; - rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv in |- *; + unfold R_dist; unfold Rminus; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv; reflexivity. apply Rabs_no_R0; assumption. replace (S n) with (n + 1)%nat; [ idtac | ring ]; rewrite pow_add; - unfold Rdiv in |- *; rewrite Rinv_mult_distr. + unfold Rdiv; rewrite Rinv_mult_distr. replace (An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)) with (An (n + 1)%nat * x ^ 1 * / An n * (x ^ n * / x ^ n)); [ idtac | ring ]; rewrite <- Rinv_r_sym. - simpl in |- *; ring. + simpl; ring. apply pow_nonzero; assumption. apply H0. apply pow_nonzero; assumption. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. - intro; unfold Bn in |- *; apply prod_neq_R0; + intro; unfold Bn; apply prod_neq_R0; [ apply H0 | apply pow_nonzero; assumption ]. Defined. @@ -383,14 +383,14 @@ Lemma AlembertC3_step2 : forall (An:nat -> R) (x:R), x = 0 -> { l:R | Pser An x l }. Proof. intros; exists (An 0%nat). - unfold Pser in |- *; unfold infinite_sum in |- *; intros; exists 0%nat; intros; + unfold Pser; unfold infinite_sum; intros; exists 0%nat; intros; replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat). - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + unfold R_dist; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. induction n as [| n Hrecn]. - simpl in |- *; ring. + simpl; ring. rewrite tech5; rewrite Hrecn; - [ rewrite H; simpl in |- *; ring | unfold ge in |- *; apply le_O_n ]. + [ rewrite H; simpl; ring | unfold ge; apply le_O_n ]. Qed. (** A useful criterion of convergence for power series *) @@ -404,11 +404,11 @@ Proof. elim s; intro. cut (x <> 0). intro; apply AlembertC3_step1; assumption. - red in |- *; intro; rewrite H1 in a; elim (Rlt_irrefl _ a). + red; intro; rewrite H1 in a; elim (Rlt_irrefl _ a). apply AlembertC3_step2; assumption. cut (x <> 0). intro; apply AlembertC3_step1; assumption. - red in |- *; intro; rewrite H1 in r; elim (Rlt_irrefl _ r). + red; intro; rewrite H1 in r; elim (Rlt_irrefl _ r). Defined. Lemma Alembert_C4 : @@ -428,8 +428,8 @@ Proof. elim H1; intros. elim H2; intros. elim H4; intros. - unfold bound in |- *; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)). - unfold is_upper_bound in |- *; intros; unfold EUn in H6. + unfold bound; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)). + unfold is_upper_bound; intros; unfold EUn in H6. elim H6; intros. rewrite H7. assert (H8 := lt_eq_lt_dec x2 x0). @@ -437,7 +437,7 @@ Proof. elim a; intro. replace (sum_f_R0 An x0) with (sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)). - pattern (sum_f_R0 An x2) at 1 in |- *; rewrite <- Rplus_0_r. + pattern (sum_f_R0 An x2) at 1; rewrite <- Rplus_0_r. rewrite Rplus_assoc; apply Rplus_le_compat_l. left; apply Rplus_lt_0_compat. apply tech1. @@ -446,8 +446,8 @@ Proof. apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. apply H. - symmetry in |- *; apply tech2; assumption. - rewrite b; pattern (sum_f_R0 An x0) at 1 in |- *; rewrite <- Rplus_0_r; + symmetry ; apply tech2; assumption. + rewrite b; pattern (sum_f_R0 An x0) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. left; apply Rmult_lt_0_compat. apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; @@ -465,7 +465,7 @@ Proof. rewrite <- (Rmult_comm (An (S x0))); apply Rmult_le_compat_l. left; apply H. rewrite tech3. - unfold Rdiv in |- *; apply Rmult_le_reg_l with (1 - x). + unfold Rdiv; apply Rmult_le_reg_l with (1 - x). apply Rplus_lt_reg_r with x; rewrite Rplus_0_r. replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. do 2 rewrite (Rmult_comm (1 - x)). @@ -473,17 +473,17 @@ Proof. rewrite Rmult_1_r; apply Rplus_le_reg_l with (x ^ S (x2 - S x0)). replace (x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0))) with 1; [ idtac | ring ]. - rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. left; apply pow_lt. apply Rle_lt_trans with k. elim Hyp; intros; assumption. elim H3; intros; assumption. apply Rminus_eq_contra. - red in |- *; intro. + red; intro. elim H3; intros. rewrite H10 in H12; elim (Rlt_irrefl _ H12). - red in |- *; intro. + red; intro. elim H3; intros. rewrite H10 in H12; elim (Rlt_irrefl _ H12). replace (An (S x0)) with (An (S x0 + 0)%nat). @@ -496,7 +496,7 @@ Proof. intro. replace (S x0 + S i)%nat with (S (S x0 + i)). apply H9. - unfold ge in |- *. + unfold ge. apply tech8. apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. @@ -510,21 +510,21 @@ Proof. replace (An (S n) * / An n) with (Rabs (An (S n) / An n)). apply H5; assumption. rewrite Rabs_right. - unfold Rdiv in |- *; reflexivity. - left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + unfold Rdiv; reflexivity. + left; unfold Rdiv; change (0 < An (S n) * / An n); apply Rmult_lt_0_compat. apply H. apply Rinv_0_lt_compat; apply H. - red in |- *; intro. + red; intro. assert (H11 := H n). rewrite H10 in H11; elim (Rlt_irrefl _ H11). replace (S x0 + 0)%nat with (S x0); [ reflexivity | ring ]. - symmetry in |- *; apply tech2; assumption. - exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. + symmetry ; apply tech2; assumption. + exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity. intro X; elim X; intros. exists x; apply Un_cv_crit_lub; - [ unfold Un_growing in |- *; intro; rewrite tech5; - pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + [ unfold Un_growing; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H | apply p ]. Qed. @@ -551,9 +551,9 @@ Proof. apply (Alembert_C4 (fun i:nat => Rabs (An i)) k). assumption. intro; apply Rabs_pos_lt; apply H0. - unfold Un_cv in |- *. + unfold Un_cv. unfold Un_cv in H1. - unfold Rdiv in |- *. + unfold Rdiv. intros. elim (H1 eps H2); intros. exists x; intros. @@ -590,22 +590,22 @@ Lemma Alembert_C6 : elim s; intro. eapply Alembert_C5 with (k * Rabs x). split. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. left; assumption. left; apply Rabs_pos_lt. - red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). + red; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). apply Rmult_lt_reg_l with (/ k). apply Rinv_0_lt_compat; assumption. rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite Rmult_1_r; assumption. - red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). intro; apply prod_neq_R0. apply H0. apply pow_nonzero. - red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). - unfold Un_cv in |- *; unfold Un_cv in H1. + red; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). + unfold Un_cv; unfold Un_cv in H1. intros. cut (0 < eps / Rabs x). intro. @@ -613,7 +613,7 @@ Lemma Alembert_C6 : exists x0. intros. replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). - unfold R_dist in |- *. + unfold R_dist. rewrite Rabs_mult. replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. @@ -621,18 +621,18 @@ Lemma Alembert_C6 : rewrite Rabs_Rabsolu. apply Rmult_lt_reg_l with (/ Rabs x). apply Rinv_0_lt_compat; apply Rabs_pos_lt. - red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). + red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite <- (Rmult_comm eps). unfold R_dist in H5. - unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. + unfold Rdiv; unfold Rdiv in H5; apply H5; assumption. apply Rabs_no_R0. - red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). - unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). + unfold Rdiv; replace (S n) with (n + 1)%nat; [ idtac | ring ]. rewrite pow_add. - simpl in |- *. + simpl. rewrite Rmult_1_r. rewrite Rinv_mult_distr. replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with @@ -641,46 +641,46 @@ Lemma Alembert_C6 : rewrite <- Rinv_r_sym. rewrite Rmult_1_r; reflexivity. apply pow_nonzero. - red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). + red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). apply H0. apply pow_nonzero. - red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat; apply Rabs_pos_lt. - red in |- *; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a). + red; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a). exists (An 0%nat). - unfold Un_cv in |- *. + unfold Un_cv. intros. exists 0%nat. intros. - unfold R_dist in |- *. + unfold R_dist. replace (sum_f_R0 (fun i:nat => An i * x ^ i) n) with (An 0%nat). - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. induction n as [| n Hrecn]. - simpl in |- *; ring. + simpl; ring. rewrite tech5. rewrite <- Hrecn. - rewrite b; simpl in |- *; ring. - unfold ge in |- *; apply le_O_n. + rewrite b; simpl; ring. + unfold ge; apply le_O_n. eapply Alembert_C5 with (k * Rabs x). split. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. left; assumption. left; apply Rabs_pos_lt. - red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). + red; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). apply Rmult_lt_reg_l with (/ k). apply Rinv_0_lt_compat; assumption. rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite Rmult_1_r; assumption. - red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). intro; apply prod_neq_R0. apply H0. apply pow_nonzero. - red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). - unfold Un_cv in |- *; unfold Un_cv in H1. + red; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). + unfold Un_cv; unfold Un_cv in H1. intros. cut (0 < eps / Rabs x). intro. @@ -688,7 +688,7 @@ Lemma Alembert_C6 : exists x0. intros. replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). - unfold R_dist in |- *. + unfold R_dist. rewrite Rabs_mult. replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. @@ -696,18 +696,18 @@ Lemma Alembert_C6 : rewrite Rabs_Rabsolu. apply Rmult_lt_reg_l with (/ Rabs x). apply Rinv_0_lt_compat; apply Rabs_pos_lt. - red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). + red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite <- (Rmult_comm eps). unfold R_dist in H5. - unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. + unfold Rdiv; unfold Rdiv in H5; apply H5; assumption. apply Rabs_no_R0. - red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). - unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). + unfold Rdiv; replace (S n) with (n + 1)%nat; [ idtac | ring ]. rewrite pow_add. - simpl in |- *. + simpl. rewrite Rmult_1_r. rewrite Rinv_mult_distr. replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with @@ -716,12 +716,12 @@ Lemma Alembert_C6 : rewrite <- Rinv_r_sym. rewrite Rmult_1_r; reflexivity. apply pow_nonzero. - red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). + red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). apply H0. apply pow_nonzero. - red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat; apply Rabs_pos_lt. - red in |- *; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r). + red; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r). Qed. diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v index 227081cab..a6d4c2314 100644 --- a/theories/Reals/AltSeries.v +++ b/theories/Reals/AltSeries.v @@ -24,13 +24,13 @@ Lemma CV_ALT_step0 : Un_decreasing Un -> Un_growing (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))). Proof. - intros; unfold Un_growing in |- *; intro. + intros; unfold Un_growing; intro. cut ((2 * S n)%nat = S (S (2 * n))). intro; rewrite H0. do 4 rewrite tech5; repeat rewrite Rplus_assoc; apply Rplus_le_compat_l. - pattern (tg_alt Un (S (2 * n))) at 1 in |- *; rewrite <- Rplus_0_r. + pattern (tg_alt Un (S (2 * n))) at 1; rewrite <- Rplus_0_r. apply Rplus_le_compat_l. - unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; + unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; rewrite Rmult_1_l. apply Rplus_le_reg_l with (Un (S (2 * S n))). rewrite Rplus_0_r; @@ -46,12 +46,12 @@ Lemma CV_ALT_step1 : Un_decreasing Un -> Un_decreasing (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)). Proof. - intros; unfold Un_decreasing in |- *; intro. + intros; unfold Un_decreasing; intro. cut ((2 * S n)%nat = S (S (2 * n))). intro; rewrite H0; do 2 rewrite tech5; repeat rewrite Rplus_assoc. - pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2 in |- *; rewrite <- Rplus_0_r. + pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2; rewrite <- Rplus_0_r. apply Rplus_le_compat_l. - unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; + unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; rewrite Rmult_1_l. apply Rplus_le_reg_l with (Un (S (2 * n))). rewrite Rplus_0_r; @@ -70,7 +70,7 @@ Lemma CV_ALT_step2 : sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N)) <= 0. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r. + simpl; unfold tg_alt; simpl; rewrite Rmult_1_r. replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ]. apply Rplus_le_reg_l with (Un 1%nat); rewrite Rplus_0_r. replace (Un 1%nat + (-1 * Un 1%nat + Un 2%nat)) with (Un 2%nat); @@ -78,10 +78,10 @@ Proof. cut (S (2 * S N) = S (S (S (2 * N)))). intro; rewrite H1; do 2 rewrite tech5. apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))). - pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2 in |- *; + pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2; rewrite <- Rplus_0_r. rewrite Rplus_assoc; apply Rplus_le_compat_l. - unfold tg_alt in |- *; rewrite <- H1. + unfold tg_alt; rewrite <- H1. rewrite pow_1_odd. cut (S (S (2 * S N)) = (2 * S (S N))%nat). intro; rewrite H2; rewrite pow_1_even; rewrite Rmult_1_l; rewrite <- H2. @@ -102,7 +102,7 @@ Lemma CV_ALT_step3 : positivity_seq Un -> sum_f_R0 (fun i:nat => tg_alt Un (S i)) N <= 0. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r. + simpl; unfold tg_alt; simpl; rewrite Rmult_1_r. apply Rplus_le_reg_l with (Un 1%nat). rewrite Rplus_0_r; replace (Un 1%nat + -1 * Un 1%nat) with 0; [ apply H0 | ring ]. @@ -112,10 +112,10 @@ Proof. rewrite H3; apply CV_ALT_step2; assumption. rewrite H3; rewrite tech5. apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))). - pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2 in |- *; + pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2; rewrite <- Rplus_0_r. apply Rplus_le_compat_l. - unfold tg_alt in |- *; simpl in |- *. + unfold tg_alt; simpl. replace (x + (x + 0))%nat with (2 * x)%nat; [ idtac | ring ]. rewrite pow_1_even. replace (-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x))))) with @@ -133,15 +133,15 @@ Lemma CV_ALT_step4 : positivity_seq Un -> has_ub (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))). Proof. - intros; unfold has_ub in |- *; unfold bound in |- *. + intros; unfold has_ub; unfold bound. exists (Un 0%nat). - unfold is_upper_bound in |- *; intros; elim H1; intros. + unfold is_upper_bound; intros; elim H1; intros. rewrite H2; rewrite decomp_sum. replace (tg_alt Un 0) with (Un 0%nat). - pattern (Un 0%nat) at 2 in |- *; rewrite <- Rplus_0_r. + pattern (Un 0%nat) at 2; rewrite <- Rplus_0_r. apply Rplus_le_compat_l. apply CV_ALT_step3; assumption. - unfold tg_alt in |- *; simpl in |- *; ring. + unfold tg_alt; simpl; ring. apply lt_O_Sn. Qed. @@ -159,11 +159,11 @@ Proof. assert (X := growing_cv _ H2 H3). elim X; intros. exists x. - unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold Un_cv; unfold R_dist; unfold Un_cv in H1; unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p. intros; cut (0 < eps / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H1 (eps / 2) H5); intros N2 H6. elim (p (eps / 2) H5); intros N1 H7. @@ -180,32 +180,32 @@ Proof. apply Rabs_triang. rewrite (double_var eps); apply Rplus_lt_compat. rewrite H12; apply H7; assumption. - rewrite Rabs_Ropp; unfold tg_alt in |- *; rewrite Rabs_mult; + rewrite Rabs_Ropp; unfold tg_alt; rewrite Rabs_mult; rewrite pow_1_abs; rewrite Rmult_1_l; unfold Rminus in H6; rewrite Ropp_0 in H6; rewrite <- (Rplus_0_r (Un (S n))); apply H6. - unfold ge in |- *; apply le_trans with n. - apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ]. + unfold ge; apply le_trans with n. + apply le_trans with N; [ unfold N; apply le_max_r | assumption ]. apply le_n_Sn. rewrite tech5; ring. rewrite H12; apply Rlt_trans with (eps / 2). apply H7; assumption. - unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. + unfold Rdiv; apply Rmult_lt_reg_l with 2. prove_sup0. rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ]. rewrite double. - pattern eps at 1 in |- *; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l; + pattern eps at 1; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l; assumption. elim H10; intro; apply le_double. rewrite <- H11; apply le_trans with N. - unfold N in |- *; apply le_trans with (S (2 * N1)); + unfold N; apply le_trans with (S (2 * N1)); [ apply le_n_Sn | apply le_max_l ]. assumption. apply lt_n_Sm_le. rewrite <- H11. apply lt_le_trans with N. - unfold N in |- *; apply lt_le_trans with (S (2 * N1)). + unfold N; apply lt_le_trans with (S (2 * N1)). apply lt_n_Sn. apply le_max_l. assumption. @@ -222,7 +222,7 @@ Theorem alternated_series : Proof. intros; apply CV_ALT. assumption. - unfold positivity_seq in |- *; apply decreasing_ineq; assumption. + unfold positivity_seq; apply decreasing_ineq; assumption. assumption. Qed. @@ -243,31 +243,31 @@ Proof. apply (decreasing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N))). apply CV_ALT_step1; assumption. assumption. - unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold Un_cv; unfold R_dist; unfold Un_cv in H1; unfold R_dist in H1; intros. elim (H1 eps H2); intros. exists x; intros. apply H3. - unfold ge in |- *; apply le_trans with (2 * n)%nat. + unfold ge; apply le_trans with (2 * n)%nat. apply le_trans with n. assumption. assert (H5 := mult_O_le n 2). elim H5; intro. cut (0%nat <> 2%nat); - [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. + [ intro; elim H7; symmetry ; assumption | discriminate ]. assumption. apply le_n_Sn. - unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold Un_cv; unfold R_dist; unfold Un_cv in H1; unfold R_dist in H1; intros. elim (H1 eps H2); intros. exists x; intros. apply H3. - unfold ge in |- *; apply le_trans with n. + unfold ge; apply le_trans with n. assumption. assert (H5 := mult_O_le n 2). elim H5; intro. cut (0%nat <> 2%nat); - [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. + [ intro; elim H7; symmetry ; assumption | discriminate ]. assumption. Qed. @@ -279,13 +279,13 @@ Definition PI_tg (n:nat) := / INR (2 * n + 1). Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n. Proof. - intro; unfold PI_tg in |- *; left; apply Rinv_0_lt_compat; apply lt_INR_0; + intro; unfold PI_tg; left; apply Rinv_0_lt_compat; apply lt_INR_0; replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. Qed. Lemma PI_tg_decreasing : Un_decreasing PI_tg. Proof. - unfold PI_tg, Un_decreasing in |- *; intro. + unfold PI_tg, Un_decreasing; intro. apply Rmult_le_reg_l with (INR (2 * n + 1)). apply lt_INR_0. replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. @@ -306,7 +306,7 @@ Qed. Lemma PI_tg_cv : Un_cv PI_tg 0. Proof. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. cut (0 < 2 * eps); [ intro | apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ]. assert (H1 := archimed (/ (2 * eps))). @@ -316,9 +316,9 @@ Proof. cut (0 < N)%nat. intro; exists N; intros. cut (0 < n)%nat. - intro; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + intro; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_right. - unfold PI_tg in |- *; apply Rlt_trans with (/ INR (2 * n)). + unfold PI_tg; apply Rlt_trans with (/ INR (2 * n)). apply Rmult_lt_reg_l with (INR (2 * n)). apply lt_INR_0. replace (2 * n)%nat with (n + n)%nat; [ idtac | ring ]. @@ -337,27 +337,27 @@ Proof. [ discriminate | ring ]. replace n with (S (pred n)). apply not_O_INR; discriminate. - symmetry in |- *; apply S_pred with 0%nat. + symmetry ; apply S_pred with 0%nat. assumption. apply Rle_lt_trans with (/ INR (2 * N)). apply Rmult_le_reg_l with (INR (2 * N)). rewrite mult_INR; apply Rmult_lt_0_compat; - [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. + [ simpl; prove_sup0 | apply lt_INR_0; assumption ]. rewrite <- Rinv_r_sym. apply Rmult_le_reg_l with (INR (2 * n)). rewrite mult_INR; apply Rmult_lt_0_compat; - [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. + [ simpl; prove_sup0 | apply lt_INR_0; assumption ]. rewrite (Rmult_comm (INR (2 * n))); rewrite Rmult_assoc; rewrite <- Rinv_l_sym. do 2 rewrite Rmult_1_r; apply le_INR. apply (fun m n p:nat => mult_le_compat_l p n m); assumption. replace n with (S (pred n)). apply not_O_INR; discriminate. - symmetry in |- *; apply S_pred with 0%nat. + symmetry ; apply S_pred with 0%nat. assumption. replace N with (S (pred N)). apply not_O_INR; discriminate. - symmetry in |- *; apply S_pred with 0%nat. + symmetry ; apply S_pred with 0%nat. assumption. rewrite mult_INR. rewrite Rinv_mult_distr. @@ -377,14 +377,14 @@ Proof. rewrite Rmult_1_r; replace (INR N) with (IZR (Z.of_nat N)). rewrite <- H4. elim H1; intros; assumption. - symmetry in |- *; apply INR_IZR_INZ. + symmetry ; apply INR_IZR_INZ. apply prod_neq_R0; - [ discrR | red in |- *; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ]. + [ discrR | red; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ]. apply not_O_INR. - red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). + red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). replace (INR 2) with 2; [ discrR | reflexivity ]. apply not_O_INR. - red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). + red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). apply Rle_ge; apply PI_tg_pos. apply lt_le_trans with N; assumption. elim H1; intros H5 _. @@ -399,7 +399,7 @@ Proof. elim (Rlt_irrefl _ (Rlt_trans _ _ _ H7 H5)). elim (lt_n_O _ b). apply le_IZR. - simpl in |- *. + simpl. left; apply Rlt_trans with (/ (2 * eps)). apply Rinv_0_lt_compat; assumption. elim H1; intros; assumption. @@ -425,10 +425,10 @@ Proof. intro; apply alternated_series_ineq. apply PI_tg_decreasing. apply PI_tg_cv. - unfold Alt_PI in |- *; case exist_PI; intro. + unfold Alt_PI; case exist_PI; intro. replace (4 * x / 4) with x. trivial. - unfold Rdiv in |- *; rewrite (Rmult_comm 4); rewrite Rmult_assoc; + unfold Rdiv; rewrite (Rmult_comm 4); rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | discrR ]. Qed. @@ -441,14 +441,14 @@ Proof. elim H; clear H; intros H _. unfold Rdiv in H; apply Rlt_le_trans with (sum_f_R0 (tg_alt PI_tg) (S (2 * 0))). - simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_l; + simpl; unfold tg_alt; simpl; rewrite Rmult_1_l; rewrite Rmult_1_r; apply Rplus_lt_reg_r with (PI_tg 1). rewrite Rplus_0_r; replace (PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)) with (PI_tg 0); - [ unfold PI_tg in |- * | ring ]. - simpl in |- *; apply Rinv_lt_contravar. + [ unfold PI_tg | ring ]. + simpl; apply Rinv_lt_contravar. rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ]. - rewrite Rplus_comm; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + rewrite Rplus_comm; pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; prove_sup0. assumption. Qed. diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v index 359b2855c..2580cc3a7 100644 --- a/theories/Reals/ArithProp.v +++ b/theories/Reals/ArithProp.v @@ -17,7 +17,7 @@ Local Open Scope R_scope. Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat. Proof. - intros; red in |- *; intro. + intros; red; intro. cut (forall n m:nat, (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m). intro; assert (H2 := H1 _ _ (lt_le_weak _ _ H) H0); rewrite H2 in H; elim (lt_irrefl _ H). @@ -27,11 +27,11 @@ Proof. forall n0 m:nat, (m <= n0)%nat -> (n0 - m)%nat = 0%nat -> n0 = m). intro; apply H1. apply nat_double_ind. - unfold R in |- *; intros; inversion H2; reflexivity. - unfold R in |- *; intros; simpl in H3; assumption. - unfold R in |- *; intros; simpl in H4; assert (H5 := le_S_n _ _ H3); + unfold R; intros; inversion H2; reflexivity. + unfold R; intros; simpl in H3; assumption. + unfold R; intros; simpl in H4; assert (H5 := le_S_n _ _ H3); assert (H6 := H2 H5 H4); rewrite H6; reflexivity. - unfold R in |- *; intros; apply H1; assumption. + unfold R; intros; apply H1; assumption. Qed. Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat. @@ -41,20 +41,20 @@ Proof. ((forall m n:nat, R m n) -> forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat). intro; apply H. apply nat_double_ind. - unfold R in |- *; intros; simpl in |- *; apply le_n. - unfold R in |- *; intros; simpl in |- *; apply le_n. - unfold R in |- *; intros; simpl in |- *; apply le_trans with n. + unfold R; intros; simpl; apply le_n. + unfold R; intros; simpl; apply le_n. + unfold R; intros; simpl; apply le_trans with n. apply H0; apply le_S_n; assumption. apply le_n_Sn. - unfold R in |- *; intros; apply H; assumption. + unfold R; intros; apply H; assumption. Qed. Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat. Proof. - intros n m; pattern n, m in |- *; apply nat_double_ind; + intros n m; pattern n, m; apply nat_double_ind; [ intros; rewrite <- minus_n_O; assumption | intros; elim (lt_n_O _ H) - | intros; simpl in |- *; apply H; apply lt_S_n; assumption ]. + | intros; simpl; apply H; apply lt_S_n; assumption ]. Qed. Lemma even_odd_cor : @@ -73,7 +73,7 @@ Proof. apply H3; assumption. right. apply H4; assumption. - unfold double in |- *;ring. + unfold double;ring. Qed. (* 2m <= 2n => m<=n *) @@ -105,9 +105,9 @@ Proof. exists (x - IZR k0 * y). split. ring. - unfold k0 in |- *; case (Rcase_abs y); intro. - assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl in |- *; - unfold Rminus in |- *. + unfold k0; case (Rcase_abs y); intro. + assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl; + unfold Rminus. replace (- ((1 + - IZR (up (x / - y))) * y)) with ((IZR (up (x / - y)) - 1) * y); [ idtac | ring ]. split. @@ -118,7 +118,7 @@ Proof. rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]. apply Rplus_le_reg_l with (IZR (up (x / - y)) - x / - y). - rewrite Rplus_0_r; unfold Rdiv in |- *; pattern (/ - y) at 4 in |- *; + rewrite Rplus_0_r; unfold Rdiv; pattern (/ - y) at 4; rewrite <- Ropp_inv_permute; [ idtac | assumption ]. replace (IZR (up (x * / - y)) - x * - / y + @@ -138,11 +138,11 @@ Proof. replace (IZR (up (x / - y)) - 1 + (- (x * / y) + - (IZR (up (x / - y)) - 1))) with (- (x * / y)); [ idtac | ring ]. rewrite <- Ropp_mult_distr_r_reverse; rewrite (Ropp_inv_permute _ H); elim H0; - unfold Rdiv in |- *; intros H1 _; exact H1. + unfold Rdiv; intros H1 _; exact H1. apply Ropp_neq_0_compat; assumption. - assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl in |- *; + assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl; cut (0 < y). - intro; unfold Rminus in |- *; + intro; unfold Rminus; replace (- ((IZR (up (x / y)) + -1) * y)) with ((1 - IZR (up (x / y))) * y); [ idtac | ring ]. split. @@ -152,7 +152,7 @@ Proof. rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]; apply Rplus_le_reg_l with (IZR (up (x / y)) - x / y); - rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite Rplus_0_r; unfold Rdiv; replace (IZR (up (x * / y)) - x * / y + (x * / y + (1 - IZR (up (x * / y))))) with 1; [ idtac | ring ]; elim H0; intros _ H2; unfold Rdiv in H2; @@ -166,12 +166,12 @@ Proof. replace (IZR (up (x / y)) - 1 + 1) with (IZR (up (x / y))); [ idtac | ring ]; replace (IZR (up (x / y)) - 1 + (x * / y + (1 - IZR (up (x / y))))) with - (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv in |- *; + (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv; intros H2 _; exact H2. case (total_order_T 0 y); intro. elim s; intro. assumption. - elim H; symmetry in |- *; exact b. + elim H; symmetry ; exact b. assert (H1 := Rge_le _ _ r); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r0)). Qed. diff --git a/theories/Reals/Binomial.v b/theories/Reals/Binomial.v index a7bb4d9a6..65ddf5437 100644 --- a/theories/Reals/Binomial.v +++ b/theories/Reals/Binomial.v @@ -16,7 +16,7 @@ Definition C (n p:nat) : R := Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i). Proof. - intros; unfold C in |- *; replace (n - (n - i))%nat with i. + intros; unfold C; replace (n - (n - i))%nat with i. rewrite Rmult_comm. reflexivity. apply plus_minus; rewrite plus_comm; apply le_plus_minus; assumption. @@ -26,10 +26,10 @@ Lemma pascal_step2 : forall n i:nat, (i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i. Proof. - intros; unfold C in |- *; replace (S n - i)%nat with (S (n - i)). + intros; unfold C; replace (S n - i)%nat with (S (n - i)). cut (forall n:nat, fact (S n) = (S n * fact n)%nat). intro; repeat rewrite H0. - unfold Rdiv in |- *; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. + unfold Rdiv; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. ring. apply INR_fact_neq_0. apply INR_fact_neq_0. @@ -46,13 +46,13 @@ Qed. Lemma pascal_step3 : forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i. Proof. - intros; unfold C in |- *. + intros; unfold C. cut (forall n:nat, fact (S n) = (S n * fact n)%nat). intro. cut ((n - i)%nat = S (n - S i)). intro. - pattern (n - i)%nat at 2 in |- *; rewrite H1. - repeat rewrite H0; unfold Rdiv in |- *; repeat rewrite mult_INR; + pattern (n - i)%nat at 2; rewrite H1. + repeat rewrite H0; unfold Rdiv; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. rewrite <- H1; rewrite (Rmult_comm (/ INR (n - i))); repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (n - i))); @@ -68,7 +68,7 @@ Proof. apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. apply INR_fact_neq_0. rewrite minus_Sn_m. - simpl in |- *; reflexivity. + simpl; reflexivity. apply lt_le_S; assumption. intro; reflexivity. Qed. @@ -95,13 +95,13 @@ Proof. rewrite <- minus_Sn_m. cut ((n - (n - i))%nat = i). intro; rewrite H0; reflexivity. - symmetry in |- *; apply plus_minus. + symmetry ; apply plus_minus. rewrite plus_comm; rewrite le_plus_minus_r. reflexivity. apply lt_le_weak; assumption. apply le_minusni_n; apply lt_le_weak; assumption. apply lt_le_weak; assumption. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite S_INR. rewrite minus_INR. cut (INR i + 1 <> 0). @@ -125,18 +125,18 @@ Lemma binomial : (x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n. Proof. intros; induction n as [| n Hrecn]. - unfold C in |- *; simpl in |- *; unfold Rdiv in |- *; + unfold C; simpl; unfold Rdiv; repeat rewrite Rmult_1_r; rewrite Rinv_1; ring. - pattern (S n) at 1 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + pattern (S n) at 1; replace (S n) with (n + 1)%nat; [ idtac | ring ]. rewrite pow_add; rewrite Hrecn. - replace ((x + y) ^ 1) with (x + y); [ idtac | simpl in |- *; ring ]. + replace ((x + y) ^ 1) with (x + y); [ idtac | simpl; ring ]. rewrite tech5. cut (forall p:nat, C p p = 1). cut (forall p:nat, C p 0 = 1). intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l. - replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl in |- *; reflexivity ]. + replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl; reflexivity ]. induction n as [| n Hrecn0]. - simpl in |- *; do 2 rewrite H; ring. + simpl; do 2 rewrite H; ring. (* N >= 1 *) set (N := S n). rewrite Rmult_plus_distr_l. @@ -158,7 +158,7 @@ Proof. rewrite (Rplus_comm (sum_f_R0 An n)). repeat rewrite Rplus_assoc. rewrite <- tech5. - fold N in |- *. + fold N. set (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)). cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i). intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n). @@ -166,42 +166,42 @@ Proof. rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N). replace (pred N) with n. ring. - unfold N in |- *; simpl in |- *; reflexivity. - unfold N in |- *; apply lt_O_Sn. - unfold Cn in |- *; rewrite H; simpl in |- *; ring. + unfold N; simpl; reflexivity. + unfold N; apply lt_O_Sn. + unfold Cn; rewrite H; simpl; ring. apply sum_eq. intros; apply H1. - unfold N in |- *; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ]. - intros; unfold Bn, Cn in |- *. + unfold N; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ]. + intros; unfold Bn, Cn. replace (S N - S i)%nat with (N - i)%nat; reflexivity. - unfold An in |- *; fold N in |- *; rewrite <- minus_n_n; rewrite H0; - simpl in |- *; ring. + unfold An; fold N; rewrite <- minus_n_n; rewrite H0; + simpl; ring. apply sum_eq. - intros; unfold An, Bn in |- *; replace (S N - S i)%nat with (N - i)%nat; + intros; unfold An, Bn; replace (S N - S i)%nat with (N - i)%nat; [ idtac | reflexivity ]. rewrite <- pascal; [ ring - | apply le_lt_trans with n; [ assumption | unfold N in |- *; apply lt_n_Sn ] ]. - unfold N in |- *; reflexivity. - unfold N in |- *; apply lt_O_Sn. + | apply le_lt_trans with n; [ assumption | unfold N; apply lt_n_Sn ] ]. + unfold N; reflexivity. + unfold N; apply lt_O_Sn. rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq. intros; replace (S N - i)%nat with (S (N - i)). replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ]. - rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl in |- *; ring ]; + rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl; ring ]; ring. apply minus_Sn_m; assumption. rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq. intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add; - replace (x ^ 1) with x; [ idtac | simpl in |- *; ring ]; + replace (x ^ 1) with x; [ idtac | simpl; ring ]; ring. - intro; unfold C in |- *. + intro; unfold C. replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. replace (p - 0)%nat with p; [ idtac | apply minus_n_O ]. - rewrite Rmult_1_l; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + rewrite Rmult_1_l; unfold Rdiv; rewrite <- Rinv_r_sym; [ reflexivity | apply INR_fact_neq_0 ]. - intro; unfold C in |- *. + intro; unfold C. replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ]. replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. - rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym; [ reflexivity | apply INR_fact_neq_0 ]. Qed. diff --git a/theories/Reals/Cauchy_prod.v b/theories/Reals/Cauchy_prod.v index 3019be9e8..41aaf8dc6 100644 --- a/theories/Reals/Cauchy_prod.v +++ b/theories/Reals/Cauchy_prod.v @@ -21,7 +21,7 @@ Proof. replace N with (S (pred N)). rewrite tech5. reflexivity. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. Qed. (**********) @@ -51,7 +51,7 @@ Proof. elim (lt_irrefl _ H). cut (N = 0%nat \/ (0 < N)%nat). intro; elim H0; intro. - rewrite H1; simpl in |- *; ring. + rewrite H1; simpl; ring. replace (pred (S N)) with (S (pred N)). do 5 rewrite tech5. rewrite Rmult_plus_distr_r; rewrite Rmult_plus_distr_l; rewrite (HrecN H1). @@ -66,7 +66,7 @@ Proof. repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. rewrite <- minus_n_n; cut (N = 1%nat \/ (2 <= N)%nat). intro; elim H2; intro. - rewrite H3; simpl in |- *; ring. + rewrite H3; simpl; ring. replace (sum_f_R0 (fun k:nat => @@ -147,7 +147,7 @@ Proof. (pred (pred N))). repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. replace (pred (N - pred N)) with 0%nat. - simpl in |- *; rewrite <- minus_n_O. + simpl; rewrite <- minus_n_O. replace (S (pred N)) with N. replace (sum_f_R0 (fun k:nat => An (S N) * Bn (S k)) (pred (pred N))) with (sum_f_R0 (fun k:nat => Bn (S k) * An (S N)) (pred (pred N))). @@ -161,11 +161,11 @@ Proof. apply S_pred with 0%nat; assumption. replace (N - pred N)%nat with 1%nat. reflexivity. - pattern N at 1 in |- *; replace N with (S (pred N)). + pattern N at 1; replace N with (S (pred N)). rewrite <- minus_Sn_m. rewrite <- minus_n_n; reflexivity. apply le_n. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. apply sum_eq; intros; rewrite (sum_N_predN (fun l:nat => An (S (S (l + i))) * Bn (N - l)%nat) @@ -259,7 +259,7 @@ Proof. apply le_n. apply (fun p n m:nat => plus_le_reg_l n m p) with 1%nat. rewrite le_plus_minus_r. - simpl in |- *; assumption. + simpl; assumption. apply le_trans with 2%nat; [ apply le_n_Sn | assumption ]. apply le_trans with 2%nat; [ apply le_n_Sn | assumption ]. simpl; ring. @@ -274,7 +274,7 @@ Proof. apply le_trans with (pred (pred N)). assumption. apply le_pred_n. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. apply INR_eq; rewrite S_INR; rewrite plus_INR; reflexivity. apply le_trans with (pred (pred N)). assumption. @@ -427,7 +427,7 @@ Proof. apply le_trans with (pred (pred N)). assumption. apply le_pred_n. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. apply INR_eq; rewrite S_INR; rewrite plus_INR; simpl; ring. apply le_trans with (pred (pred N)). assumption. @@ -441,11 +441,11 @@ Proof. inversion H1. left; reflexivity. right; apply le_n_S; assumption. - simpl in |- *. + simpl. replace (S (pred N)) with N. reflexivity. apply S_pred with 0%nat; assumption. - simpl in |- *. + simpl. cut ((N - pred N)%nat = 1%nat). intro; rewrite H2; reflexivity. rewrite pred_of_minus. @@ -453,7 +453,7 @@ Proof. simpl; ring. apply lt_le_S; assumption. rewrite <- pred_of_minus; apply le_pred_n. - simpl in |- *; symmetry in |- *; apply S_pred with 0%nat; assumption. + simpl; symmetry ; apply S_pred with 0%nat; assumption. inversion H. left; reflexivity. right; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | exact H1 ]. diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v index e966357ce..b679d9c41 100644 --- a/theories/Reals/Cos_plus.v +++ b/theories/Reals/Cos_plus.v @@ -29,23 +29,23 @@ Proof. intro. assert (H1 := cv_speed_pow_fact C0). unfold Un_cv in H1; unfold R_dist in H1. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. cut (0 < eps / C0); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; assumption ] ]. elim (H1 (eps / C0) H3); intros N0 H4. exists N0; intros. replace (Majxy x y n) with (C0 ^ S n / INR (fact n)). - simpl in |- *. + simpl. apply Rmult_lt_reg_l with (Rabs (/ C0)). apply Rabs_pos_lt. apply Rinv_neq_0_compat. - red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). + red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). rewrite <- Rabs_mult. - unfold Rminus in |- *; rewrite Rmult_plus_distr_l. + unfold Rminus; rewrite Rmult_plus_distr_l. rewrite Ropp_0; rewrite Rmult_0_r. - unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. + unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite (Rabs_right (/ C0)). @@ -54,15 +54,15 @@ Proof. [ idtac | ring ]. unfold Rdiv in H4; apply H4; assumption. apply Rle_ge; left; apply Rinv_0_lt_compat; assumption. - red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). - unfold Majxy in |- *. - unfold C0 in |- *. + red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). + unfold Majxy. + unfold C0. rewrite pow_mult. - unfold C in |- *; reflexivity. - unfold C0 in |- *; apply pow_lt; assumption. + unfold C; reflexivity. + unfold C0; apply pow_lt; assumption. apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *. + unfold C. apply RmaxLess1. Qed. @@ -72,7 +72,7 @@ Lemma reste1_maj : Proof. intros. set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). - unfold Reste1 in |- *. + unfold Reste1. apply Rle_trans with (sum_f_R0 (fun k:nat => @@ -120,7 +120,7 @@ Proof. C ^ (2 * S (N + k))) (pred (N - k))) (pred N)). apply sum_Rle; intros. apply sum_Rle; intros. - unfold Rdiv in |- *; repeat rewrite Rabs_mult. + unfold Rdiv; repeat rewrite Rabs_mult. do 2 rewrite pow_1_abs. do 2 rewrite Rmult_1_l. rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))). @@ -142,7 +142,7 @@ Proof. apply pow_incr. split. apply Rabs_pos. - unfold C in |- *. + unfold C. apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))). do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))). @@ -150,11 +150,11 @@ Proof. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. apply pow_incr. split. apply Rabs_pos. - unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). + unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). apply RmaxLess1. apply RmaxLess2. right. @@ -203,7 +203,7 @@ Proof. left; apply Rinv_0_lt_compat. rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. apply Rle_pow. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ]. apply (fun m n p:nat => mult_le_compat_l p n m). replace (2 * N)%nat with (S (N + pred N)). @@ -223,33 +223,33 @@ Proof. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))). apply Rle_trans with (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))). - unfold Rdiv in |- *; + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))). apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. apply C_maj. omega. right. - unfold Rdiv in |- *; rewrite Rmult_comm. - unfold Binomial.C in |- *. - unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. + unfold Rdiv; rewrite Rmult_comm. + unfold Binomial.C. + unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)). rewrite Rinv_mult_distr. - unfold Rsqr in |- *; reflexivity. + unfold Rsqr; reflexivity. apply INR_fact_neq_0. apply INR_fact_neq_0. omega. apply INR_fact_neq_0. - unfold Rdiv in |- *; rewrite Rmult_comm. - unfold Binomial.C in |- *. - unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. + unfold Rdiv; rewrite Rmult_comm. + unfold Binomial.C. + unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat. @@ -271,17 +271,17 @@ Proof. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N). apply Rmult_le_compat_l. apply Rle_0_sqr. apply le_INR. omega. - rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. + rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l. apply pos_INR. apply Rle_trans with (/ INR (fact (S (N + n)))). - pattern (/ INR (fact (S (N + n)))) at 2 in |- *; rewrite <- Rmult_1_r. - unfold Rsqr in |- *. + pattern (/ INR (fact (S (N + n)))) at 2; rewrite <- Rmult_1_r. + unfold Rsqr. apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. apply Rmult_le_reg_l with (INR (fact (S (N + n)))). @@ -313,14 +313,14 @@ Proof. rewrite sum_cte. apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))). rewrite <- (Rmult_comm (C ^ (4 * N))). - unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. + unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. cut (S (pred N) = N). intro; rewrite H0. - pattern N at 2 in |- *; rewrite <- H0. + pattern N at 2; rewrite <- H0. do 2 rewrite fact_simpl. rewrite H0. repeat rewrite mult_INR. @@ -329,7 +329,7 @@ Proof. repeat rewrite <- Rmult_assoc. rewrite <- Rinv_r_sym. rewrite Rmult_1_l. - pattern (/ INR (fact (pred N))) at 2 in |- *; rewrite <- Rmult_1_r. + pattern (/ INR (fact (pred N))) at 2; rewrite <- Rmult_1_r. rewrite Rmult_assoc. apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. @@ -340,19 +340,19 @@ Proof. apply le_INR; apply le_n_Sn. apply not_O_INR; discriminate. apply not_O_INR. - red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). + red; intro; rewrite H1 in H; elim (lt_irrefl _ H). apply not_O_INR. - red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). + red; intro; rewrite H1 in H; elim (lt_irrefl _ H). apply INR_fact_neq_0. apply not_O_INR; discriminate. apply prod_neq_R0. apply not_O_INR. - red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). + red; intro; rewrite H1 in H; elim (lt_irrefl _ H). apply INR_fact_neq_0. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. right. - unfold Majxy in |- *. - unfold C in |- *. + unfold Majxy. + unfold C. replace (S (pred N)) with N. reflexivity. apply S_pred with 0%nat; assumption. @@ -363,7 +363,7 @@ Lemma reste2_maj : Proof. intros. set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). - unfold Reste2 in |- *. + unfold Reste2. apply Rle_trans with (sum_f_R0 (fun k:nat => @@ -415,7 +415,7 @@ Proof. pred N)). apply sum_Rle; intros. apply sum_Rle; intros. - unfold Rdiv in |- *; repeat rewrite Rabs_mult. + unfold Rdiv; repeat rewrite Rabs_mult. do 2 rewrite pow_1_abs. do 2 rewrite Rmult_1_l. rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))). @@ -437,7 +437,7 @@ Proof. apply pow_incr. split. apply Rabs_pos. - unfold C in |- *. + unfold C. apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)). do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))). @@ -445,11 +445,11 @@ Proof. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. apply pow_incr. split. apply Rabs_pos. - unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). + unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). apply RmaxLess1. apply RmaxLess2. right. @@ -477,7 +477,7 @@ Proof. left; apply Rinv_0_lt_compat. rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. apply Rle_pow. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ]. apply (fun m n p:nat => mult_le_compat_l p n m). replace (2 * S N)%nat with (S (S (N + N))). @@ -500,14 +500,14 @@ Proof. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with (Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) / INR (fact (2 * S (S (N + n))))). apply Rle_trans with (Binomial.C (2 * S (S (N + n))) (S (S (N + n))) / INR (fact (2 * S (S (N + n))))). - unfold Rdiv in |- *; + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))). apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. @@ -518,21 +518,21 @@ Proof. ring. omega. right. - unfold Rdiv in |- *; rewrite Rmult_comm. - unfold Binomial.C in |- *. - unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. + unfold Rdiv; rewrite Rmult_comm. + unfold Binomial.C. + unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))). rewrite Rinv_mult_distr. - unfold Rsqr in |- *; reflexivity. + unfold Rsqr; reflexivity. apply INR_fact_neq_0. apply INR_fact_neq_0. omega. apply INR_fact_neq_0. - unfold Rdiv in |- *; rewrite Rmult_comm. - unfold Binomial.C in |- *. - unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. + unfold Rdiv; rewrite Rmult_comm. + unfold Binomial.C. + unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with @@ -556,7 +556,7 @@ Proof. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N). apply Rmult_le_compat_l. apply Rle_0_sqr. @@ -564,11 +564,11 @@ Proof. apply le_INR. omega. omega. - rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. + rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l. apply pos_INR. apply Rle_trans with (/ INR (fact (S (S (N + n))))). - pattern (/ INR (fact (S (S (N + n))))) at 2 in |- *; rewrite <- Rmult_1_r. - unfold Rsqr in |- *. + pattern (/ INR (fact (S (S (N + n))))) at 2; rewrite <- Rmult_1_r. + unfold Rsqr. apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). @@ -599,11 +599,11 @@ Proof. rewrite sum_cte. apply Rle_trans with (C ^ (4 * S N) / INR (fact N)). rewrite <- (Rmult_comm (C ^ (4 * S N))). - unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. + unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l. apply pow_le. left; apply Rlt_le_trans with 1. apply Rlt_0_1. - unfold C in |- *; apply RmaxLess1. + unfold C; apply RmaxLess1. cut (S (pred N) = N). intro; rewrite H0. do 2 rewrite fact_simpl. @@ -642,10 +642,10 @@ Proof. apply INR_fact_neq_0. apply not_O_INR; discriminate. apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. right. - unfold Majxy in |- *. - unfold C in |- *. + unfold Majxy. + unfold C. reflexivity. Qed. @@ -654,10 +654,10 @@ Proof. intros. assert (H := Majxy_cv_R0 x y). unfold Un_cv in H; unfold R_dist in H. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H eps H0); intros N0 H1. exists (S N0); intros. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r. apply Rle_lt_trans with (Rabs (Majxy x y (pred n))). rewrite (Rabs_right (Majxy x y (pred n))). apply reste1_maj. @@ -665,8 +665,8 @@ Proof. apply lt_O_Sn. assumption. apply Rle_ge. - unfold Majxy in |- *. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Majxy. + unfold Rdiv; apply Rmult_le_pos. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. @@ -674,7 +674,7 @@ Proof. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ]. apply H1. - unfold ge in |- *; apply le_S_n. + unfold ge; apply le_S_n. replace (S (pred n)) with n. assumption. apply S_pred with 0%nat. @@ -686,10 +686,10 @@ Proof. intros. assert (H := Majxy_cv_R0 x y). unfold Un_cv in H; unfold R_dist in H. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H eps H0); intros N0 H1. exists (S N0); intros. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r. apply Rle_lt_trans with (Rabs (Majxy x y n)). rewrite (Rabs_right (Majxy x y n)). apply reste2_maj. @@ -697,8 +697,8 @@ Proof. apply lt_O_Sn. assumption. apply Rle_ge. - unfold Majxy in |- *. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Majxy. + unfold Rdiv; apply Rmult_le_pos. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. @@ -706,7 +706,7 @@ Proof. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ]. apply H1. - unfold ge in |- *; apply le_trans with (S N0). + unfold ge; apply le_trans with (S N0). apply le_n_Sn. exact H2. Qed. @@ -714,7 +714,7 @@ Qed. Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0. Proof. intros. - unfold Reste in |- *. + unfold Reste. set (An := fun n:nat => Reste2 x y n). set (Bn := fun n:nat => Reste1 x y (S n)). cut @@ -723,21 +723,21 @@ Proof. intro. apply H. apply CV_minus. - unfold An in |- *. + unfold An. replace (fun n:nat => Reste2 x y n) with (Reste2 x y). apply reste2_cv_R0. reflexivity. - unfold Bn in |- *. + unfold Bn. assert (H0 := reste1_cv_R0 x y). unfold Un_cv in H0; unfold R_dist in H0. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H0 eps H1); intros N0 H2. exists N0; intros. apply H2. - unfold ge in |- *; apply le_trans with (S N0). + unfold ge; apply le_trans with (S N0). apply le_n_Sn. apply le_n_S; assumption. - unfold An, Bn in |- *. + unfold An, Bn. intro. replace 0 with (0 - 0); [ idtac | ring ]. exact H. @@ -751,7 +751,7 @@ Proof. intros. apply UL_sequence with (C1 x y); assumption. apply C1_cvg. - unfold Un_cv in |- *; unfold R_dist in |- *. + unfold Un_cv; unfold R_dist. intros. assert (H0 := A1_cvg x). assert (H1 := A1_cvg y). @@ -764,7 +764,7 @@ Proof. unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6. cut (0 < eps / 3); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H4 (eps / 3) H7); intros N1 H8. elim (H5 (eps / 3) H7); intros N2 H9. @@ -788,8 +788,8 @@ Proof. replace eps with (eps / 3 + (eps / 3 + eps / 3)). apply Rplus_lt_compat. apply H8. - unfold ge in |- *; apply le_trans with N. - unfold N in |- *. + unfold ge; apply le_trans with N. + unfold N. apply le_trans with (max N1 N2). apply le_max_l. apply le_trans with (max (max N1 N2) N3). @@ -804,12 +804,12 @@ Proof. rewrite <- Rabs_Ropp. rewrite Ropp_minus_distr. apply H9. - unfold ge in |- *; apply le_trans with (max N1 N2). + unfold ge; apply le_trans with (max N1 N2). apply le_max_r. apply le_S_n. rewrite <- H12. apply le_trans with N. - unfold N in |- *. + unfold N. apply le_n_S. apply le_trans with (max (max N1 N2) N3). apply le_max_l. @@ -817,35 +817,35 @@ Proof. assumption. replace (Reste x y (pred n)) with (Reste x y (pred n) - 0). apply H10. - unfold ge in |- *. + unfold ge. apply le_S_n. rewrite <- H12. apply le_trans with N. - unfold N in |- *. + unfold N. apply le_n_S. apply le_trans with (max (max N1 N2) N3). apply le_max_r. apply le_n_Sn. assumption. ring. - pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)). + pattern eps at 4; replace eps with (3 * (eps / 3)). ring. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Rmult_assoc. apply Rinv_r_simpl_m. discrR. apply lt_le_trans with (pred N). - unfold N in |- *; simpl in |- *; apply lt_O_Sn. + unfold N; simpl; apply lt_O_Sn. apply le_S_n. rewrite <- H12. replace (S (pred N)) with N. assumption. - unfold N in |- *; simpl in |- *; reflexivity. + unfold N; simpl; reflexivity. cut (0 < N)%nat. intro. cut (0 < n)%nat. intro. apply S_pred with 0%nat; assumption. apply lt_le_trans with N; assumption. - unfold N in |- *; apply lt_O_Sn. + unfold N; apply lt_O_Sn. Qed. diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v index 204e2009d..cf501917f 100644 --- a/theories/Reals/Cos_rel.v +++ b/theories/Reals/Cos_rel.v @@ -50,7 +50,7 @@ Theorem cos_plus_form : (0 < n)%nat -> A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n). intros. -unfold A1, B1 in |- *. +unfold A1, B1. rewrite (cauchy_finite (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * y ^ (2 * k)) ( @@ -60,7 +60,7 @@ rewrite (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1)) (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * y ^ (2 * k + 1)) n H) . -unfold Reste in |- *. +unfold Reste. replace (sum_f_R0 (fun k:nat => @@ -119,13 +119,13 @@ replace ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) * y ^ (2 * (k - p) + 1))) k) n) with (sum_f_R0 sin_nnn (S n)). rewrite <- sum_plus. -unfold C1 in |- *. +unfold C1. apply sum_eq; intros. induction i as [| i Hreci]. -simpl in |- *. -unfold C in |- *; simpl in |- *. +simpl. +unfold C; simpl. field; discrR. -unfold sin_nnn in |- *. +unfold sin_nnn. rewrite <- Rmult_plus_distr_l. apply Rmult_eq_compat_l. rewrite binomial. @@ -141,13 +141,13 @@ replace (sum_f_R0 (fun l:nat => Wn (S (2 * l))) i). apply sum_decomposition. apply sum_eq; intros. -unfold Wn in |- *. +unfold Wn. apply Rmult_eq_compat_l. replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))). reflexivity. omega. apply sum_eq; intros. -unfold Wn in |- *. +unfold Wn. apply Rmult_eq_compat_l. replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat. reflexivity. @@ -177,11 +177,11 @@ change (pred (S n)) with n. (* replace (pred (S n)) with n; [ idtac | reflexivity ]. *) apply sum_eq; intros. rewrite Rmult_comm. -unfold sin_nnn in |- *. +unfold sin_nnn. rewrite scal_sum. rewrite scal_sum. apply sum_eq; intros. -unfold Rdiv in |- *. +unfold Rdiv. (*repeat rewrite Rmult_assoc.*) (* rewrite (Rmult_comm (/ INR (fact (2 * S i)))). *) repeat rewrite <- Rmult_assoc. @@ -193,13 +193,13 @@ replace (S (2 * i0)) with (2 * i0 + 1)%nat; [ idtac | ring ]. replace (S (2 * (i - i0))) with (2 * (i - i0) + 1)%nat; [ idtac | ring ]. replace ((-1) ^ S i) with (-1 * (-1) ^ i0 * (-1) ^ (i - i0)). ring. -simpl in |- *. -pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat. +simpl. +pattern i at 2; replace i with (i0 + (i - i0))%nat. rewrite pow_add. ring. -symmetry in |- *; apply le_plus_minus; assumption. -unfold C in |- *. -unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +symmetry ; apply le_plus_minus; assumption. +unfold C. +unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite Rinv_mult_distr. @@ -217,7 +217,7 @@ apply lt_O_Sn. apply sum_eq; intros. rewrite scal_sum. apply sum_eq; intros. -unfold Rdiv in |- *. +unfold Rdiv. repeat rewrite <- Rmult_assoc. rewrite <- (Rmult_comm (/ INR (fact (2 * i)))). repeat rewrite <- Rmult_assoc. @@ -225,12 +225,12 @@ replace (/ INR (fact (2 * i)) * C (2 * i) (2 * i0)) with (/ INR (fact (2 * i0)) * / INR (fact (2 * (i - i0)))). replace ((-1) ^ i) with ((-1) ^ i0 * (-1) ^ (i - i0)). ring. -pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat. +pattern i at 2; replace i with (i0 + (i - i0))%nat. rewrite pow_add. ring. -symmetry in |- *; apply le_plus_minus; assumption. -unfold C in |- *. -unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +symmetry ; apply le_plus_minus; assumption. +unfold C. +unfold Rdiv; repeat rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_l. rewrite Rinv_mult_distr. @@ -240,12 +240,12 @@ omega. apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. -unfold Reste2 in |- *; apply sum_eq; intros. +unfold Reste2; apply sum_eq; intros. apply sum_eq; intros. -unfold Rdiv in |- *; ring. -unfold Reste1 in |- *; apply sum_eq; intros. +unfold Rdiv; ring. +unfold Reste1; apply sum_eq; intros. apply sum_eq; intros. -unfold Rdiv in |- *; ring. +unfold Rdiv; ring. apply lt_O_Sn. Qed. @@ -266,10 +266,10 @@ unfold R_dist in p. cut (cos x = x0). intro. rewrite H0. -unfold Un_cv in |- *; unfold R_dist in |- *; intros. +unfold Un_cv; unfold R_dist; intros. elim (p eps H1); intros. exists x1; intros. -unfold A1 in |- *. +unfold A1. replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) n) with (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * (x * x) ^ i) n). @@ -279,9 +279,9 @@ intros. replace ((x * x) ^ i) with (x ^ (2 * i)). reflexivity. apply pow_sqr. -unfold cos in |- *. +unfold cos. case (exist_cos (Rsqr x)). -unfold Rsqr in |- *; intros. +unfold Rsqr; intros. unfold cos_in in p_i. unfold cos_in in c. apply uniqueness_sum with (fun i:nat => cos_n i * (x * x) ^ i); assumption. @@ -298,10 +298,10 @@ unfold R_dist in p. cut (cos (x + y) = x0). intro. rewrite H0. -unfold Un_cv in |- *; unfold R_dist in |- *; intros. +unfold Un_cv; unfold R_dist; intros. elim (p eps H1); intros. exists x1; intros. -unfold C1 in |- *. +unfold C1. replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) n) with @@ -313,9 +313,9 @@ intros. replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)). reflexivity. apply pow_sqr. -unfold cos in |- *. +unfold cos. case (exist_cos (Rsqr (x + y))). -unfold Rsqr in |- *; intros. +unfold Rsqr; intros. unfold cos_in in p_i. unfold cos_in in c. apply uniqueness_sum with (fun i:nat => cos_n i * ((x + y) * (x + y)) ^ i); @@ -327,17 +327,17 @@ intro. case (Req_dec x 0); intro. rewrite H. rewrite sin_0. -unfold B1 in |- *. -unfold Un_cv in |- *; unfold R_dist in |- *; intros; exists 0%nat; intros. +unfold B1. +unfold Un_cv; unfold R_dist; intros; exists 0%nat; intros. replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k + 1)) n) with 0. -unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. induction n as [| n Hrecn]. -simpl in |- *; ring. +simpl; ring. rewrite tech5; rewrite <- Hrecn. -simpl in |- *; ring. -unfold ge in |- *; apply le_O_n. +simpl; ring. +unfold ge; apply le_O_n. assert (H0 := exist_sin (x * x)). elim H0; intros. assert (p_i := p). @@ -347,14 +347,14 @@ unfold R_dist in p. cut (sin x = x * x0). intro. rewrite H1. -unfold Un_cv in |- *; unfold R_dist in |- *; intros. +unfold Un_cv; unfold R_dist; intros. cut (0 < eps / Rabs x); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ]. elim (p (eps / Rabs x) H3); intros. exists x1; intros. -unfold B1 in |- *. +unfold B1. replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1)) n) with @@ -380,11 +380,11 @@ apply sum_eq. intros. rewrite pow_add. rewrite pow_sqr. -simpl in |- *. +simpl. ring. -unfold sin in |- *. +unfold sin. case (exist_sin (Rsqr x)). -unfold Rsqr in |- *; intros. +unfold Rsqr; intros. unfold sin_in in p_i. unfold sin_in in s. assert diff --git a/theories/Reals/DiscrR.v b/theories/Reals/DiscrR.v index c21421b94..6eb73fa71 100644 --- a/theories/Reals/DiscrR.v +++ b/theories/Reals/DiscrR.v @@ -21,7 +21,7 @@ intros; rewrite H; reflexivity. Qed. Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2. -intros; red in |- *; intro; elim H; apply eq_IZR; assumption. +intros; red; intro; elim H; apply eq_IZR; assumption. Qed. Ltac discrR := @@ -45,7 +45,7 @@ Ltac prove_sup0 := repeat (apply Rmult_lt_0_compat || apply Rplus_lt_pos; try apply Rlt_0_1 || apply Rlt_R0_R2) - | |- (?X1 > 0) => change (0 < X1) in |- *; prove_sup0 + | |- (?X1 > 0) => change (0 < X1); prove_sup0 end. Ltac omega_sup := @@ -59,7 +59,7 @@ Ltac omega_sup := Ltac prove_sup := match goal with - | |- (?X1 > ?X2) => change (X2 < X1) in |- *; prove_sup + | |- (?X1 > ?X2) => change (X2 < X1); prove_sup | |- (0 < ?X1) => prove_sup0 | |- (- ?X1 < 0) => rewrite <- Ropp_0; prove_sup | |- (- ?X1 < - ?X2) => apply Ropp_lt_gt_contravar; prove_sup diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index 4b1b09406..c6b37383a 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.v @@ -23,9 +23,9 @@ Definition E1 (x:R) (N:nat) : R := Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x). Proof. - intro; unfold exp in |- *; unfold projT1 in |- *. + intro; unfold exp; unfold projT1. case (exist_exp x); intro. - unfold exp_in, Un_cv in |- *; unfold infinite_sum, E1 in |- *; trivial. + unfold exp_in, Un_cv; unfold infinite_sum, E1; trivial. Qed. Definition Reste_E (x y:R) (N:nat) : R := @@ -41,14 +41,14 @@ Lemma exp_form : forall (x y:R) (n:nat), (0 < n)%nat -> E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n. Proof. - intros; unfold E1 in |- *. + intros; unfold E1. rewrite cauchy_finite. - unfold Reste_E in |- *; unfold Rminus in |- *; rewrite Rplus_assoc; + unfold Reste_E; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r; apply sum_eq; intros. rewrite binomial. rewrite scal_sum; apply sum_eq; intros. - unfold C in |- *; unfold Rdiv in |- *; repeat rewrite Rmult_assoc; + unfold C; unfold Rdiv; repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (fact i))); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite Rinv_mult_distr. @@ -70,7 +70,7 @@ Proof. intro; induction N as [| N HrecN]. reflexivity. replace (2 * S N)%nat with (S (S (2 * N))). - simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. + simpl; simpl in HrecN; rewrite HrecN; reflexivity. ring. Qed. @@ -79,7 +79,7 @@ Proof. intro; induction N as [| N HrecN]. reflexivity. replace (2 * S N)%nat with (S (S (2 * N))). - simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. + simpl; simpl in HrecN; rewrite HrecN; reflexivity. ring. Qed. @@ -93,7 +93,7 @@ Proof. elim H2; intro. rewrite <- (even_div2 _ a); apply HrecN; assumption. rewrite <- (odd_div2 _ b); apply lt_O_Sn. - rewrite H1; simpl in |- *; apply lt_O_Sn. + rewrite H1; simpl; apply lt_O_Sn. inversion H. right; reflexivity. left; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | apply H1 ]. @@ -110,7 +110,7 @@ Proof. (fun k:nat => sum_f_R0 (fun l:nat => / Rsqr (INR (fact (div2 (S N))))) (pred (N - k))) (pred N)). - unfold Reste_E in |- *. + unfold Reste_E. apply Rle_trans with (sum_f_R0 (fun k:nat => @@ -189,25 +189,25 @@ Proof. apply Rabs_pos. apply Rle_trans with (Rmax (Rabs x) (Rabs y)). apply RmaxLess1. - unfold M in |- *; apply RmaxLess2. + unfold M; apply RmaxLess2. apply Rle_trans with (M ^ S (n0 + n) * M ^ (N - n0)). apply Rmult_le_compat_l. apply pow_le; apply Rle_trans with 1. left; apply Rlt_0_1. - unfold M in |- *; apply RmaxLess1. + unfold M; apply RmaxLess1. apply pow_incr; split. apply Rabs_pos. apply Rle_trans with (Rmax (Rabs x) (Rabs y)). apply RmaxLess2. - unfold M in |- *; apply RmaxLess2. + unfold M; apply RmaxLess2. rewrite <- pow_add; replace (S (n0 + n) + (N - n0))%nat with (N + S n)%nat. apply Rle_pow. - unfold M in |- *; apply RmaxLess1. + unfold M; apply RmaxLess1. replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]. apply plus_le_compat_l. replace N with (S (pred N)). apply le_n_S; apply H0. - symmetry in |- *; apply S_pred with 0%nat; apply H. + symmetry ; apply S_pred with 0%nat; apply H. apply INR_eq; do 2 rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR; rewrite minus_INR. ring. @@ -246,7 +246,7 @@ Proof. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. - unfold M in |- *; apply RmaxLess1. + unfold M; apply RmaxLess1. assert (H2 := even_odd_cor N). elim H2; intros N0 H3. elim H3; intro. @@ -262,9 +262,9 @@ Proof. apply le_n_Sn. replace (/ INR (fact n0) * / INR (fact (N - n0))) with (C N n0 / INR (fact N)). - pattern N at 1 in |- *; rewrite H4. + pattern N at 1; rewrite H4. apply Rle_trans with (C N N0 / INR (fact N)). - unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact N))). + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact N))). apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. rewrite H4. @@ -294,7 +294,7 @@ Proof. apply le_pred_n. replace (C N N0 / INR (fact N)) with (/ Rsqr (INR (fact N0))). rewrite H4; rewrite div2_S_double; right; reflexivity. - unfold Rsqr, C, Rdiv in |- *. + unfold Rsqr, C, Rdiv. repeat rewrite Rinv_mult_distr. rewrite (Rmult_comm (INR (fact N))). repeat rewrite Rmult_assoc. @@ -302,7 +302,7 @@ Proof. rewrite Rmult_1_r; replace (N - N0)%nat with N0. ring. replace N with (N0 + N0)%nat. - symmetry in |- *; apply minus_plus. + symmetry ; apply minus_plus. rewrite H4. ring. apply INR_fact_neq_0. @@ -310,7 +310,7 @@ Proof. apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. - unfold C, Rdiv in |- *. + unfold C, Rdiv. rewrite (Rmult_comm (INR (fact N))). repeat rewrite Rmult_assoc. rewrite <- Rinv_r_sym. @@ -322,7 +322,7 @@ Proof. replace (/ INR (fact (S n0)) * / INR (fact (N - n0))) with (C (S N) (S n0) / INR (fact (S N))). apply Rle_trans with (C (S N) (S N0) / INR (fact (S N))). - unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))). + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))). apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. cut (S N = (2 * S N0)%nat). @@ -357,7 +357,7 @@ Proof. replace (C (S N) (S N0) / INR (fact (S N))) with (/ Rsqr (INR (fact (S N0)))). rewrite H5; rewrite div2_double. right; reflexivity. - unfold Rsqr, C, Rdiv in |- *. + unfold Rsqr, C, Rdiv. repeat rewrite Rinv_mult_distr. replace (S N - S N0)%nat with (S N0). rewrite (Rmult_comm (INR (fact (S N)))). @@ -366,14 +366,14 @@ Proof. rewrite Rmult_1_r; reflexivity. apply INR_fact_neq_0. replace (S N) with (S N0 + S N0)%nat. - symmetry in |- *; apply minus_plus. + symmetry ; apply minus_plus. rewrite H5; ring. apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. rewrite H4; ring. - unfold C, Rdiv in |- *. + unfold C, Rdiv. rewrite (Rmult_comm (INR (fact (S N)))). rewrite Rmult_assoc; rewrite <- Rinv_r_sym. rewrite Rmult_1_r; rewrite Rinv_mult_distr. @@ -381,8 +381,8 @@ Proof. apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. - unfold maj_Reste_E in |- *. - unfold Rdiv in |- *; rewrite (Rmult_comm 4). + unfold maj_Reste_E. + unfold Rdiv; rewrite (Rmult_comm 4). rewrite Rmult_assoc. apply Rmult_le_compat_l. apply pow_le. @@ -433,7 +433,7 @@ Proof. cut (INR N <= INR (2 * div2 (S N))). intro; apply Rmult_le_reg_l with (Rsqr (INR (div2 (S N)))). apply Rsqr_pos_lt. - apply not_O_INR; red in |- *; intro. + apply not_O_INR; red; intro. cut (1 < S N)%nat. intro; assert (H4 := div2_not_R0 _ H3). rewrite H2 in H4; elim (lt_n_O _ H4). @@ -456,17 +456,17 @@ Proof. apply lt_INR_0; apply div2_not_R0. apply lt_n_S; apply H. cut (1 < S N)%nat. - intro; unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; intro; + intro; unfold Rsqr; apply prod_neq_R0; apply not_O_INR; intro; assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4; elim (lt_n_O _ H4). apply lt_n_S; apply H. assert (H1 := even_odd_cor N). elim H1; intros N0 H2. elim H2; intro. - pattern N at 2 in |- *; rewrite H3. + pattern N at 2; rewrite H3. rewrite div2_S_double. right; rewrite H3; reflexivity. - pattern N at 2 in |- *; rewrite H3. + pattern N at 2; rewrite H3. replace (S (S (2 * N0))) with (2 * S N0)%nat. rewrite div2_double. rewrite H3. @@ -475,12 +475,12 @@ Proof. rewrite Rmult_plus_distr_l. apply Rplus_le_compat_l. rewrite Rmult_1_r. - simpl in |- *. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + simpl. + pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply Rlt_0_1. ring. - unfold Rsqr in |- *; apply prod_neq_R0; apply INR_fact_neq_0. - unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; discriminate. + unfold Rsqr; apply prod_neq_R0; apply INR_fact_neq_0. + unfold Rsqr; apply prod_neq_R0; apply not_O_INR; discriminate. assert (H0 := even_odd_cor N). elim H0; intros N0 H1. elim H1; intro. @@ -506,15 +506,15 @@ Qed. Lemma maj_Reste_cv_R0 : forall x y:R, Un_cv (maj_Reste_E x y) 0. Proof. intros; assert (H := Majxy_cv_R0 x y). - unfold Un_cv in H; unfold Un_cv in |- *; intros. + unfold Un_cv in H; unfold Un_cv; intros. cut (0 < eps / 4); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H _ H1); intros N0 H2. exists (max (2 * S N0) 2); intros. - unfold R_dist in H2; unfold R_dist in |- *; rewrite Rminus_0_r; - unfold Majxy in H2; unfold maj_Reste_E in |- *. + unfold R_dist in H2; unfold R_dist; rewrite Rminus_0_r; + unfold Majxy in H2; unfold maj_Reste_E. rewrite Rabs_right. apply Rle_lt_trans with (4 * @@ -522,7 +522,7 @@ Proof. INR (fact (div2 (pred n))))). apply Rmult_le_compat_l. left; prove_sup0. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr. rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n))); rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))))) @@ -530,7 +530,7 @@ Proof. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. apply Rle_trans with (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)). rewrite Rmult_comm; - pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2 in |- *; + pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2; rewrite <- Rmult_1_r; apply Rmult_le_compat_l. apply pow_le; apply Rle_trans with 1. left; apply Rlt_0_1. @@ -584,11 +584,11 @@ Proof. (Rabs (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / INR (fact (div2 (pred n))) - 0)). - apply H2; unfold ge in |- *. + apply H2; unfold ge. cut (2 * S N0 <= n)%nat. intro; apply le_S_n. apply INR_le; apply Rmult_le_reg_l with (INR 2). - simpl in |- *; prove_sup0. + simpl; prove_sup0. do 2 rewrite <- mult_INR; apply le_INR. apply le_trans with n. apply H4. @@ -606,12 +606,12 @@ Proof. apply S_pred with 0%nat; apply H8. replace (2 * N1)%nat with (S (S (2 * pred N1))). reflexivity. - pattern N1 at 2 in |- *; replace N1 with (S (pred N1)). + pattern N1 at 2; replace N1 with (S (pred N1)). ring. - symmetry in |- *; apply S_pred with 0%nat; apply H8. + symmetry ; apply S_pred with 0%nat; apply H8. apply INR_lt. apply Rmult_lt_reg_l with (INR 2). - simpl in |- *; prove_sup0. + simpl; prove_sup0. rewrite Rmult_0_r; rewrite <- mult_INR. apply lt_INR_0. rewrite <- H7. @@ -632,7 +632,7 @@ Proof. apply H3. rewrite Rminus_0_r; apply Rabs_right. apply Rle_ge. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. apply pow_le. apply Rle_trans with 1. left; apply Rlt_0_1. @@ -640,7 +640,7 @@ Proof. left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. discrR. apply Rle_ge. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. left; prove_sup0. apply Rmult_le_pos. apply pow_le. @@ -654,9 +654,9 @@ Qed. Lemma Reste_E_cv : forall x y:R, Un_cv (Reste_E x y) 0. Proof. intros; assert (H := maj_Reste_cv_R0 x y). - unfold Un_cv in H; unfold Un_cv in |- *; intros; elim (H _ H0); intros. + unfold Un_cv in H; unfold Un_cv; intros; elim (H _ H0); intros. exists (max x0 1); intros. - unfold R_dist in |- *; rewrite Rminus_0_r. + unfold R_dist; rewrite Rminus_0_r. apply Rle_lt_trans with (maj_Reste_E x y n). apply Reste_E_maj. apply lt_le_trans with 1%nat. @@ -666,10 +666,10 @@ Proof. apply H2. replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) 0). apply H1. - unfold ge in |- *; apply le_trans with (max x0 1). + unfold ge; apply le_trans with (max x0 1). apply le_max_l. apply H2. - unfold R_dist in |- *; rewrite Rminus_0_r; apply Rabs_right. + unfold R_dist; rewrite Rminus_0_r; apply Rabs_right. apply Rle_ge; apply Rle_trans with (Rabs (Reste_E x y n)). apply Rabs_pos. apply Reste_E_maj. @@ -690,13 +690,13 @@ Proof. apply H1. assert (H2 := CV_mult _ _ _ _ H0 H). assert (H3 := CV_minus _ _ _ _ H2 (Reste_E_cv x y)). - unfold Un_cv in |- *; unfold Un_cv in H3; intros. + unfold Un_cv; unfold Un_cv in H3; intros. elim (H3 _ H4); intros. exists (S x0); intros. rewrite <- (exp_form x y n). rewrite Rminus_0_r in H5. apply H5. - unfold ge in |- *; apply le_trans with (S x0). + unfold ge; apply le_trans with (S x0). apply le_n_Sn. apply H6. apply lt_le_trans with (S x0). @@ -710,15 +710,15 @@ Proof. intros; set (An := fun N:nat => / INR (fact N) * x ^ N). cut (Un_cv (fun n:nat => sum_f_R0 An n) (exp x)). intro; apply Rlt_le_trans with (sum_f_R0 An 0). - unfold An in |- *; simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; + unfold An; simpl; rewrite Rinv_1; rewrite Rmult_1_r; apply Rlt_0_1. apply sum_incr. assumption. - intro; unfold An in |- *; left; apply Rmult_lt_0_compat. + intro; unfold An; left; apply Rmult_lt_0_compat. apply Rinv_0_lt_compat; apply INR_fact_lt_0. apply (pow_lt _ n H). - unfold exp in |- *; unfold projT1 in |- *; case (exist_exp x); intro. - unfold exp_in in |- *; unfold infinite_sum, Un_cv in |- *; trivial. + unfold exp; unfold projT1; case (exist_exp x); intro. + unfold exp_in; unfold infinite_sum, Un_cv; trivial. Qed. (**********) @@ -729,12 +729,12 @@ Proof. apply (exp_pos_pos _ a). rewrite <- b; rewrite exp_0; apply Rlt_0_1. replace (exp x) with (1 / exp (- x)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply Rlt_0_1. apply Rinv_0_lt_compat; apply exp_pos_pos. apply (Ropp_0_gt_lt_contravar _ r). cut (exp (- x) <> 0). - intro; unfold Rdiv in |- *; apply Rmult_eq_reg_l with (exp (- x)). + intro; unfold Rdiv; apply Rmult_eq_reg_l with (exp (- x)). rewrite Rmult_1_l; rewrite <- Rinv_r_sym. rewrite <- exp_plus. rewrite Rplus_opp_l; rewrite exp_0; reflexivity. @@ -742,7 +742,7 @@ Proof. apply H. assert (H := exp_plus x (- x)). rewrite Rplus_opp_r in H; rewrite exp_0 in H. - red in |- *; intro; rewrite H0 in H. + red; intro; rewrite H0 in H. rewrite Rmult_0_r in H. elim R1_neq_R0; assumption. Qed. @@ -750,7 +750,7 @@ Qed. (* ((exp h)-1)/h -> 0 quand h->0 *) Lemma derivable_pt_lim_exp_0 : derivable_pt_lim exp 0 1. Proof. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. set (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))). cut (CVN_R fn). intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }). @@ -768,41 +768,41 @@ Proof. replace 1 with (SFL fn cv 0). apply H5. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. apply (not_eq_sym H6). rewrite Rminus_0_r; apply H7. - unfold SFL in |- *. + unfold SFL. case (cv 0); intros. eapply UL_sequence. apply u. - unfold Un_cv, SP in |- *. + unfold Un_cv, SP. intros; exists 1%nat; intros. - unfold R_dist in |- *; rewrite decomp_sum. + unfold R_dist; rewrite decomp_sum. rewrite (Rplus_comm (fn 0%nat 0)). replace (fn 0%nat 0) with 1. - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r; + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r. replace (sum_f_R0 (fun i:nat => fn (S i) 0) (pred n)) with 0. rewrite Rabs_R0; apply H8. - symmetry in |- *; apply sum_eq_R0; intros. - unfold fn in |- *. - simpl in |- *. - unfold Rdiv in |- *; do 2 rewrite Rmult_0_l; reflexivity. - unfold fn in |- *; simpl in |- *. - unfold Rdiv in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. + symmetry ; apply sum_eq_R0; intros. + unfold fn. + simpl. + unfold Rdiv; do 2 rewrite Rmult_0_l; reflexivity. + unfold fn; simpl. + unfold Rdiv; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ]. - unfold SFL, exp in |- *. + unfold SFL, exp. case (cv h); case (exist_exp h); simpl; intros. eapply UL_sequence. apply u. - unfold Un_cv in |- *; intros. + unfold Un_cv; intros. unfold exp_in in e. unfold infinite_sum in e. cut (0 < eps0 * Rabs h). intro; elim (e _ H9); intros N0 H10. exists N0; intros. - unfold R_dist in |- *. + unfold R_dist. apply Rmult_lt_reg_l with (Rabs h). apply Rabs_pos_lt; assumption. rewrite <- Rabs_mult. @@ -813,47 +813,47 @@ Proof. (sum_f_R0 (fun i:nat => / INR (fact i) * h ^ i) (S n) - x). rewrite (Rmult_comm (Rabs h)). apply H10. - unfold ge in |- *. + unfold ge. apply le_trans with (S N0). apply le_n_Sn. apply le_n_S; apply H11. rewrite decomp_sum. replace (/ INR (fact 0) * h ^ 0) with 1. - unfold Rminus in |- *. + unfold Rminus. rewrite Ropp_plus_distr. rewrite Ropp_involutive. rewrite <- (Rplus_comm (- x)). rewrite <- (Rplus_comm (- x + 1)). rewrite Rplus_assoc; repeat apply Rplus_eq_compat_l. replace (pred (S n)) with n; [ idtac | reflexivity ]. - unfold SP in |- *. + unfold SP. rewrite scal_sum. apply sum_eq; intros. - unfold fn in |- *. + unfold fn. replace (h ^ S i) with (h * h ^ i). - unfold Rdiv in |- *; ring. - simpl in |- *; ring. - simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. + unfold Rdiv; ring. + simpl; ring. + simpl; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. apply lt_O_Sn. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Rmult_assoc. - symmetry in |- *; apply Rinv_r_simpl_m. + symmetry ; apply Rinv_r_simpl_m. assumption. apply Rmult_lt_0_compat. apply H8. apply Rabs_pos_lt; assumption. apply SFL_continuity; assumption. - intro; unfold fn in |- *. + intro; unfold fn. replace (fun x:R => x ^ n / INR (fact (S n))) with (pow_fct n / fct_cte (INR (fact (S n))))%F; [ idtac | reflexivity ]. apply continuity_div. apply derivable_continuous; apply (derivable_pow n). apply derivable_continuous; apply derivable_const. - intro; unfold fct_cte in |- *; apply INR_fact_neq_0. + intro; unfold fct_cte; apply INR_fact_neq_0. apply (CVN_R_CVS _ X). assert (H0 := Alembert_exp). - unfold CVN_R in |- *. - intro; unfold CVN_r in |- *. + unfold CVN_R. + intro; unfold CVN_r. exists (fun N:nat => r ^ N / INR (fact (S N))). cut { l:R | @@ -865,10 +865,10 @@ Proof. exists x; intros. split. apply p. - unfold Boule in |- *; intros. + unfold Boule; intros. rewrite Rminus_0_r in H1. - unfold fn in |- *. - unfold Rdiv in |- *; rewrite Rabs_mult. + unfold fn. + unfold Rdiv; rewrite Rabs_mult. cut (0 < INR (fact (S n))). intro. rewrite (Rabs_right (/ INR (fact (S n)))). @@ -883,14 +883,14 @@ Proof. cut ((r:R) <> 0). intro; apply Alembert_C2. intro; apply Rabs_no_R0. - unfold Rdiv in |- *; apply prod_neq_R0. + unfold Rdiv; apply prod_neq_R0. apply pow_nonzero; assumption. apply Rinv_neq_0_compat; apply INR_fact_neq_0. unfold Un_cv in H0. - unfold Un_cv in |- *; intros. + unfold Un_cv; intros. cut (0 < eps0 / r); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply (cond_pos r) ] ]. elim (H0 _ H3); intros N0 H4. exists N0; intros. @@ -899,7 +899,7 @@ Proof. assert (H6 := H4 _ hyp_sn). unfold R_dist in H6; rewrite Rminus_0_r in H6. rewrite Rabs_Rabsolu in H6. - unfold R_dist in |- *; rewrite Rminus_0_r. + unfold R_dist; rewrite Rminus_0_r. rewrite Rabs_Rabsolu. replace (Rabs (r ^ S n / INR (fact (S (S n)))) / Rabs (r ^ n / INR (fact (S n)))) @@ -913,7 +913,7 @@ Proof. apply H6. assumption. apply Rle_ge; left; apply (cond_pos r). - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rabs_mult. repeat rewrite Rabs_Rinv. rewrite Rinv_mult_distr. @@ -926,7 +926,7 @@ Proof. rewrite (Rmult_comm r). rewrite <- Rmult_assoc; rewrite <- (Rmult_comm (INR (fact (S n)))). apply Rmult_eq_compat_l. - simpl in |- *. + simpl. rewrite Rmult_assoc; rewrite <- Rinv_r_sym. ring. apply pow_nonzero; assumption. @@ -939,10 +939,10 @@ Proof. apply Rinv_neq_0_compat; apply Rabs_no_R0; apply INR_fact_neq_0. apply INR_fact_neq_0. apply INR_fact_neq_0. - unfold ge in |- *; apply le_trans with n. + unfold ge; apply le_trans with n. apply H5. apply le_n_Sn. - assert (H1 := cond_pos r); red in |- *; intro; rewrite H2 in H1; + assert (H1 := cond_pos r); red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). Qed. @@ -950,10 +950,10 @@ Qed. Lemma derivable_pt_lim_exp : forall x:R, derivable_pt_lim exp x (exp x). Proof. intro; assert (H0 := derivable_pt_lim_exp_0). - unfold derivable_pt_lim in H0; unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim in H0; unfold derivable_pt_lim; intros. cut (0 < eps / exp x); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply exp_pos ] ]. elim (H0 _ H1); intros del H2. exists del; intros. @@ -967,11 +967,11 @@ Proof. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). apply H5. - assert (H6 := exp_pos x); red in |- *; intro; rewrite H7 in H6; + assert (H6 := exp_pos x); red; intro; rewrite H7 in H6; elim (Rlt_irrefl _ H6). apply Rle_ge; left; apply exp_pos. rewrite Rmult_minus_distr_l. - rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rmult_minus_distr_l. rewrite Rmult_1_r; rewrite exp_plus; reflexivity. Qed. diff --git a/theories/Reals/LegacyRfield.v b/theories/Reals/LegacyRfield.v index 49a94021a..8a62eb866 100644 --- a/theories/Reals/LegacyRfield.v +++ b/theories/Reals/LegacyRfield.v @@ -17,9 +17,9 @@ Open Scope R_scope. Lemma RLegacyTheory : Ring_Theory Rplus Rmult 1 0 Ropp (fun x y:R => false). split. exact Rplus_comm. - symmetry in |- *; apply Rplus_assoc. + symmetry ; apply Rplus_assoc. exact Rmult_comm. - symmetry in |- *; apply Rmult_assoc. + symmetry ; apply Rmult_assoc. intro; apply Rplus_0_l. intro; apply Rmult_1_l. exact Rplus_opp_r. diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v index b872e3fe3..59dcf1344 100644 --- a/theories/Reals/MVT.v +++ b/theories/Reals/MVT.v @@ -55,13 +55,13 @@ 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; 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; rewrite double; apply Rplus_lt_compat_l; apply H. @@ -103,7 +103,7 @@ Proof. inversion H13. apply H14. rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity. - intros; unfold h in |- *; + intros; unfold h; replace (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P)) with @@ -115,11 +115,11 @@ Proof. rewrite derive_pt_minus; do 2 rewrite derive_pt_mult; do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l; do 2 rewrite Rplus_0_l; reflexivity. - unfold h in |- *; ring. - intros; unfold h in |- *; + unfold h; ring. + intros; unfold h; change (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) - c) in |- *. + c). apply continuity_pt_minus; apply continuity_pt_mult. apply derivable_continuous_pt; apply derivable_const. apply H0; apply H3. @@ -128,7 +128,7 @@ Proof. intros; change (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) - c) in |- *. + c). apply derivable_pt_minus; apply derivable_pt_mult. apply derivable_pt_const. apply (pr1 _ H3). @@ -178,7 +178,7 @@ Proof. cut (derive_pt id x (X2 x x0) = 1). cut (derive_pt f x (X0 x x0) = f' x). intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3; - rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry in |- *; + rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry ; assumption. apply derive_pt_eq_0; apply H0; elim x0; intros; split; left; assumption. apply derive_pt_eq_0; apply derivable_pt_lim_id. @@ -188,7 +188,7 @@ Proof. intros; apply derivable_pt_id. intros; apply derivable_continuous_pt; apply X; assumption. intros; elim H1; intros; apply X; split; left; assumption. - intros; unfold derivable_pt in |- *; exists (f' c); apply H0; + intros; unfold derivable_pt; exists (f' c); apply H0; apply H1. Qed. @@ -221,7 +221,7 @@ Proof. unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6; rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a); [ rewrite Rmult_0_r; apply H6 - | apply Rminus_eq_contra; red in |- *; intro; rewrite H7 in H0; + | apply Rminus_eq_contra; red; intro; rewrite H7 in H0; elim (Rlt_irrefl _ H0) ]. Qed. @@ -231,7 +231,7 @@ Lemma nonneg_derivative_1 : (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f. Proof. intros. - unfold increasing in |- *. + unfold increasing. intros. case (total_order_T x y); intro. elim s; intro. @@ -268,12 +268,12 @@ Proof. intro; decompose [and] H8; intros; generalize (H7 (delta / 2) H9 H12); cut ((f (x + delta / 2) - f x) / (delta / 2) <= 0). intro; cut (0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)). - intro; unfold Rabs in |- *; + intro; unfold Rabs; case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)). intros; generalize (Rplus_lt_compat_r (- l) (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) - (l / 2) H14); unfold Rminus in |- *. + (l / 2) H14); unfold Rminus. replace (l / 2 + - l) with (- (l / 2)). replace (- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) + - l) with (- ((f (x + delta / 2) + - f x) / (delta / 2))). @@ -290,7 +290,7 @@ Proof. (Rlt_irrefl 0 (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)). ring. - pattern l at 3 in |- *; rewrite double_var. + pattern l at 3; rewrite double_var. ring. intros. generalize @@ -303,22 +303,22 @@ Proof. H15)). replace (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) with ((f x - f (x + delta / 2)) / (delta / 2) + l). - unfold Rminus in |- *. + unfold Rminus. apply Rplus_le_lt_0_compat. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. cut (x <= x + delta * / 2). intro; generalize (H0 x (x + delta * / 2) H13); intro; generalize (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14); rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. - pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; left; assumption. left; apply Rinv_0_lt_compat; assumption. assumption. rewrite Ropp_minus_distr. - unfold Rminus in |- *. + unfold Rminus. rewrite (Rplus_comm l). - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Ropp_mult_distr_l_reverse. rewrite Ropp_plus_distr. rewrite Ropp_involutive. @@ -329,38 +329,38 @@ Proof. rewrite <- Ropp_0. apply Ropp_ge_le_contravar. apply Rle_ge. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. cut (x <= x + delta * / 2). intro; generalize (H0 x (x + delta * / 2) H10); intro. generalize (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13); rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. - pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; left; assumption. left; apply Rinv_0_lt_compat; assumption. - unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. + unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse. rewrite Ropp_minus_distr. reflexivity. split. - unfold Rdiv in |- *; apply prod_neq_R0. - generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H8; + unfold Rdiv; apply prod_neq_R0. + generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H8; elim (Rlt_irrefl 0 H8). apply Rinv_neq_0_compat; discrR. split. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. rewrite Rabs_right. - unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. + unfold Rdiv; apply Rmult_lt_reg_l with 2. prove_sup0. rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. - rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1 in |- *; + rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1; rewrite <- Rplus_0_r. apply Rplus_lt_compat_l; apply (cond_pos delta). discrR. - apply Rle_ge; unfold Rdiv in |- *; left; apply Rmult_lt_0_compat. + apply Rle_ge; unfold Rdiv; left; apply Rmult_lt_0_compat. apply (cond_pos delta). apply Rinv_0_lt_compat; prove_sup0. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. @@ -368,7 +368,7 @@ Qed. Lemma increasing_decreasing_opp : forall f:R -> R, increasing f -> decreasing (- f)%F. Proof. - unfold increasing, decreasing, opp_fct in |- *; intros; generalize (H x y H0); + unfold increasing, decreasing, opp_fct; intros; generalize (H x y H0); intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption. Qed. @@ -381,8 +381,8 @@ Proof. cut (forall h:R, - - f h = f h). intro. generalize (increasing_decreasing_opp (- f)%F). - unfold decreasing in |- *. - unfold opp_fct in |- *. + unfold decreasing. + unfold opp_fct. intros. rewrite <- (H0 x); rewrite <- (H0 y). apply H1. @@ -410,7 +410,7 @@ Lemma positive_derivative : (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f. Proof. intros. - unfold strict_increasing in |- *. + unfold strict_increasing. intros. apply Rplus_lt_reg_r with (- f x). rewrite Rplus_opp_l; rewrite Rplus_comm. @@ -429,7 +429,7 @@ Qed. Lemma strictincreasing_strictdecreasing_opp : forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F. Proof. - unfold strict_increasing, strict_decreasing, opp_fct in |- *; intros; + unfold strict_increasing, strict_decreasing, opp_fct; intros; generalize (H x y H0); intro; apply Ropp_lt_gt_contravar; assumption. Qed. @@ -443,7 +443,7 @@ Proof. cut (forall h:R, - - f h = f h). intros. generalize (strictincreasing_strictdecreasing_opp (- f)%F). - unfold strict_decreasing, opp_fct in |- *. + unfold strict_decreasing, opp_fct. intros. rewrite <- (H0 x). rewrite <- (H0 y). @@ -470,8 +470,8 @@ Proof. intros. unfold constant in H. apply derive_pt_eq_0. - intros; exists (mkposreal 1 Rlt_0_1); simpl in |- *; intros. - rewrite (H x (x + h)); unfold Rminus in |- *; unfold Rdiv in |- *; + intros; exists (mkposreal 1 Rlt_0_1); simpl; intros. + rewrite (H x (x + h)); unfold Rminus; unfold Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. Qed. @@ -480,13 +480,13 @@ Qed. Lemma increasing_decreasing : forall f:R -> R, increasing f -> decreasing f -> constant f. Proof. - unfold increasing, decreasing, constant in |- *; intros; + unfold increasing, decreasing, constant; intros; case (Rtotal_order x y); intro. generalize (Rlt_le x y H1); intro; apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)). elim H1; intro. rewrite H2; reflexivity. - generalize (Rlt_le y x H2); intro; symmetry in |- *; + generalize (Rlt_le y x H2); intro; symmetry ; apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). Qed. @@ -502,7 +502,7 @@ Proof. assert (H2 := nonneg_derivative_1 f pr H0). assert (H3 := nonpos_derivative_1 f pr H1). apply increasing_decreasing; assumption. - intro; right; symmetry in |- *; apply (H x). + intro; right; symmetry ; apply (H x). intro; right; apply (H x). Qed. @@ -601,7 +601,7 @@ Proof. elim H4; intros. split; left; assumption. rewrite b0. - unfold Rminus in |- *; do 2 rewrite Rplus_opp_r. + unfold Rminus; do 2 rewrite Rplus_opp_r. rewrite Rmult_0_r; right; reflexivity. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). Qed. @@ -648,7 +648,7 @@ Lemma null_derivative_loc : (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) -> constant_D_eq f (fun x:R => a <= x <= b) (f a). Proof. - intros; unfold constant_D_eq in |- *; intros; case (total_order_T a b); intro. + intros; unfold constant_D_eq; intros; case (total_order_T a b); intro. elim s; intro. assert (H2 : forall y:R, a < y < x -> derivable_pt id y). intros; apply derivable_pt_id. @@ -674,7 +674,7 @@ Proof. assert (H12 : derive_pt id x0 (H2 x0 x1) = 1). apply derive_pt_eq_0; apply derivable_pt_lim_id. rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9; - rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry in |- *; + rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry ; assumption. rewrite H1; reflexivity. assert (H2 : x = a). @@ -691,15 +691,15 @@ Lemma antiderivative_Ucte : antiderivative f g2 a b -> exists c : R, (forall x:R, a <= x <= b -> g1 x = g2 x + c). Proof. - unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0; + unfold antiderivative; intros; elim H; clear H; intros; elim H0; clear H0; intros H0 _; exists (g1 a - g2 a); intros; assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x). - intros; unfold derivable_pt in |- *; exists (f x0); elim (H x0 H3); - intros; eapply derive_pt_eq_1; symmetry in |- *; + intros; unfold derivable_pt; exists (f x0); elim (H x0 H3); + intros; eapply derive_pt_eq_1; symmetry ; apply H4. assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x). - intros; unfold derivable_pt in |- *; exists (f x0); - elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry in |- *; + intros; unfold derivable_pt; exists (f x0); + elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry ; apply H5. assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x). intros; elim H5; intros; apply derivable_pt_minus; @@ -713,7 +713,7 @@ Proof. assert (H9 : a <= x0 <= b). split; left; assumption. apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros; - eapply derive_pt_eq_1; symmetry in |- *; apply H10. + eapply derive_pt_eq_1; symmetry ; apply H10. assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7); unfold constant_D_eq in H8; assert (H9 := H8 _ H2); unfold minus_fct in H9; rewrite <- H9; ring. diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v index 35192d0c4..bfe18df28 100644 --- a/theories/Reals/NewtonInt.v +++ b/theories/Reals/NewtonInt.v @@ -28,8 +28,8 @@ Lemma FTCN_step1 : forall (f:Differential) (a b:R), Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b. Proof. - intros f a b; unfold Newton_integrable in |- *; exists (d1 f); - unfold antiderivative in |- *; intros; case (Rle_dec a b); + intros f a b; unfold Newton_integrable; exists (d1 f); + unfold antiderivative; intros; case (Rle_dec a b); intro; [ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ] | right; split; @@ -42,26 +42,26 @@ Lemma FTC_Newton : NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b (FTCN_step1 f a b) = f b - f a. Proof. - intros; unfold NewtonInt in |- *; reflexivity. + intros; unfold NewtonInt; reflexivity. Qed. (* $\int_a^a f$ exists forall a:R and f:R->R *) Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a. Proof. - intros f a; unfold Newton_integrable in |- *; + intros f a; unfold Newton_integrable; exists (fct_cte (f a) * id)%F; left; - unfold antiderivative in |- *; split. + unfold antiderivative; split. intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x). apply derivable_pt_mult. apply derivable_pt_const. apply derivable_pt_id. exists H1; assert (H2 : x = a). elim H; intros; apply Rle_antisym; assumption. - symmetry in |- *; apply derive_pt_eq_0; + symmetry ; apply derive_pt_eq_0; replace (f x) with (0 * id x + fct_cte (f a) x * 1); [ apply (derivable_pt_lim_mult (fct_cte (f a)) id x); [ apply derivable_pt_lim_const | apply derivable_pt_lim_id ] - | unfold id, fct_cte in |- *; rewrite H2; ring ]. + | unfold id, fct_cte; rewrite H2; ring ]. right; reflexivity. Defined. @@ -69,8 +69,8 @@ Defined. Lemma NewtonInt_P2 : forall (f:R -> R) (a:R), NewtonInt f a a (NewtonInt_P1 f a) = 0. Proof. - intros; unfold NewtonInt in |- *; simpl in |- *; - unfold mult_fct, fct_cte, id in |- *; ring. + intros; unfold NewtonInt; simpl; + unfold mult_fct, fct_cte, id; ring. Qed. (* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *) @@ -78,7 +78,7 @@ Lemma NewtonInt_P3 : forall (f:R -> R) (a b:R) (X:Newton_integrable f a b), Newton_integrable f b a. Proof. - unfold Newton_integrable in |- *; intros; elim X; intros g H; + unfold Newton_integrable; intros; elim X; intros g H; exists g; tauto. Defined. @@ -88,7 +88,7 @@ Lemma NewtonInt_P4 : NewtonInt f a b pr = - NewtonInt f b a (NewtonInt_P3 f a b pr). Proof. intros; unfold Newton_integrable in pr; elim pr; intros; elim p; intro. - unfold NewtonInt in |- *; + unfold NewtonInt; case (NewtonInt_P3 f a b (exist @@ -106,7 +106,7 @@ Proof. assert (H4 : a <= b <= b). split; [ assumption | right; reflexivity ]. assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring. - unfold NewtonInt in |- *; + unfold NewtonInt; case (NewtonInt_P3 f a b (exist @@ -132,37 +132,37 @@ Lemma NewtonInt_P5 : Newton_integrable g a b -> Newton_integrable (fun x:R => l * f x + g x) a b. Proof. - unfold Newton_integrable in |- *; intros f g l a b X X0; + unfold Newton_integrable; intros f g l a b X X0; elim X; intros; elim X0; intros; exists (fun y:R => l * x y + x0 y). elim p; intro. elim p0; intro. - left; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + left; unfold antiderivative; unfold antiderivative in H, H0; elim H; clear H; intros; elim H0; clear H0; intros H0 _. split. intros; elim (H _ H2); elim (H0 _ H2); intros. assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). reg. - exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. + exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity. assumption. unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). - left; rewrite <- H5; unfold antiderivative in |- *; split. + left; rewrite <- H5; unfold antiderivative; split. intros; elim H6; intros; assert (H9 : x1 = a). apply Rle_antisym; assumption. assert (H10 : a <= x1 <= b). - split; right; [ symmetry in |- *; assumption | rewrite <- H5; assumption ]. + split; right; [ symmetry ; assumption | rewrite <- H5; assumption ]. assert (H11 : b <= x1 <= a). - split; right; [ rewrite <- H5; symmetry in |- *; assumption | assumption ]. + split; right; [ rewrite <- H5; symmetry ; assumption | assumption ]. assert (H12 : derivable_pt x x1). - unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H10); intros; - eapply derive_pt_eq_1; symmetry in |- *; apply H12. + unfold derivable_pt; exists (f x1); elim (H3 _ H10); intros; + eapply derive_pt_eq_1; symmetry ; apply H12. assert (H13 : derivable_pt x0 x1). - unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H11); intros; - eapply derive_pt_eq_1; symmetry in |- *; apply H13. + unfold derivable_pt; exists (g x1); elim (H1 _ H11); intros; + eapply derive_pt_eq_1; symmetry ; apply H13. assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). reg. - exists H14; symmetry in |- *; reg. + exists H14; symmetry ; reg. assert (H15 : derive_pt x0 x1 H13 = g x1). elim (H1 _ H11); intros; rewrite H15; apply pr_nu. assert (H16 : derive_pt x x1 H12 = f x1). @@ -172,34 +172,34 @@ Proof. elim p0; intro. unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). - left; rewrite H5; unfold antiderivative in |- *; split. + left; rewrite H5; unfold antiderivative; split. intros; elim H6; intros; assert (H9 : x1 = a). apply Rle_antisym; assumption. assert (H10 : a <= x1 <= b). - split; right; [ symmetry in |- *; assumption | rewrite H5; assumption ]. + split; right; [ symmetry ; assumption | rewrite H5; assumption ]. assert (H11 : b <= x1 <= a). - split; right; [ rewrite H5; symmetry in |- *; assumption | assumption ]. + split; right; [ rewrite H5; symmetry ; assumption | assumption ]. assert (H12 : derivable_pt x x1). - unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H11); intros; - eapply derive_pt_eq_1; symmetry in |- *; apply H12. + unfold derivable_pt; exists (f x1); elim (H3 _ H11); intros; + eapply derive_pt_eq_1; symmetry ; apply H12. assert (H13 : derivable_pt x0 x1). - unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H10); intros; - eapply derive_pt_eq_1; symmetry in |- *; apply H13. + unfold derivable_pt; exists (g x1); elim (H1 _ H10); intros; + eapply derive_pt_eq_1; symmetry ; apply H13. assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). reg. - exists H14; symmetry in |- *; reg. + exists H14; symmetry ; reg. assert (H15 : derive_pt x0 x1 H13 = g x1). elim (H1 _ H10); intros; rewrite H15; apply pr_nu. assert (H16 : derive_pt x x1 H12 = f x1). elim (H3 _ H11); intros; rewrite H16; apply pr_nu. rewrite H15; rewrite H16; ring. right; reflexivity. - right; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + right; unfold antiderivative; unfold antiderivative in H, H0; elim H; clear H; intros; elim H0; clear H0; intros H0 _; split. intros; elim (H _ H2); elim (H0 _ H2); intros. assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). reg. - exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. + exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity. assumption. Defined. @@ -210,12 +210,12 @@ Lemma antiderivative_P1 : antiderivative g G a b -> antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b. Proof. - unfold antiderivative in |- *; intros; elim H; elim H0; clear H H0; intros; + unfold antiderivative; intros; elim H; elim H0; clear H H0; intros; split. intros; elim (H _ H3); elim (H1 _ H3); intros. assert (H6 : derivable_pt (fun x:R => l * F x + G x) x). reg. - exists H6; symmetry in |- *; reg; rewrite <- H4; rewrite <- H5; ring. + exists H6; symmetry ; reg; rewrite <- H4; rewrite <- H5; ring. assumption. Qed. @@ -226,7 +226,7 @@ Lemma NewtonInt_P6 : NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) = l * NewtonInt f a b pr1 + NewtonInt g a b pr2. Proof. - intros f g l a b pr1 pr2; unfold NewtonInt in |- *; + intros f g l a b pr1 pr2; unfold NewtonInt; case (NewtonInt_P5 f g l a b pr1 pr2); intros; case pr1; intros; case pr2; intros; case (total_order_T a b); intro. @@ -277,7 +277,7 @@ Lemma antiderivative_P2 : | right _ => F1 x + (F0 b - F1 b) end) a c. Proof. - unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0; + unfold antiderivative; intros; elim H; clear H; intros; elim H0; clear H0; intros; split. 2: apply Rle_trans with b; assumption. intros; elim H3; clear H3; intros; case (total_order_T x b); intro. @@ -293,25 +293,25 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x (f x)). - unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F0 x x0 = f x). - symmetry in |- *; assumption. + unfold derivable_pt_lim; assert (H7 : derive_pt F0 x x0 = f x). + symmetry ; assumption. assert (H8 := derive_pt_eq_1 F0 x (f x) x0 H7); unfold derivable_pt_lim in H8; intros; elim (H8 _ H9); intros; set (D := Rmin x1 (b - x)). assert (H11 : 0 < D). - unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - x)); intro. + unfold D; unfold Rmin; case (Rle_dec x1 (b - x)); intro. apply (cond_pos x1). apply Rlt_Rminus; assumption. exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. case (Rle_dec (x + h) b); intro. apply H10. assumption. - apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. + apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ]. elim n; left; apply Rlt_le_trans with (x + D). apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h). apply RRle_abs. apply H13. apply Rplus_le_reg_l with (- x); rewrite <- Rplus_assoc; rewrite Rplus_opp_l; - rewrite Rplus_0_l; rewrite Rplus_comm; unfold D in |- *; + rewrite Rplus_0_l; rewrite Rplus_comm; unfold D; apply Rmin_r. elim n; left; assumption. assert @@ -322,16 +322,16 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x). - unfold derivable_pt in |- *; exists (f x); apply H7. - exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. + unfold derivable_pt; exists (f x); apply H7. + exists H8; symmetry ; apply derive_pt_eq_0; apply H7. assert (H5 : a <= x <= b). split; [ assumption | right; assumption ]. assert (H6 : b <= x <= c). - split; [ right; symmetry in |- *; assumption | assumption ]. + split; [ right; symmetry ; assumption | assumption ]. elim (H _ H5); elim (H0 _ H6); intros; assert (H9 : derive_pt F0 x x1 = f x). - symmetry in |- *; assumption. + symmetry ; assumption. assert (H10 : derive_pt F1 x x0 = f x). - symmetry in |- *; assumption. + symmetry ; assumption. assert (H11 := derive_pt_eq_1 F0 x (f x) x1 H9); assert (H12 := derive_pt_eq_1 F1 x (f x) x0 H10); assert @@ -342,21 +342,21 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x (f x)). - unfold derivable_pt_lim in |- *; unfold derivable_pt_lim in H11, H12; intros; + unfold derivable_pt_lim; unfold derivable_pt_lim in H11, H12; intros; elim (H11 _ H13); elim (H12 _ H13); intros; set (D := Rmin x2 x3); assert (H16 : 0 < D). - unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x2 x3); intro. + unfold D; unfold Rmin; case (Rle_dec x2 x3); intro. apply (cond_pos x2). apply (cond_pos x3). exists (mkposreal _ H16); intros; case (Rle_dec x b); intro. case (Rle_dec (x + h) b); intro. apply H15. assumption. - apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_r ]. + apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_r ]. replace (F1 (x + h) + (F0 b - F1 b) - F0 x) with (F1 (x + h) - F1 x). apply H14. assumption. - apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. + apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ]. rewrite b0; ring. elim n; right; assumption. assert @@ -367,8 +367,8 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x). - unfold derivable_pt in |- *; exists (f x); apply H13. - exists H14; symmetry in |- *; apply derive_pt_eq_0; apply H13. + unfold derivable_pt; exists (f x); apply H13. + exists H14; symmetry ; apply derive_pt_eq_0; apply H13. assert (H5 : b <= x <= c). split; [ left; assumption | assumption ]. assert (H6 := H0 _ H5); elim H6; clear H6; intros; @@ -380,12 +380,12 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x (f x)). - unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F1 x x0 = f x). - symmetry in |- *; assumption. + unfold derivable_pt_lim; assert (H7 : derive_pt F1 x x0 = f x). + symmetry ; assumption. assert (H8 := derive_pt_eq_1 F1 x (f x) x0 H7); unfold derivable_pt_lim in H8; intros; elim (H8 _ H9); intros; set (D := Rmin x1 (x - b)); assert (H11 : 0 < D). - unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (x - b)); intro. + unfold D; unfold Rmin; case (Rle_dec x1 (x - b)); intro. apply (cond_pos x1). apply Rlt_Rminus; assumption. exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. @@ -399,13 +399,13 @@ Proof. rewrite <- Rabs_Ropp; apply RRle_abs. apply Rlt_le_trans with D. apply H13. - unfold D in |- *; apply Rmin_r. + unfold D; apply Rmin_r. replace (F1 (x + h) + (F0 b - F1 b) - (F1 x + (F0 b - F1 b))) with (F1 (x + h) - F1 x); [ idtac | ring ]; apply H10. assumption. apply Rlt_le_trans with D. assumption. - unfold D in |- *; apply Rmin_l. + unfold D; apply Rmin_l. assert (H8 : derivable_pt @@ -414,8 +414,8 @@ Proof. | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) end) x). - unfold derivable_pt in |- *; exists (f x); apply H7. - exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. + unfold derivable_pt; exists (f x); apply H7. + exists H8; symmetry ; apply derive_pt_eq_0; apply H7. Qed. Lemma antiderivative_P3 : @@ -427,15 +427,15 @@ Proof. intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; intros; case (total_order_T a c); intro. elim s; intro. - right; unfold antiderivative in |- *; split. + right; unfold antiderivative; split. intros; apply H1; elim H3; intros; split; [ assumption | apply Rle_trans with c; assumption ]. left; assumption. - right; unfold antiderivative in |- *; split. + right; unfold antiderivative; split. intros; apply H1; elim H3; intros; split; [ assumption | apply Rle_trans with c; assumption ]. right; assumption. - left; unfold antiderivative in |- *; split. + left; unfold antiderivative; split. intros; apply H; elim H3; intros; split; [ assumption | apply Rle_trans with a; assumption ]. left; assumption. @@ -450,15 +450,15 @@ Proof. intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; intros; case (total_order_T c b); intro. elim s; intro. - right; unfold antiderivative in |- *; split. + right; unfold antiderivative; split. intros; apply H1; elim H3; intros; split; [ apply Rle_trans with c; assumption | assumption ]. left; assumption. - right; unfold antiderivative in |- *; split. + right; unfold antiderivative; split. intros; apply H1; elim H3; intros; split; [ apply Rle_trans with c; assumption | assumption ]. right; assumption. - left; unfold antiderivative in |- *; split. + left; unfold antiderivative; split. intros; apply H; elim H3; intros; split; [ apply Rle_trans with b; assumption | assumption ]. left; assumption. @@ -471,7 +471,7 @@ Lemma NewtonInt_P7 : Newton_integrable f a b -> Newton_integrable f b c -> Newton_integrable f a c. Proof. - unfold Newton_integrable in |- *; intros f a b c Hab Hbc X X0; elim X; + unfold Newton_integrable; intros f a b c Hab Hbc X X0; elim X; clear X; intros F0 H0; elim X0; clear X0; intros F1 H1; set (g := @@ -479,7 +479,7 @@ Proof. match Rle_dec x b with | left _ => F0 x | right _ => F1 x + (F0 b - F1 b) - end); exists g; left; unfold g in |- *; + end); exists g; left; unfold g; apply antiderivative_P2. elim H0; intro. assumption. @@ -504,7 +504,7 @@ Proof. case (total_order_T b c); intro. elim s0; intro. (* a<b & b<c *) - unfold Newton_integrable in |- *; + unfold Newton_integrable; exists (fun x:R => match Rle_dec x b with @@ -523,7 +523,7 @@ Proof. (* a<b & b>c *) case (total_order_T a c); intro. elim s0; intro. - unfold Newton_integrable in |- *; exists F0. + unfold Newton_integrable; exists F0. left. elim H1; intro. unfold antiderivative in H; elim H; clear H; intros _ H. @@ -537,7 +537,7 @@ Proof. unfold antiderivative in H2; elim H2; clear H2; intros _ H2. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). rewrite b0; apply NewtonInt_P1. - unfold Newton_integrable in |- *; exists F1. + unfold Newton_integrable; exists F1. right. elim H1; intro. unfold antiderivative in H; elim H; clear H; intros _ H. @@ -557,7 +557,7 @@ Proof. (* a>b & b<c *) case (total_order_T a c); intro. elim s0; intro. - unfold Newton_integrable in |- *; exists F1. + unfold Newton_integrable; exists F1. left. elim H1; intro. (*****************) @@ -572,7 +572,7 @@ Proof. unfold antiderivative in H; elim H; clear H; intros _ H. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). rewrite b0; apply NewtonInt_P1. - unfold Newton_integrable in |- *; exists F0. + unfold Newton_integrable; exists F0. right. elim H0; intro. unfold antiderivative in H; elim H; clear H; intros _ H. @@ -601,7 +601,7 @@ Lemma NewtonInt_P9 : NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) = NewtonInt f a b pr1 + NewtonInt f b c pr2. Proof. - intros; unfold NewtonInt in |- *. + intros; unfold NewtonInt. case (NewtonInt_P8 f a b c pr1 pr2); intros. case pr1; intros. case pr2; intros. @@ -641,7 +641,7 @@ Proof. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). (* a<b & b=c *) rewrite <- b0. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r. rewrite <- b0 in o. elim o0; intro. elim o; intro. @@ -759,7 +759,7 @@ Proof. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). (* a>b & b=c *) rewrite <- b0. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r. rewrite <- b0 in o. elim o0; intro. unfold antiderivative in H; elim H; clear H; intros _ H. diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v index 38de00030..d426f06b5 100644 --- a/theories/Reals/PSeries_reg.v +++ b/theories/Reals/PSeries_reg.v @@ -44,7 +44,7 @@ Lemma CVN_CVU : (cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l }) (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r. Proof. - intros; unfold CVU in |- *; intros. + intros; unfold CVU; intros. unfold CVN_r in X. elim X; intros An X0. elim X0; intros s H0. @@ -58,7 +58,7 @@ Proof. rewrite Ropp_minus_distr'; rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)). eapply sum_maj1. - unfold SFL in |- *; case (cv y); intro. + unfold SFL; case (cv y); intro. trivial. apply H1. intro; elim H0; intros. @@ -69,7 +69,7 @@ Proof. apply H8; apply H6. apply Rle_ge; apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n). - rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm s); + rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm s); rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l; apply sum_incr. apply H1. @@ -77,10 +77,10 @@ Proof. unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4. assert (H7 := H4 n H5). rewrite Rplus_0_r in H7; apply H7. - unfold Un_cv in H1; unfold Un_cv in |- *; intros. + unfold Un_cv in H1; unfold Un_cv; intros. elim (H1 _ H3); intros. exists x; intros. - unfold R_dist in |- *; unfold R_dist in H4. + unfold R_dist; unfold R_dist in H4. rewrite Rminus_0_r; apply H4; assumption. Qed. @@ -91,13 +91,13 @@ Lemma CVU_continuity : (forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) -> forall y:R, Boule x r y -> continuity_pt f y. Proof. - intros; unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + intros; unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros. unfold CVU in H. cut (0 < eps / 3); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H _ H3); intros N0 H4. assert (H5 := H0 N0 y H1). @@ -110,7 +110,7 @@ Proof. set (del := Rmin del1 del2). exists del; intros. split. - unfold del in |- *; unfold Rmin in |- *; case (Rle_dec del1 del2); intro. + unfold del; unfold Rmin; case (Rle_dec del1 del2); intro. apply (cond_pos del1). elim H8; intros; assumption. intros; @@ -130,27 +130,27 @@ Proof. elim H9; intros. apply Rlt_le_trans with del. assumption. - unfold del in |- *; apply Rmin_l. + unfold del; apply Rmin_l. elim H8; intros. apply H11. split. elim H9; intros; assumption. elim H9; intros; apply Rlt_le_trans with del. assumption. - unfold del in |- *; apply Rmin_r. + unfold del; apply Rmin_r. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4. apply le_n. assumption. apply Rmult_eq_reg_l with 3. - do 2 rewrite Rmult_plus_distr_l; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + do 2 rewrite Rmult_plus_distr_l; unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m. ring. discrR. discrR. cut (0 < r - Rabs (x - y)). intro; exists (mkposreal _ H6). - simpl in |- *; intros. - unfold Boule in |- *; replace (y + h - x) with (h + (y - x)); + simpl; intros. + unfold Boule; replace (y + h - x) with (h + (y - x)); [ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)). apply Rabs_triang. apply Rplus_lt_reg_r with (- Rabs (x - y)). @@ -173,8 +173,8 @@ Lemma continuity_pt_finite_SF : continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply (H 0%nat); apply le_n. - simpl in |- *; + simpl; apply (H 0%nat); apply le_n. + simpl; replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with ((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F; [ idtac | reflexivity ]. @@ -197,7 +197,7 @@ Proof. intros; eapply CVU_continuity. apply CVN_CVU. apply X. - intros; unfold SP in |- *; apply continuity_pt_finite_SF. + intros; unfold SP; apply continuity_pt_finite_SF. intros; apply H. apply H1. apply H0. @@ -208,7 +208,7 @@ Lemma SFL_continuity : (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }), CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv). Proof. - intros; unfold continuity in |- *; intro. + intros; unfold continuity; intro. cut (0 < Rabs x + 1); [ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ]. cut (Boule 0 (mkposreal _ H0) x). @@ -216,8 +216,8 @@ Proof. apply X. intros; apply (H n y). apply H1. - unfold Boule in |- *; simpl in |- *; rewrite Rminus_0_r; - pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; + unfold Boule; simpl; rewrite Rminus_0_r; + pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. Qed. @@ -227,10 +227,10 @@ Lemma CVN_R_CVS : CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }. Proof. intros; apply R_complete. - unfold SP in |- *; set (An := fun N:nat => fn N x). - change (Cauchy_crit_series An) in |- *. + unfold SP; set (An := fun N:nat => fn N x). + change (Cauchy_crit_series An). apply cauchy_abs. - unfold Cauchy_crit_series in |- *; apply CV_Cauchy. + unfold Cauchy_crit_series; apply CV_Cauchy. unfold CVN_R in X; cut (0 < Rabs x + 1). intro; assert (H0 := X (mkposreal _ H)). unfold CVN_r in H0; elim H0; intros Bn H1. @@ -239,13 +239,13 @@ Proof. apply Rseries_CV_comp with Bn. intro; split. apply Rabs_pos. - unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; + unfold An; apply H4; unfold Boule; simpl; rewrite Rminus_0_r. - pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. exists l. cut (forall n:nat, 0 <= Bn n). - intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros. + intro; unfold Un_cv in H3; unfold Un_cv; intros. elim (H3 _ H6); intros. exists x0; intros. replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n). @@ -253,8 +253,8 @@ Proof. apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5. intro; apply Rle_trans with (Rabs (An n)). apply Rabs_pos. - unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; - rewrite Rminus_0_r; pattern (Rabs x) at 1 in |- *; + unfold An; apply H4; unfold Boule; simpl; + rewrite Rminus_0_r; pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ]. Qed. diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v index 3d9314b7d..801bfa399 100644 --- a/theories/Reals/PartSum.v +++ b/theories/Reals/PartSum.v @@ -18,8 +18,8 @@ Lemma tech1 : (forall n:nat, (n <= N)%nat -> 0 < An n) -> 0 < sum_f_R0 An N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H; apply le_n. - simpl in |- *; apply Rplus_lt_0_compat. + simpl; apply H; apply le_n. + simpl; apply Rplus_lt_0_compat. apply HrecN; intros; apply H; apply le_S; assumption. apply H; apply le_n. Qed. @@ -52,7 +52,7 @@ Proof. repeat rewrite S_INR; ring. apply le_n_S; apply lt_le_weak; assumption. apply lt_le_S; assumption. - rewrite H1; rewrite <- minus_n_n; simpl in |- *. + rewrite H1; rewrite <- minus_n_n; simpl. replace (n + 0)%nat with n; [ reflexivity | ring ]. inversion H. right; reflexivity. @@ -66,7 +66,7 @@ Lemma tech3 : Proof. intros; cut (1 - k <> 0). intro; induction N as [| N HrecN]. - simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. + simpl; rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym. reflexivity. apply H0. replace (sum_f_R0 (fun i:nat => k ^ i) (S N)) with @@ -75,15 +75,15 @@ Proof. replace ((1 - k ^ S N) / (1 - k) + k ^ S N) with ((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)). apply Rmult_eq_reg_l with (1 - k). - unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ (1 - k))); + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ (1 - k))); repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ do 2 rewrite Rmult_1_l; simpl in |- *; ring | apply H0 ]. + [ do 2 rewrite Rmult_1_l; simpl; ring | apply H0 ]. apply H0. - unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k)); + unfold Rdiv; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. rewrite Rmult_1_r; reflexivity. apply H0. - apply Rminus_eq_contra; red in |- *; intro; elim H; symmetry in |- *; + apply Rminus_eq_contra; red; intro; elim H; symmetry ; assumption. Qed. @@ -92,11 +92,11 @@ Lemma tech4 : 0 <= k -> (forall i:nat, An (S i) < k * An i) -> An N <= An 0%nat * k ^ N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; right; ring. + simpl; right; ring. apply Rle_trans with (k * An N). left; apply (H0 N). replace (S N) with (N + 1)%nat; [ idtac | ring ]. - rewrite pow_add; simpl in |- *; rewrite Rmult_1_r; + rewrite pow_add; simpl; rewrite Rmult_1_r; replace (An 0%nat * (k ^ N * k)) with (k * (An 0%nat * k ^ N)); [ idtac | ring ]; apply Rmult_le_compat_l. assumption. @@ -116,7 +116,7 @@ Lemma tech6 : sum_f_R0 An N <= An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; right; ring. + simpl; right; ring. apply Rle_trans with (An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N + An (S N)). rewrite tech5; do 2 rewrite <- (Rplus_comm (An (S N))); apply Rplus_le_compat_l. @@ -127,13 +127,13 @@ Qed. Lemma tech7 : forall r1 r2:R, r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2. Proof. - intros; red in |- *; intro. + intros; red; intro. assert (H3 := Rmult_eq_compat_l r1 _ _ H2). rewrite <- Rinv_r_sym in H3; [ idtac | assumption ]. assert (H4 := Rmult_eq_compat_l r2 _ _ H3). rewrite Rmult_1_r in H4; rewrite <- Rmult_assoc in H4. rewrite Rinv_r_simpl_m in H4; [ idtac | assumption ]. - elim H1; symmetry in |- *; assumption. + elim H1; symmetry ; assumption. Qed. Lemma tech11 : @@ -142,7 +142,7 @@ Lemma tech11 : sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H. + simpl; apply H. do 3 rewrite tech5; rewrite HrecN; rewrite (H (S N)); ring. Qed. @@ -151,7 +151,7 @@ Lemma tech12 : Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l -> Pser An x l. Proof. - intros; unfold Pser in |- *; unfold infinite_sum in |- *; unfold Un_cv in H; + intros; unfold Pser; unfold infinite_sum; unfold Un_cv in H; assumption. Qed. @@ -160,7 +160,7 @@ Lemma scal_sum : x * sum_f_R0 An N = sum_f_R0 (fun i:nat => An i * x) N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; ring. + simpl; ring. do 2 rewrite tech5. rewrite Rmult_plus_distr_l; rewrite <- HrecN; ring. Qed. @@ -179,14 +179,14 @@ Proof. do 2 rewrite tech5. replace (S (S (pred N))) with (S N). rewrite (HrecN H1); ring. - rewrite H2; simpl in |- *; reflexivity. + rewrite H2; simpl; reflexivity. assert (H2 := O_or_S N). elim H2; intros. elim a; intros. rewrite <- p. - simpl in |- *; reflexivity. + simpl; reflexivity. rewrite <- b in H1; elim (lt_irrefl _ H1). - rewrite H1; simpl in |- *; reflexivity. + rewrite H1; simpl; reflexivity. inversion H. right; reflexivity. left; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. @@ -197,7 +197,7 @@ Lemma plus_sum : sum_f_R0 (fun i:nat => An i + Bn i) N = sum_f_R0 An N + sum_f_R0 Bn N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; ring. + simpl; ring. do 3 rewrite tech5; rewrite HrecN; ring. Qed. @@ -207,7 +207,7 @@ Lemma sum_eq : sum_f_R0 An N = sum_f_R0 Bn N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H; apply le_n. + simpl; apply H; apply le_n. do 2 rewrite tech5; rewrite HrecN. rewrite (H (S N)); [ reflexivity | apply le_n ]. intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ]. @@ -218,7 +218,7 @@ Lemma uniqueness_sum : forall (An:nat -> R) (l1 l2:R), infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2. Proof. - unfold infinite_sum in |- *; intros. + unfold infinite_sum; intros. case (Req_dec l1 l2); intro. assumption. cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ]. @@ -235,19 +235,19 @@ Proof. intro; rewrite H12 in H11; assert (H13 := double_var); unfold Rdiv in H13; rewrite <- H13 in H11. elim (Rlt_irrefl _ H11). - apply Rabs_right; left; change (0 < / 2) in |- *; apply Rinv_0_lt_compat; + apply Rabs_right; left; change (0 < / 2); apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); - [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR in |- *; + [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR; intro; assumption | discriminate ]. - unfold R_dist in |- *; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1)); + unfold R_dist; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1)); rewrite Ropp_minus_distr'. replace (l1 - l2) with (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2)); [ idtac | ring ]. apply Rabs_triang. - unfold ge in |- *; unfold N in |- *; apply le_max_r. - unfold ge in |- *; unfold N in |- *; apply le_max_l. - unfold Rdiv in |- *; apply prod_neq_R0. + unfold ge; unfold N; apply le_max_r. + unfold ge; unfold N; apply le_max_l. + unfold Rdiv; apply prod_neq_R0. apply Rminus_eq_contra; assumption. apply Rinv_neq_0_compat; discrR. Qed. @@ -257,7 +257,7 @@ Lemma minus_sum : sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; ring. + simpl; ring. do 3 rewrite tech5; rewrite HrecN; ring. Qed. @@ -268,7 +268,7 @@ Lemma sum_decomposition : Proof. intros. induction N as [| N HrecN]. - simpl in |- *; ring. + simpl; ring. rewrite tech5. rewrite (tech5 (fun l:nat => An (S (2 * l))) N). replace (2 * S (S N))%nat with (S (S (2 * S N))). @@ -286,7 +286,7 @@ Lemma sum_Rle : Proof. intros. induction N as [| N HrecN]. - simpl in |- *; apply H. + simpl; apply H. apply le_n. do 2 rewrite tech5. apply Rle_trans with (sum_f_R0 An N + Bn (S N)). @@ -306,7 +306,7 @@ Lemma Rsum_abs : Proof. intros. induction N as [| N HrecN]. - simpl in |- *. + simpl. right; reflexivity. do 2 rewrite tech5. apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). @@ -321,7 +321,7 @@ Lemma sum_cte : Proof. intros. induction N as [| N HrecN]. - simpl in |- *; ring. + simpl; ring. rewrite tech5. rewrite HrecN; repeat rewrite S_INR; ring. Qed. @@ -333,7 +333,7 @@ Lemma sum_growing : Proof. intros. induction N as [| N HrecN]. - simpl in |- *; apply H. + simpl; apply H. do 2 rewrite tech5. apply Rle_trans with (sum_f_R0 An N + Bn (S N)). apply Rplus_le_compat_l; apply H. @@ -348,7 +348,7 @@ Lemma Rabs_triang_gen : Proof. intros. induction N as [| N HrecN]. - simpl in |- *. + simpl. right; reflexivity. do 2 rewrite tech5. apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). @@ -364,7 +364,7 @@ Lemma cond_pos_sum : Proof. intros. induction N as [| N HrecN]. - simpl in |- *; apply H. + simpl; apply H. rewrite tech5. apply Rplus_le_le_0_compat. apply HrecN. @@ -380,7 +380,7 @@ Lemma cauchy_abs : forall An:nat -> R, Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An. Proof. - unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. + unfold Cauchy_crit_series; unfold Cauchy_crit. intros. elim (H eps H0); intros. exists x. @@ -400,8 +400,8 @@ Proof. elim a; intro. rewrite (tech2 An n m); [ idtac | assumption ]. rewrite (tech2 (fun i:nat => Rabs (An i)) n m); [ idtac | assumption ]. - unfold R_dist in |- *. - unfold Rminus in |- *. + unfold R_dist. + unfold Rminus. do 2 rewrite Ropp_plus_distr. do 2 rewrite <- Rplus_assoc. do 2 rewrite Rplus_opp_r. @@ -414,18 +414,18 @@ Proof. replace (fun i:nat => Rabs (An (S n + i)%nat)) with (fun i:nat => Rabs (Bn i)). apply Rabs_triang_gen. - unfold Bn in |- *; reflexivity. + unfold Bn; reflexivity. apply Rle_ge. apply cond_pos_sum. intro; apply Rabs_pos. rewrite b. - unfold R_dist in |- *. - unfold Rminus in |- *; do 2 rewrite Rplus_opp_r. + unfold R_dist. + unfold Rminus; do 2 rewrite Rplus_opp_r. rewrite Rabs_R0; right; reflexivity. rewrite (tech2 An m n); [ idtac | assumption ]. rewrite (tech2 (fun i:nat => Rabs (An i)) m n); [ idtac | assumption ]. - unfold R_dist in |- *. - unfold Rminus in |- *. + unfold R_dist. + unfold Rminus. do 2 rewrite Rplus_assoc. rewrite (Rplus_comm (sum_f_R0 An m)). rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (An i)) m)). @@ -439,7 +439,7 @@ Proof. replace (fun i:nat => Rabs (An (S m + i)%nat)) with (fun i:nat => Rabs (Bn i)). apply Rabs_triang_gen. - unfold Bn in |- *; reflexivity. + unfold Bn; reflexivity. apply Rle_ge. apply cond_pos_sum. intro; apply Rabs_pos. @@ -454,7 +454,7 @@ Proof. intros An X. elim X; intros. unfold Un_cv in p. - unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. + unfold Cauchy_crit_series; unfold Cauchy_crit. intros. cut (0 < eps / 2). intro. @@ -462,7 +462,7 @@ Proof. exists x0. intros. apply Rle_lt_trans with (R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x). - unfold R_dist in |- *. + unfold R_dist. replace (sum_f_R0 An n - sum_f_R0 An m) with (sum_f_R0 An n - x + - (sum_f_R0 An m - x)); [ idtac | ring ]. rewrite <- (Rabs_Ropp (sum_f_R0 An m - x)). @@ -471,8 +471,8 @@ Proof. apply Rplus_lt_compat. apply H1; assumption. apply H1; assumption. - right; symmetry in |- *; apply double_var. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + right; symmetry ; apply double_var. + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. @@ -493,7 +493,7 @@ Lemma sum_eq_R0 : (forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H; apply le_n. + simpl; apply H; apply le_n. rewrite tech5; rewrite HrecN; [ rewrite Rplus_0_l; apply H; apply le_n | intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ] ]. @@ -530,15 +530,15 @@ Proof. [ idtac | ring ]; apply Rle_trans with l1. left; apply r. apply H6. - unfold l1 in |- *; apply Rge_le; + unfold l1; apply Rge_le; apply (growing_prop (fun k:nat => sum_f_R0 An k)). apply H1. - unfold ge, N0 in |- *; apply le_max_r. - unfold ge, N0 in |- *; apply le_max_l. + unfold ge, N0; apply le_max_r. + unfold ge, N0; apply le_max_l. apply Rplus_lt_reg_r with l; rewrite Rplus_0_r; replace (l + (l1 - l)) with l1; [ apply r | ring ]. - unfold Un_growing in |- *; intro; simpl in |- *; - pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + unfold Un_growing; intro; simpl; + pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; apply H0. Qed. @@ -572,7 +572,7 @@ Proof. apply Rlt_trans with (Rabs l1). 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_l_sym. rewrite Rmult_1_r; rewrite double; apply Rplus_lt_compat_l; apply r. discrR. @@ -581,18 +581,18 @@ Proof. apply Rplus_lt_reg_r with ((Rabs l1 - l2) / 2 - Rabs (SP fn N x)). replace ((Rabs l1 - l2) / 2 - Rabs (SP fn N x) + (Rabs l1 + l2) / 2) with (Rabs l1 - Rabs (SP fn N x)). - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; apply H7. - unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + unfold Rdiv; rewrite Rmult_plus_distr_r; rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l; - repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1 in |- *; - rewrite double_var; unfold Rdiv in |- *; ring. + repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1; + rewrite double_var; unfold Rdiv; ring. case (Rcase_abs (sum_f_R0 An N - l2)); intro. apply Rlt_trans with l2. apply (Rminus_lt _ _ r0). apply Rmult_lt_reg_l with 2. prove_sup0. - rewrite (double l2); unfold Rdiv in |- *; rewrite (Rmult_comm 2); + rewrite (double l2); unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite (Rplus_comm (Rabs l1)); apply Rplus_lt_compat_l; apply r. @@ -600,23 +600,23 @@ Proof. rewrite (Rabs_right _ r0) in H6; apply Rplus_lt_reg_r with (- l2). replace (- l2 + (Rabs l1 + l2) / 2) with ((Rabs l1 - l2) / 2). rewrite Rplus_comm; apply H6. - unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l; rewrite Rmult_plus_distr_r; - pattern l2 at 2 in |- *; rewrite double_var; + pattern l2 at 2; rewrite double_var; repeat rewrite (Rmult_comm (/ 2)); rewrite Ropp_plus_distr; - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. apply Rle_lt_trans with (Rabs (SP fn N x - l1)). rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply Rabs_triang_inv2. - apply H4; unfold ge, N in |- *; apply le_max_l. - apply H5; unfold ge, N in |- *; apply le_max_r. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + apply H4; unfold ge, N; apply le_max_l. + apply H5; unfold ge, N; apply le_max_r. + unfold Rdiv; apply Rmult_lt_0_compat. apply Rplus_lt_reg_r with l2. rewrite Rplus_0_r; replace (l2 + (Rabs l1 - l2)) with (Rabs l1); [ apply r | ring ]. apply Rinv_0_lt_compat; prove_sup0. intros; induction n0 as [| n0 Hrecn0]. - unfold SP in |- *; simpl in |- *; apply H1. - unfold SP in |- *; simpl in |- *. + unfold SP; simpl; apply H1. + unfold SP; simpl. apply Rle_trans with (Rabs (sum_f_R0 (fun k:nat => fn k x) n0) + Rabs (fn (S n0) x)). apply Rabs_triang. diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index 2f58201f7..cc8c478ab 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -52,8 +52,8 @@ Proof. exact Rlt_irrefl. Qed. Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2. Proof. - red in |- *; intros r1 r2 H H0; apply (Rlt_irrefl r1). - pattern r1 at 2 in |- *; rewrite H0; trivial. + red; intros r1 r2 H H0; apply (Rlt_irrefl r1). + pattern r1 at 2; rewrite H0; trivial. Qed. Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2. @@ -104,7 +104,7 @@ Qed. Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2. Proof. - intros; red in |- *; tauto. + intros; red; tauto. Qed. Hint Resolve Rlt_le: real. @@ -116,14 +116,14 @@ Qed. (**********) Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1. Proof. - destruct 1; red in |- *; auto with real. + destruct 1; red; auto with real. Qed. Hint Immediate Rle_ge: real. Hint Resolve Rle_ge: rorders. Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1. Proof. - destruct 1; red in |- *; auto with real. + destruct 1; red; auto with real. Qed. Hint Resolve Rge_le: real. Hint Immediate Rge_le: rorders. @@ -145,7 +145,7 @@ Hint Immediate Rgt_lt: rorders. Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1. Proof. - intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle in |- *; tauto. + intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle; tauto. Qed. Hint Immediate Rnot_le_lt: real. @@ -176,7 +176,7 @@ Proof. eauto using Rnot_gt_ge with rorders. Qed. (**********) Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2. Proof. - generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle in |- *. + generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle. unfold not; intuition eauto 3. Qed. Hint Immediate Rlt_not_le: real. @@ -194,7 +194,7 @@ Proof. exact Rlt_not_ge. Qed. Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2. Proof. intros r1 r2. generalize (Rlt_asym r1 r2) (Rlt_dichotomy_converse r1 r2). - unfold Rle in |- *; intuition. + unfold Rle; intuition. Qed. Lemma Rge_not_lt : forall r1 r2, r1 >= r2 -> ~ r1 < r2. @@ -209,25 +209,25 @@ Proof. do 2 intro; apply Rge_not_lt. Qed. (**********) Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2. Proof. - unfold Rle in |- *; tauto. + unfold Rle; tauto. Qed. Hint Immediate Req_le: real. Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2. Proof. - unfold Rge in |- *; tauto. + unfold Rge; tauto. Qed. Hint Immediate Req_ge: real. Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2. Proof. - unfold Rle in |- *; auto. + unfold Rle; auto. Qed. Hint Immediate Req_le_sym: real. Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2. Proof. - unfold Rge in |- *; auto. + unfold Rge; auto. Qed. Hint Immediate Req_ge_sym: real. @@ -242,7 +242,7 @@ Proof. do 2 intro; apply Rlt_asym. Qed. Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2. Proof. - intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle in |- *; intuition. + intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle; intuition. Qed. Hint Resolve Rle_antisym: real. @@ -293,13 +293,13 @@ Proof. eauto using Rlt_trans with rorders. Qed. Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3. Proof. generalize Rlt_trans Rlt_eq_compat. - unfold Rle in |- *. + unfold Rle. intuition eauto 2. Qed. Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3. Proof. - generalize Rlt_trans Rlt_eq_compat; unfold Rle in |- *; intuition eauto 2. + generalize Rlt_trans Rlt_eq_compat; unfold Rle; intuition eauto 2. Qed. Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3. @@ -432,7 +432,7 @@ Hint Resolve Rplus_eq_reg_l: real. (**********) Lemma Rplus_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0. Proof. - intros r b; pattern r at 2 in |- *; replace r with (r + 0); eauto with real. + intros r b; pattern r at 2; replace r with (r + 0); eauto with real. Qed. (***********) @@ -443,7 +443,7 @@ Proof. absurd (0 < a + b). rewrite H1; auto with real. apply Rle_lt_trans with (a + 0). - rewrite Rplus_0_r in |- *; assumption. + rewrite Rplus_0_r; assumption. auto using Rplus_lt_compat_l with real. rewrite <- H0, Rplus_0_r in H1; assumption. Qed. @@ -572,14 +572,14 @@ Qed. (**********) Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0. Proof. - intros r1 r2 H; split; red in |- *; intro; apply H; auto with real. + intros r1 r2 H; split; red; intro; apply H; auto with real. Qed. (**********) Lemma Rmult_integral_contrapositive : forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0. Proof. - red in |- *; intros r1 r2 [H1 H2] H. + red; intros r1 r2 [H1 H2] H. case (Rmult_integral r1 r2); auto with real. Qed. Hint Resolve Rmult_integral_contrapositive: real. @@ -606,12 +606,12 @@ Notation "r ²" := (Rsqr r) (at level 1, format "r ²") : R_scope. (***********) Lemma Rsqr_0 : Rsqr 0 = 0. - unfold Rsqr in |- *; auto with real. + unfold Rsqr; auto with real. Qed. (***********) Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0. - unfold Rsqr in |- *; intros; elim (Rmult_integral r r H); trivial. + unfold Rsqr; intros; elim (Rmult_integral r r H); trivial. Qed. (*********************************************************) @@ -649,7 +649,7 @@ Hint Resolve Ropp_involutive: real. (*********) Lemma Ropp_neq_0_compat : forall r, r <> 0 -> - r <> 0. Proof. - red in |- *; intros r H H0. + red; intros r H H0. apply H. transitivity (- - r); auto with real. Qed. @@ -722,7 +722,7 @@ Hint Resolve Rminus_diag_eq: real. (**********) Lemma Rminus_diag_uniq : forall r1 r2, r1 - r2 = 0 -> r1 = r2. Proof. - intros r1 r2; unfold Rminus in |- *; rewrite Rplus_comm; intro. + intros r1 r2; unfold Rminus; rewrite Rplus_comm; intro. rewrite <- (Ropp_involutive r2); apply (Rplus_opp_r_uniq (- r2) r1 H). Qed. Hint Immediate Rminus_diag_uniq: real. @@ -743,20 +743,20 @@ Hint Resolve Rplus_minus: real. (**********) Lemma Rminus_eq_contra : forall r1 r2, r1 <> r2 -> r1 - r2 <> 0. Proof. - red in |- *; intros r1 r2 H H0. + red; intros r1 r2 H H0. apply H; auto with real. Qed. Hint Resolve Rminus_eq_contra: real. Lemma Rminus_not_eq : forall r1 r2, r1 - r2 <> 0 -> r1 <> r2. Proof. - red in |- *; intros; elim H; apply Rminus_diag_eq; auto. + red; intros; elim H; apply Rminus_diag_eq; auto. Qed. Hint Resolve Rminus_not_eq: real. Lemma Rminus_not_eq_right : forall r1 r2, r2 - r1 <> 0 -> r1 <> r2. Proof. - red in |- *; intros; elim H; rewrite H0; ring. + red; intros; elim H; rewrite H0; ring. Qed. Hint Resolve Rminus_not_eq_right: real. @@ -780,7 +780,7 @@ Hint Resolve Rinv_1: real. (*********) Lemma Rinv_neq_0_compat : forall r, r <> 0 -> / r <> 0. Proof. - red in |- *; intros; apply R1_neq_R0. + red; intros; apply R1_neq_R0. replace 1 with (/ r * r); auto with real. Qed. Hint Resolve Rinv_neq_0_compat: real. @@ -860,7 +860,7 @@ Proof. do 3 intro; apply Rplus_lt_compat_r. Qed. (**********) Lemma Rplus_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2. Proof. - unfold Rle in |- *; intros; elim H; intro. + unfold Rle; intros; elim H; intro. left; apply (Rplus_lt_compat_l r r1 r2 H0). right; rewrite <- H0; auto with zarith real. Qed. @@ -872,7 +872,7 @@ Hint Resolve Rplus_ge_compat_l: real. (**********) Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r. Proof. - unfold Rle in |- *; intros; elim H; intro. + unfold Rle; intros; elim H; intro. left; apply (Rplus_lt_compat_r r r1 r2 H0). right; rewrite <- H0; auto with real. Qed. @@ -933,7 +933,7 @@ Lemma Rplus_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 + r2. Proof. intros x y; intros; apply Rlt_trans with x; [ assumption - | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; + | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; assumption ]. Qed. @@ -941,7 +941,7 @@ Lemma Rplus_le_lt_0_compat : forall r1 r2, 0 <= r1 -> 0 < r2 -> 0 < r1 + r2. Proof. intros x y; intros; apply Rle_lt_trans with x; [ assumption - | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; + | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; assumption ]. Qed. @@ -955,7 +955,7 @@ Lemma Rplus_le_le_0_compat : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 + r2. Proof. intros x y; intros; apply Rle_trans with x; [ assumption - | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; assumption ]. Qed. @@ -983,7 +983,7 @@ Qed. Lemma Rplus_le_reg_l : forall r r1 r2, r + r1 <= r + r2 -> r1 <= r2. Proof. - unfold Rle in |- *; intros; elim H; intro. + unfold Rle; intros; elim H; intro. left; apply (Rplus_lt_reg_r r r1 r2 H0). right; apply (Rplus_eq_reg_l r r1 r2 H0). Qed. @@ -997,7 +997,7 @@ Qed. Lemma Rplus_gt_reg_l : forall r r1 r2, r + r1 > r + r2 -> r1 > r2. Proof. - unfold Rgt in |- *; intros; apply (Rplus_lt_reg_r r r2 r1 H). + unfold Rgt; intros; apply (Rplus_lt_reg_r r r2 r1 H). Qed. Lemma Rplus_ge_reg_l : forall r r1 r2, r + r1 >= r + r2 -> r1 >= r2. @@ -1048,7 +1048,7 @@ Qed. Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2. Proof. - unfold Rgt in |- *; intros. + unfold Rgt; intros. apply (Rplus_lt_reg_r (r2 + r1)). replace (r2 + r1 + - r1) with r2. replace (r2 + r1 + - r2) with r1. @@ -1060,7 +1060,7 @@ Hint Resolve Ropp_gt_lt_contravar. Lemma Ropp_lt_gt_contravar : forall r1 r2, r1 < r2 -> - r1 > - r2. Proof. - unfold Rgt in |- *; auto with real. + unfold Rgt; auto with real. Qed. Hint Resolve Ropp_lt_gt_contravar: real. @@ -1185,7 +1185,7 @@ Proof. eauto using Rmult_lt_compat_l with rorders. Qed. Lemma Rmult_le_compat_l : forall r r1 r2, 0 <= r -> r1 <= r2 -> r * r1 <= r * r2. Proof. - intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle in |- *; + intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle; auto with real. right; rewrite <- H; do 2 rewrite Rmult_0_l; reflexivity. Qed. @@ -1344,7 +1344,7 @@ Qed. (**********) Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0. Proof. - destruct 1; unfold Rle in |- *; auto with real. + destruct 1; unfold Rle; auto with real. Qed. Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0. @@ -1358,7 +1358,7 @@ Qed. Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2. Proof. intros; replace r1 with (r1 - r2 + r2). - pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. + pattern r2 at 3; replace r2 with (0 + r2); auto with real. ring. Qed. @@ -1374,7 +1374,7 @@ Qed. Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2. Proof. intros; replace r1 with (r1 - r2 + r2). - pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. + pattern r2 at 3; replace r2 with (0 + r2); auto with real. ring. Qed. @@ -1400,7 +1400,7 @@ Hint Immediate tech_Rplus: real. Lemma Rle_0_sqr : forall r, 0 <= Rsqr r. Proof. - intro; case (Rlt_le_dec r 0); unfold Rsqr in |- *; intro. + intro; case (Rlt_le_dec r 0); unfold Rsqr; intro. replace (r * r) with (- r * - r); auto with real. replace 0 with (- r * 0); auto with real. replace 0 with (0 * r); auto with real. @@ -1409,7 +1409,7 @@ Qed. (***********) Lemma Rlt_0_sqr : forall r, r <> 0 -> 0 < Rsqr r. Proof. - intros; case (Rdichotomy r 0); trivial; unfold Rsqr in |- *; intro. + intros; case (Rdichotomy r 0); trivial; unfold Rsqr; intro. replace (r * r) with (- r * - r); auto with real. replace 0 with (- r * 0); auto with real. replace 0 with (0 * r); auto with real. @@ -1439,7 +1439,7 @@ Qed. Lemma Rlt_0_1 : 0 < 1. Proof. replace 1 with (Rsqr 1); auto with real. - unfold Rsqr in |- *; auto with real. + unfold Rsqr; auto with real. Qed. Hint Resolve Rlt_0_1: real. @@ -1455,7 +1455,7 @@ Qed. Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r. Proof. - intros; apply Rnot_le_lt; red in |- *; intros. + intros; apply Rnot_le_lt; red; intros. absurd (1 <= 0); auto with real. replace 1 with (r * / r); auto with real. replace 0 with (r * 0); auto with real. @@ -1465,7 +1465,7 @@ Hint Resolve Rinv_0_lt_compat: real. (*********) Lemma Rinv_lt_0_compat : forall r, r < 0 -> / r < 0. Proof. - intros; apply Rnot_le_lt; red in |- *; intros. + intros; apply Rnot_le_lt; red; intros. absurd (1 <= 0); auto with real. replace 1 with (r * / r); auto with real. replace 0 with (r * 0); auto with real. @@ -1479,8 +1479,8 @@ Proof. case (Rmult_neq_0_reg r1 r2); intros; auto with real. replace (r1 * r2 * / r2) with r1. replace (r1 * r2 * / r1) with r2; trivial. - symmetry in |- *; auto with real. - symmetry in |- *; auto with real. + symmetry ; auto with real. + symmetry ; auto with real. Qed. Lemma Rinv_1_lt_contravar : forall r1 r2, 1 <= r1 -> r1 < r2 -> / r2 < / r1. @@ -1497,7 +1497,7 @@ Proof. rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite (Rmult_comm y (/ y)); rewrite Rinv_l; auto with real. apply Rlt_dichotomy_converse; right. - red in |- *; apply Rlt_trans with (r2 := x); auto with real. + red; apply Rlt_trans with (r2 := x); auto with real. Qed. Hint Resolve Rinv_1_lt_contravar: real. @@ -1510,7 +1510,7 @@ Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1. Proof. intros. apply Rlt_le_trans with 1; auto with real. - pattern 1 at 1 in |- *; replace 1 with (0 + 1); auto with real. + pattern 1 at 1; replace 1 with (0 + 1); auto with real. Qed. Hint Resolve Rle_lt_0_plus_1: real. @@ -1518,15 +1518,15 @@ Hint Resolve Rle_lt_0_plus_1: real. Lemma Rlt_plus_1 : forall r, r < r + 1. Proof. intros. - pattern r at 1 in |- *; replace r with (r + 0); auto with real. + pattern r at 1; replace r with (r + 0); auto with real. Qed. Hint Resolve Rlt_plus_1: real. (**********) Lemma tech_Rgt_minus : forall r1 r2, 0 < r2 -> r1 > r1 - r2. Proof. - red in |- *; unfold Rminus in |- *; intros. - pattern r1 at 2 in |- *; replace r1 with (r1 + 0); auto with real. + red; unfold Rminus; intros. + pattern r1 at 2; replace r1 with (r1 + 0); auto with real. Qed. (*********************************************************) @@ -1542,14 +1542,14 @@ Qed. (**********) Lemma S_O_plus_INR : forall n:nat, INR (1 + n) = INR 1 + INR n. Proof. - intro; simpl in |- *; case n; intros; auto with real. + intro; simpl; case n; intros; auto with real. Qed. (**********) Lemma plus_INR : forall n m:nat, INR (n + m) = INR n + INR m. Proof. intros n m; induction n as [| n Hrecn]. - simpl in |- *; auto with real. + simpl; auto with real. replace (S n + m)%nat with (S (n + m)); auto with arith. repeat rewrite S_INR. rewrite Hrecn; ring. @@ -1559,9 +1559,9 @@ Hint Resolve plus_INR: real. (**********) Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m. Proof. - intros n m le; pattern m, n in |- *; apply le_elim_rel; auto with real. + intros n m le; pattern m, n; apply le_elim_rel; auto with real. intros; rewrite <- minus_n_O; auto with real. - intros; repeat rewrite S_INR; simpl in |- *. + intros; repeat rewrite S_INR; simpl. rewrite H0; ring. Qed. Hint Resolve minus_INR: real. @@ -1570,8 +1570,8 @@ Hint Resolve minus_INR: real. Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m. Proof. intros n m; induction n as [| n Hrecn]. - simpl in |- *; auto with real. - intros; repeat rewrite S_INR; simpl in |- *. + simpl; auto with real. + intros; repeat rewrite S_INR; simpl. rewrite plus_INR; rewrite Hrecn; ring. Qed. Hint Resolve mult_INR: real. @@ -1602,7 +1602,7 @@ Hint Resolve lt_1_INR: real. Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (Pos.to_nat p). Proof. intro; apply lt_0_INR. - simpl in |- *; auto with real. + simpl; auto with real. apply Pos2Nat.is_pos. Qed. Hint Resolve pos_INR_nat_of_P: real. @@ -1611,7 +1611,7 @@ Hint Resolve pos_INR_nat_of_P: real. Lemma pos_INR : forall n:nat, 0 <= INR n. Proof. intro n; case n. - simpl in |- *; auto with real. + simpl; auto with real. auto with arith real. Qed. Hint Resolve pos_INR: real. @@ -1619,10 +1619,10 @@ Hint Resolve pos_INR: real. Lemma INR_lt : forall n m:nat, INR n < INR m -> (n < m)%nat. Proof. double induction n m; intros. - simpl in |- *; exfalso; apply (Rlt_irrefl 0); auto. + simpl; exfalso; apply (Rlt_irrefl 0); auto. auto with arith. generalize (pos_INR (S n0)); intro; cut (INR 0 = 0); - [ intro H2; rewrite H2 in H0; idtac | simpl in |- *; trivial ]. + [ intro H2; rewrite H2 in H0; idtac | simpl; trivial ]. generalize (Rle_lt_trans 0 (INR (S n0)) 0 H1 H0); intro; exfalso; apply (Rlt_irrefl 0); auto. do 2 rewrite S_INR in H1; cut (INR n1 < INR n0). @@ -1644,7 +1644,7 @@ Hint Resolve le_INR: real. (**********) Lemma INR_not_0 : forall n:nat, INR n <> 0 -> n <> 0%nat. Proof. - red in |- *; intros n H H1. + red; intros n H H1. apply H. rewrite H1; trivial. Qed. @@ -1656,7 +1656,7 @@ Proof. intro n; case n. intro; absurd (0%nat = 0%nat); trivial. intros; rewrite S_INR. - apply Rgt_not_eq; red in |- *; auto with real. + apply Rgt_not_eq; red; auto with real. Qed. Hint Resolve not_0_INR: real. @@ -1677,7 +1677,7 @@ Proof. cut (n <> m). intro H3; generalize (not_INR n m H3); intro H4; exfalso; auto. omega. - symmetry in |- *; cut (m <> n). + symmetry ; cut (m <> n). intro H3; generalize (not_INR m n H3); intro H4; exfalso; auto. omega. Qed. @@ -1712,7 +1712,7 @@ Qed. Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z.of_nat n). Proof. simple induction n; auto with real. - intros; simpl in |- *; rewrite SuccNat2Pos.id_succ; + intros; simpl; rewrite SuccNat2Pos.id_succ; auto with real. Qed. @@ -1744,7 +1744,7 @@ Qed. (**********) Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m. Proof. - intros z t; case z; case t; simpl in |- *; auto with real. + intros z t; case z; case t; simpl; auto with real. intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real. rewrite Rmult_comm. @@ -1775,7 +1775,7 @@ Qed. (**********) Lemma opp_IZR : forall n:Z, IZR (- n) = - IZR n. Proof. - intro z; case z; simpl in |- *; auto with real. + intro z; case z; simpl; auto with real. Qed. Definition Ropp_Ropp_IZR := opp_IZR. @@ -1790,16 +1790,16 @@ Qed. (**********) Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m). Proof. - intros z1 z2; unfold Rminus in |- *; unfold Z.sub in |- *. - rewrite <- (Ropp_Ropp_IZR z2); symmetry in |- *; apply plus_IZR. + intros z1 z2; unfold Rminus; unfold Z.sub. + rewrite <- (Ropp_Ropp_IZR z2); symmetry ; apply plus_IZR. Qed. (**********) Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z. Proof. - intro z; case z; simpl in |- *; intros. + intro z; case z; simpl; intros. absurd (0 < 0); auto with real. - unfold Z.lt in |- *; simpl in |- *; trivial. + unfold Z.lt; simpl; trivial. case Rlt_not_le with (1 := H). replace 0 with (-0); auto with real. Qed. @@ -1816,10 +1816,10 @@ Qed. (**********) Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z. Proof. - intro z; destruct z; simpl in |- *; intros; auto with zarith. + intro z; destruct z; simpl; intros; auto with zarith. case (Rlt_not_eq 0 (INR (Pos.to_nat p))); auto with real. case (Rlt_not_eq (- INR (Pos.to_nat p)) 0); auto with real. - apply Ropp_lt_gt_0_contravar. unfold Rgt in |- *; apply pos_INR_nat_of_P. + apply Ropp_lt_gt_0_contravar. unfold Rgt; apply pos_INR_nat_of_P. Qed. (**********) @@ -1833,22 +1833,22 @@ Qed. (**********) Lemma not_0_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0. Proof. - intros z H; red in |- *; intros H0; case H. + intros z H; red; intros H0; case H. apply eq_IZR; auto. Qed. (*********) Lemma le_0_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z. Proof. - unfold Rle in |- *; intros z [H| H]. - red in |- *; intro; apply (Z.lt_le_incl 0 z (lt_0_IZR z H)); assumption. + unfold Rle; intros z [H| H]. + red; intro; apply (Z.lt_le_incl 0 z (lt_0_IZR z H)); assumption. rewrite (eq_IZR_R0 z); auto with zarith real. Qed. (**********) Lemma le_IZR : forall n m:Z, IZR n <= IZR m -> (n <= m)%Z. Proof. - unfold Rle in |- *; intros z1 z2 [H| H]. + unfold Rle; intros z1 z2 [H| H]. apply (Z.lt_le_incl z1 z2); auto with real. apply lt_IZR; trivial. rewrite (eq_IZR z1 z2); auto with zarith real. @@ -1857,20 +1857,20 @@ Qed. (**********) Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z. Proof. - pattern 1 at 1 in |- *; replace 1 with (IZR 1); intros; auto. + pattern 1 at 1; replace 1 with (IZR 1); intros; auto. apply le_IZR; trivial. Qed. (**********) Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m. Proof. - intros m n H; apply Rnot_lt_ge; red in |- *; intro. + intros m n H; apply Rnot_lt_ge; red; intro. generalize (lt_IZR m n H0); intro; omega. Qed. Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m. Proof. - intros m n H; apply Rnot_gt_le; red in |- *; intro. + intros m n H; apply Rnot_gt_le; red; intro. unfold Rgt in H0; generalize (lt_IZR n m H0); intro; omega. Qed. @@ -1899,10 +1899,10 @@ Proof. apply one_IZR_lt1. rewrite <- Z_R_minus; split. replace (-1) with (r - (r + 1)). - unfold Rminus in |- *; apply Rplus_lt_le_compat; auto with real. + unfold Rminus; apply Rplus_lt_le_compat; auto with real. ring. replace 1 with (r + 1 - r). - unfold Rminus in |- *; apply Rplus_le_lt_compat; auto with real. + unfold Rminus; apply Rplus_le_lt_compat; auto with real. ring. Qed. @@ -1943,8 +1943,8 @@ Proof. rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; apply H1. - red in |- *; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0). - red in |- *; intro; rewrite H2 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0). + red; intro; rewrite H2 in H; elim (Rlt_irrefl _ H). Qed. Lemma double : forall r1, 2 * r1 = r1 + r1. @@ -1954,10 +1954,10 @@ Qed. Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2. Proof. - intro; rewrite <- double; unfold Rdiv in |- *; rewrite <- Rmult_assoc; - symmetry in |- *; apply Rinv_r_simpl_m. + intro; rewrite <- double; unfold Rdiv; rewrite <- Rmult_assoc; + symmetry ; apply Rinv_r_simpl_m. replace 2 with (INR 2); - [ apply not_0_INR; discriminate | unfold INR in |- *; ring ]. + [ apply not_0_INR; discriminate | unfold INR; ring ]. Qed. (*********************************************************) @@ -1992,22 +1992,22 @@ Proof. rewrite (Rplus_comm y); intro H5; apply Rplus_le_reg_l with x; assumption. ring. replace 2 with (INR 2); [ apply not_0_INR; discriminate | reflexivity ]. - pattern y at 2 in |- *; replace y with (y / 2 + y / 2). - unfold Rminus, Rdiv in |- *. + pattern y at 2; replace y with (y / 2 + y / 2). + unfold Rminus, Rdiv. repeat rewrite Rmult_plus_distr_r. ring. cut (forall z:R, 2 * z = z + z). intro. rewrite <- (H4 (y / 2)). - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. replace 2 with (INR 2). apply not_0_INR. discriminate. - unfold INR in |- *; reflexivity. + unfold INR; reflexivity. intro; ring. cut (0%nat <> 2%nat); - [ intro H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR in |- *; + [ intro H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR; intro; assumption | discriminate ]. Qed. diff --git a/theories/Reals/RList.v b/theories/Reals/RList.v index 246d6dea9..4d140cdea 100644 --- a/theories/Reals/RList.v +++ b/theories/Reals/RList.v @@ -52,19 +52,19 @@ Proof. simpl in H; elim H. induction l as [| r0 l Hrecl0]. simpl in H; elim H; intro. - simpl in |- *; right; assumption. + simpl; right; assumption. elim H0. replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))). simpl in H; decompose [or] H. rewrite H0; apply RmaxLess1. - unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro. - apply Hrecl; simpl in |- *; tauto. + unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro. + apply Hrecl; simpl; tauto. apply Rle_trans with (MaxRlist (cons r0 l)); - [ apply Hrecl; simpl in |- *; tauto | left; auto with real ]. - unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro. - apply Hrecl; simpl in |- *; tauto. + [ apply Hrecl; simpl; tauto | left; auto with real ]. + unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro. + apply Hrecl; simpl; tauto. apply Rle_trans with (MaxRlist (cons r0 l)); - [ apply Hrecl; simpl in |- *; tauto | left; auto with real ]. + [ apply Hrecl; simpl; tauto | left; auto with real ]. reflexivity. Qed. @@ -80,19 +80,19 @@ Proof. simpl in H; elim H. induction l as [| r0 l Hrecl0]. simpl in H; elim H; intro. - simpl in |- *; right; symmetry in |- *; assumption. + simpl; right; symmetry ; assumption. elim H0. replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). simpl in H; decompose [or] H. rewrite H0; apply Rmin_l. - unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro. + unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro. apply Rle_trans with (MinRlist (cons r0 l)). assumption. - apply Hrecl; simpl in |- *; tauto. - apply Hrecl; simpl in |- *; tauto. + apply Hrecl; simpl; tauto. + apply Hrecl; simpl; tauto. apply Rle_trans with (MinRlist (cons r0 l)). apply Rmin_r. - apply Hrecl; simpl in |- *; tauto. + apply Hrecl; simpl; tauto. reflexivity. Qed. @@ -101,7 +101,7 @@ Lemma AbsList_P1 : Proof. intros; induction l as [| r l Hrecl]. elim H. - simpl in |- *; simpl in H; elim H; intro. + simpl; simpl in H; elim H; intro. left; rewrite H0; reflexivity. right; apply Hrecl; assumption. Qed. @@ -112,11 +112,11 @@ Proof. intros; induction l as [| r l Hrecl]. apply Rlt_0_1. induction l as [| r0 l Hrecl0]. - simpl in |- *; apply H; simpl in |- *; tauto. + simpl; apply H; simpl; tauto. replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). - unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro. - apply H; simpl in |- *; tauto. - apply Hrecl; intros; apply H; simpl in |- *; simpl in H0; tauto. + unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro. + apply H; simpl; tauto. + apply Hrecl; intros; apply H; simpl; simpl in H0; tauto. reflexivity. Qed. @@ -128,10 +128,10 @@ Proof. elim H. elim H; intro. exists r; split. - simpl in |- *; tauto. + simpl; tauto. assumption. assert (H1 := Hrecl H0); elim H1; intros; elim H2; clear H2; intros; - exists x0; simpl in |- *; simpl in H2; tauto. + exists x0; simpl; simpl in H2; tauto. Qed. Lemma MaxRlist_P2 : @@ -140,9 +140,9 @@ Proof. intros; induction l as [| r l Hrecl]. simpl in H; elim H; trivial. induction l as [| r0 l Hrecl0]. - simpl in |- *; left; reflexivity. - change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))) in |- *; - unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); + simpl; left; reflexivity. + change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))); + unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro. right; apply Hrecl; exists r0; left; reflexivity. left; reflexivity. @@ -164,7 +164,7 @@ Lemma pos_Rl_P1 : Proof. intros; induction l as [| r l Hrecl]; [ elim (lt_n_O _ H) - | simpl in |- *; case (Rlength l); [ reflexivity | intro; reflexivity ] ]. + | simpl; case (Rlength l); [ reflexivity | intro; reflexivity ] ]. Qed. Lemma pos_Rl_P2 : @@ -177,14 +177,14 @@ Proof. split; intro. elim H; intro. exists 0%nat; split; - [ simpl in |- *; apply lt_O_Sn | simpl in |- *; apply H0 ]. + [ simpl; apply lt_O_Sn | simpl; apply H0 ]. elim Hrecl; intros; assert (H3 := H1 H0); elim H3; intros; elim H4; intros; exists (S x0); split; - [ simpl in |- *; apply lt_n_S; assumption | simpl in |- *; assumption ]. + [ simpl; apply lt_n_S; assumption | simpl; assumption ]. elim H; intros; elim H0; intros; elim (zerop x0); intro. rewrite a in H2; simpl in H2; left; assumption. right; elim Hrecl; intros; apply H4; assert (H5 : S (pred x0) = x0). - symmetry in |- *; apply S_pred with 0%nat; assumption. + symmetry ; apply S_pred with 0%nat; assumption. exists (pred x0); split; [ simpl in H1; apply lt_S_n; rewrite H5; assumption | rewrite <- H5 in H2; simpl in H2; assumption ]. @@ -201,18 +201,18 @@ Proof. exists nil; intros; split; [ reflexivity | intros; simpl in H0; elim (lt_n_O _ H0) ]. assert (H0 : In r (cons r l)). - simpl in |- *; left; reflexivity. + simpl; left; reflexivity. assert (H1 := H _ H0); assert (H2 : forall x:R, In x l -> exists y : R, P x y). - intros; apply H; simpl in |- *; right; assumption. + intros; apply H; simpl; right; assumption. assert (H3 := Hrecl H2); elim H1; intros; elim H3; intros; exists (cons x x0); intros; elim H5; clear H5; intros; split. - simpl in |- *; rewrite H5; reflexivity. + simpl; rewrite H5; reflexivity. intros; elim (zerop i); intro. - rewrite a; simpl in |- *; assumption. + rewrite a; simpl; assumption. assert (H8 : i = S (pred i)). apply S_pred with 0%nat; assumption. - rewrite H8; simpl in |- *; apply H6; simpl in H7; apply lt_S_n; rewrite <- H8; + rewrite H8; simpl; apply H6; simpl in H7; apply lt_S_n; rewrite <- H8; assumption. Qed. @@ -271,7 +271,7 @@ Lemma RList_P0 : Proof. intros; induction l as [| r l Hrecl]; [ left; reflexivity - | simpl in |- *; case (Rle_dec r a); intro; + | simpl; case (Rle_dec r a); intro; [ right; reflexivity | left; reflexivity ] ]. Qed. @@ -279,41 +279,41 @@ Lemma RList_P1 : forall (l:Rlist) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a). Proof. intros; induction l as [| r l Hrecl]. - simpl in |- *; unfold ordered_Rlist in |- *; intros; simpl in H0; + simpl; unfold ordered_Rlist; intros; simpl in H0; elim (lt_n_O _ H0). - simpl in |- *; case (Rle_dec r a); intro. + simpl; case (Rle_dec r a); intro. assert (H1 : ordered_Rlist l). - unfold ordered_Rlist in |- *; unfold ordered_Rlist in H; intros; + unfold ordered_Rlist; unfold ordered_Rlist in H; intros; assert (H1 : (S i < pred (Rlength (cons r l)))%nat); - [ simpl in |- *; replace (Rlength l) with (S (pred (Rlength l))); + [ simpl; replace (Rlength l) with (S (pred (Rlength l))); [ 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 <- H1 in H0; simpl in H0; elim (lt_n_O _ H0) ] | apply (H _ H1) ]. - assert (H2 := Hrecl H1); unfold ordered_Rlist in |- *; intros; + assert (H2 := Hrecl H1); unfold ordered_Rlist; intros; induction i as [| i Hreci]. - simpl in |- *; assert (H3 := RList_P0 l a); elim H3; intro. + simpl; assert (H3 := RList_P0 l a); elim H3; intro. rewrite H4; assumption. induction l as [| r1 l Hrecl0]; - [ simpl in |- *; assumption - | rewrite H4; apply (H 0%nat); simpl in |- *; apply lt_O_Sn ]. - simpl in |- *; apply H2; simpl in H0; apply lt_S_n; + [ simpl; assumption + | rewrite H4; apply (H 0%nat); simpl; apply lt_O_Sn ]. + simpl; apply H2; simpl in H0; apply lt_S_n; replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a)); [ 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 <- H3 in H0; elim (lt_n_O _ H0) ]. - unfold ordered_Rlist in |- *; intros; induction i as [| i Hreci]; - [ simpl in |- *; auto with real - | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)) in |- *; apply H; - simpl in H0; simpl in |- *; apply (lt_S_n _ _ H0) ]. + unfold ordered_Rlist; intros; induction i as [| i Hreci]; + [ simpl; auto with real + | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)); apply H; + simpl in H0; simpl; apply (lt_S_n _ _ H0) ]. Qed. Lemma RList_P2 : forall l1 l2:Rlist, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2). Proof. simple induction l1; - [ intros; simpl in |- *; apply H - | intros; simpl in |- *; apply H; apply RList_P1; assumption ]. + [ intros; simpl; apply H + | intros; simpl; apply H; apply RList_P1; assumption ]. Qed. Lemma RList_P3 : @@ -324,11 +324,11 @@ Proof. [ induction l as [| r l Hrecl] | induction l as [| r l Hrecl] ]. elim H. elim H; intro; - [ exists 0%nat; split; [ apply H0 | simpl in |- *; apply lt_O_Sn ] + [ exists 0%nat; split; [ apply H0 | simpl; apply lt_O_Sn ] | elim (Hrecl H0); intros; elim H1; clear H1; intros; exists (S x0); split; - [ apply H1 | simpl in |- *; apply lt_n_S; assumption ] ]. + [ apply H1 | simpl; apply lt_n_S; assumption ] ]. elim H; intros; elim H0; intros; elim (lt_n_O _ H2). - simpl in |- *; elim H; intros; elim H0; clear H0; intros; + simpl; elim H; intros; elim H0; clear H0; intros; induction x0 as [| x0 Hrecx0]; [ left; apply H0 | right; apply Hrecl; exists x0; split; @@ -338,10 +338,10 @@ Qed. Lemma RList_P4 : forall (l1:Rlist) (a:R), ordered_Rlist (cons a l1) -> ordered_Rlist l1. Proof. - intros; unfold ordered_Rlist in |- *; intros; apply (H (S i)); simpl in |- *; + intros; unfold ordered_Rlist; intros; apply (H (S i)); simpl; replace (Rlength l1) with (S (pred (Rlength l1))); [ 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 <- H1 in H0; elim (lt_n_O _ H0) ]. Qed. @@ -350,11 +350,11 @@ Lemma RList_P5 : Proof. intros; induction l as [| r l Hrecl]; [ elim H0 - | simpl in |- *; elim H0; intro; + | simpl; elim H0; intro; [ rewrite H1; right; reflexivity | apply Rle_trans with (pos_Rl l 0); - [ apply (H 0%nat); simpl in |- *; induction l as [| r0 l Hrecl0]; - [ elim H1 | simpl in |- *; apply lt_O_Sn ] + [ apply (H 0%nat); simpl; induction l as [| r0 l Hrecl0]; + [ elim H1 | simpl; apply lt_O_Sn ] | apply Hrecl; [ eapply RList_P4; apply H | assumption ] ] ] ]. Qed. @@ -366,13 +366,13 @@ Lemma RList_P6 : Proof. simple induction l; split; intro. intros; right; reflexivity. - unfold ordered_Rlist in |- *; intros; simpl in H0; elim (lt_n_O _ H0). + unfold ordered_Rlist; intros; simpl in H0; elim (lt_n_O _ H0). intros; induction i as [| i Hreci]; [ induction j as [| j Hrecj]; [ right; reflexivity - | simpl in |- *; apply Rle_trans with (pos_Rl r0 0); - [ apply (H0 0%nat); simpl in |- *; simpl in H2; apply neq_O_lt; - red in |- *; intro; rewrite <- H3 in H2; + | simpl; apply Rle_trans with (pos_Rl r0 0); + [ apply (H0 0%nat); simpl; simpl in H2; apply neq_O_lt; + red; intro; rewrite <- H3 in H2; assert (H4 := lt_S_n _ _ H2); elim (lt_n_O _ H4) | elim H; intros; apply H3; [ apply RList_P4 with r; assumption @@ -380,12 +380,12 @@ Proof. | simpl in H2; apply lt_S_n; assumption ] ] ] | induction j as [| j Hrecj]; [ elim (le_Sn_O _ H1) - | simpl in |- *; elim H; intros; apply H3; + | simpl; elim H; intros; apply H3; [ apply RList_P4 with r; assumption | apply le_S_n; assumption | simpl in H2; apply lt_S_n; assumption ] ] ]. - unfold ordered_Rlist in |- *; intros; apply H0; - [ apply le_n_Sn | simpl in |- *; simpl in H1; apply lt_n_S; assumption ]. + unfold ordered_Rlist; intros; apply H0; + [ apply le_n_Sn | simpl; simpl in H1; apply lt_n_S; assumption ]. Qed. Lemma RList_P7 : @@ -397,11 +397,11 @@ Proof. clear H1; intros; assert (H4 := H1 H0); elim H4; clear H4; intros; elim H4; clear H4; intros; rewrite H4; assert (H6 : Rlength l = S (pred (Rlength l))). - 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 <- H6 in H5; elim (lt_n_O _ H5). apply H3; [ rewrite H6 in H5; apply lt_n_Sm_le; assumption - | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H7 in H5; + | apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H7 in H5; elim (lt_n_O _ H5) ]. Qed. @@ -420,7 +420,7 @@ Proof. [ left; assumption | right; left; assumption | right; right; assumption ] ] - | simpl in |- *; case (Rle_dec r a); intro; + | simpl; case (Rle_dec r a); intro; [ simpl in H0; decompose [or] H0; [ right; elim (H a x); intros; apply H3; left | left @@ -435,14 +435,14 @@ Proof. simple induction l1. intros; split; intro; [ simpl in H; right; assumption - | simpl in |- *; elim H; intro; [ elim H0 | assumption ] ]. + | simpl; elim H; intro; [ elim H0 | assumption ] ]. intros; split. - simpl in |- *; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0); + simpl; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0); elim H3; intro; [ left; right; assumption | elim (RList_P8 l2 r x); intros H5 _; assert (H6 := H5 H4); elim H6; intro; [ left; left; assumption | right; assumption ] ]. - intro; simpl in |- *; elim (H (insert l2 r) x); intros _ H1; apply H1; + intro; simpl; elim (H (insert l2 r) x); intros _ H1; apply H1; elim H0; intro; [ elim H2; intro; [ right; elim (RList_P8 l2 r x); intros _ H4; apply H4; left; assumption @@ -455,8 +455,8 @@ Lemma RList_P10 : Proof. intros; induction l as [| r l Hrecl]; [ reflexivity - | simpl in |- *; case (Rle_dec r a); intro; - [ simpl in |- *; rewrite Hrecl; reflexivity | reflexivity ] ]. + | simpl; case (Rle_dec r a); intro; + [ simpl; rewrite Hrecl; reflexivity | reflexivity ] ]. Qed. Lemma RList_P11 : @@ -465,7 +465,7 @@ Lemma RList_P11 : Proof. simple induction l1; [ intro; reflexivity - | intros; simpl in |- *; rewrite (H (insert l2 r)); rewrite RList_P10; + | intros; simpl; rewrite (H (insert l2 r)); rewrite RList_P10; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring ]. Qed. @@ -477,7 +477,7 @@ Proof. simple induction l; [ intros; elim (lt_n_O _ H) | intros; induction i as [| i Hreci]; - [ reflexivity | simpl in |- *; apply H; apply lt_S_n; apply H0 ] ]. + [ reflexivity | simpl; apply H; apply lt_S_n; apply H0 ] ]. Qed. Lemma RList_P13 : @@ -494,13 +494,13 @@ Proof. change (pos_Rl (mid_Rlist (cons r1 r2) r) (S i) = (pos_Rl (cons r1 r2) i + pos_Rl (cons r1 r2) (S i)) / 2) - in |- *; apply H0; simpl in |- *; apply lt_S_n; assumption. + ; apply H0; simpl; apply lt_S_n; assumption. Qed. Lemma RList_P14 : forall (l:Rlist) (a:R), Rlength (mid_Rlist l a) = Rlength l. Proof. simple induction l; intros; - [ reflexivity | simpl in |- *; rewrite (H r); reflexivity ]. + [ reflexivity | simpl; rewrite (H r); reflexivity ]. Qed. Lemma RList_P15 : @@ -511,7 +511,7 @@ Lemma RList_P15 : Proof. intros; apply Rle_antisym. induction l1 as [| r l1 Hrecl1]; - [ simpl in |- *; simpl in H1; right; symmetry in |- *; assumption + [ simpl; simpl in H1; right; symmetry ; assumption | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) 0)); intros; assert (H4 : @@ -520,7 +520,7 @@ Proof. | assert (H5 := H3 H4); apply RList_P5; [ apply RList_P2; assumption | assumption ] ] ]. induction l1 as [| r l1 Hrecl1]; - [ simpl in |- *; simpl in H1; right; assumption + [ simpl; simpl in H1; right; assumption | assert (H2 : In (pos_Rl (cons_ORlist (cons r l1) l2) 0) (cons_ORlist (cons r l1) l2)); @@ -528,7 +528,7 @@ Proof. (RList_P3 (cons_ORlist (cons r l1) l2) (pos_Rl (cons_ORlist (cons r l1) l2) 0)); intros; apply H3; 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 l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) 0)); intros; assert (H5 := H3 H2); elim H5; intro; [ apply RList_P5; assumption @@ -545,7 +545,7 @@ Lemma RList_P16 : Proof. intros; apply Rle_antisym. induction l1 as [| r l1 Hrecl1]. - simpl in |- *; simpl in H1; right; symmetry in |- *; assumption. + simpl; simpl in H1; right; symmetry ; assumption. assert (H2 : In @@ -557,7 +557,7 @@ Proof. (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2))))); intros; apply H3; exists (pred (Rlength (cons_ORlist (cons r l1) l2))); - 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 l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) @@ -565,7 +565,7 @@ Proof. intros; assert (H5 := H3 H2); elim H5; intro; [ apply RList_P7; assumption | rewrite H1; apply RList_P7; assumption ] ]. induction l1 as [| r l1 Hrecl1]. - simpl in |- *; simpl in H1; right; assumption. + simpl; simpl in H1; right; assumption. elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); intros; @@ -573,10 +573,10 @@ Proof. (H4 : In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1) \/ In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2); - [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)) in |- *; + [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)); elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1))); intros; apply H5; exists (Rlength l1); split; - [ reflexivity | simpl in |- *; apply lt_n_Sn ] + [ reflexivity | simpl; apply lt_n_Sn ] | assert (H5 := H3 H4); apply RList_P7; [ apply RList_P2; assumption | elim @@ -587,7 +587,7 @@ Proof. (RList_P3 (cons r l1) (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); intros; apply H9; exists (pred (Rlength (cons r l1))); - split; [ reflexivity | simpl in |- *; apply lt_n_Sn ] ] ]. + split; [ reflexivity | simpl; apply lt_n_Sn ] ] ]. Qed. Lemma RList_P17 : @@ -599,14 +599,14 @@ Proof. simple induction l1. intros; elim H0. intros; induction i as [| i Hreci]. - simpl in |- *; elim H1; intro; + simpl; elim H1; intro; [ simpl in H2; rewrite H4 in H2; elim (Rlt_irrefl _ H2) | apply RList_P5; [ apply RList_P4 with r; assumption | assumption ] ]. - simpl in |- *; simpl in H2; elim H1; intro. + simpl; simpl in H2; elim H1; intro. rewrite H4 in H2; assert (H5 : r <= pos_Rl r0 i); [ apply Rle_trans with (pos_Rl r0 0); - [ apply (H0 0%nat); simpl in |- *; simpl in H3; apply neq_O_lt; - red in |- *; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3) + [ apply (H0 0%nat); simpl; simpl in H3; apply neq_O_lt; + red; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3) | elim (RList_P6 r0); intros; apply H5; [ apply RList_P4 with r; assumption | apply le_O_n @@ -618,7 +618,7 @@ Proof. | simpl in H3; apply lt_S_n; replace (S (pred (Rlength r0))) with (Rlength r0); [ apply H3 - | 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 <- H5 in H3; elim (lt_n_O _ H3) ] ]. Qed. @@ -626,7 +626,7 @@ Lemma RList_P18 : forall (l:Rlist) (f:R -> R), Rlength (app_Rlist l f) = Rlength l. Proof. simple induction l; intros; - [ reflexivity | simpl in |- *; rewrite H; reflexivity ]. + [ reflexivity | simpl; rewrite H; reflexivity ]. Qed. Lemma RList_P19 : @@ -666,7 +666,7 @@ Lemma RList_P23 : Rlength (cons_Rlist l1 l2) = (Rlength l1 + Rlength l2)%nat. Proof. simple induction l1; - [ intro; reflexivity | intros; simpl in |- *; rewrite H; reflexivity ]. + [ intro; reflexivity | intros; simpl; rewrite H; reflexivity ]. Qed. Lemma RList_P24 : @@ -685,9 +685,9 @@ Proof. [ replace (Rlength r0 + Rlength (cons r1 l2))%nat with (S (Rlength r0 + Rlength l2)); [ reflexivity - | simpl in |- *; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; + | simpl; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring ] - | simpl in |- *; apply INR_eq; do 3 rewrite S_INR; do 2 rewrite plus_INR; + | simpl; apply INR_eq; do 3 rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring ]. Qed. @@ -699,27 +699,27 @@ Lemma RList_P25 : ordered_Rlist (cons_Rlist l1 l2). Proof. simple induction l1. - intros; simpl in |- *; assumption. + intros; simpl; assumption. simple induction r0. - intros; simpl in |- *; simpl in H2; unfold ordered_Rlist in |- *; intros; + intros; simpl; simpl in H2; unfold ordered_Rlist; intros; simpl in H3. induction i as [| i Hreci]. - simpl in |- *; assumption. - change (pos_Rl l2 i <= pos_Rl l2 (S i)) in |- *; apply (H1 i); apply lt_S_n; + simpl; assumption. + change (pos_Rl l2 i <= pos_Rl l2 (S i)); apply (H1 i); apply lt_S_n; replace (S (pred (Rlength l2))) with (Rlength l2); [ 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 <- H4 in H3; elim (lt_n_O _ H3) ]. intros; clear H; assert (H : ordered_Rlist (cons_Rlist (cons r1 r2) l2)). apply H0; try assumption. apply RList_P4 with r; assumption. - unfold ordered_Rlist in |- *; intros; simpl in H4; + unfold ordered_Rlist; intros; simpl in H4; induction i as [| i Hreci]. - simpl in |- *; apply (H1 0%nat); simpl in |- *; apply lt_O_Sn. + simpl; apply (H1 0%nat); simpl; apply lt_O_Sn. change (pos_Rl (cons_Rlist (cons r1 r2) l2) i <= - pos_Rl (cons_Rlist (cons r1 r2) l2) (S i)) in |- *; - apply (H i); simpl in |- *; apply lt_S_n; assumption. + pos_Rl (cons_Rlist (cons r1 r2) l2) (S i)); + apply (H i); simpl; apply lt_S_n; assumption. Qed. Lemma RList_P26 : @@ -738,13 +738,13 @@ Lemma RList_P27 : cons_Rlist l1 (cons_Rlist l2 l3) = cons_Rlist (cons_Rlist l1 l2) l3. Proof. simple induction l1; intros; - [ reflexivity | simpl in |- *; rewrite (H l2 l3); reflexivity ]. + [ reflexivity | simpl; rewrite (H l2 l3); reflexivity ]. Qed. Lemma RList_P28 : forall l:Rlist, cons_Rlist l nil = l. Proof. simple induction l; - [ reflexivity | intros; simpl in |- *; rewrite H; reflexivity ]. + [ reflexivity | intros; simpl; rewrite H; reflexivity ]. Qed. Lemma RList_P29 : @@ -759,23 +759,23 @@ Proof. replace (cons_Rlist l1 (cons r r0)) with (cons_Rlist (cons_Rlist l1 (cons r nil)) r0). inversion H0. - rewrite <- minus_n_n; simpl in |- *; rewrite RList_P26. + rewrite <- minus_n_n; simpl; rewrite RList_P26. clear l2 r0 H i H0 H1 H2; induction l1 as [| r0 l1 Hrecl1]. reflexivity. - simpl in |- *; assumption. - rewrite RList_P23; rewrite plus_comm; simpl in |- *; apply lt_n_Sn. + simpl; assumption. + rewrite RList_P23; rewrite plus_comm; simpl; apply lt_n_Sn. replace (S m - Rlength l1)%nat with (S (S m - S (Rlength l1))). - rewrite H3; simpl in |- *; + rewrite H3; simpl; replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))). apply (H (cons_Rlist l1 (cons r nil)) i). - rewrite RList_P23; rewrite plus_comm; simpl in |- *; rewrite <- H3; + rewrite RList_P23; rewrite plus_comm; simpl; rewrite <- H3; apply le_n_S; assumption. - repeat rewrite RList_P23; simpl in |- *; rewrite RList_P23 in H1; + repeat rewrite RList_P23; simpl; rewrite RList_P23 in H1; rewrite plus_comm in H1; simpl in H1; rewrite (plus_comm (Rlength l1)); - simpl in |- *; rewrite plus_comm; apply H1. + simpl; rewrite plus_comm; apply H1. rewrite RList_P23; rewrite plus_comm; reflexivity. - change (S (m - Rlength l1) = (S m - Rlength l1)%nat) in |- *; + change (S (m - Rlength l1) = (S m - Rlength l1)%nat); apply minus_Sn_m; assumption. replace (cons r r0) with (cons_Rlist (cons r nil) r0); - [ symmetry in |- *; apply RList_P27 | reflexivity ]. + [ symmetry ; apply RList_P27 | reflexivity ]. Qed. diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v index 9b64ea5f0..d0d4abb30 100644 --- a/theories/Reals/R_Ifp.v +++ b/theories/Reals/R_Ifp.v @@ -45,7 +45,7 @@ Proof. intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H; rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1; cut (1 = IZR 1); auto with zarith real. - intro; generalize H1; pattern 1 at 1 in |- *; rewrite H; intro; clear H H1; + intro; generalize H1; pattern 1 at 1; rewrite H; intro; clear H H1; rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1)); auto with zarith real. Qed. @@ -53,8 +53,8 @@ Qed. (**********) Lemma fp_R0 : frac_part 0 = 0. Proof. - unfold frac_part in |- *; unfold Int_part in |- *; elim (archimed 0); intros; - unfold Rminus in |- *; elim (Rplus_ne (- IZR (up 0 - 1))); + unfold frac_part; unfold Int_part; elim (archimed 0); intros; + unfold Rminus; elim (Rplus_ne (- IZR (up 0 - 1))); intros a b; rewrite b; clear a b; rewrite <- Z_R_minus; cut (up 0 = 1%Z). intro; rewrite H1; @@ -81,21 +81,21 @@ Qed. (**********) Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1. Proof. - intro; unfold frac_part in |- *; unfold Int_part in |- *; split. + intro; unfold frac_part; unfold Int_part; split. (*sup a O*) cut (r - IZR (up r) >= -1). - rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite <- Z_R_minus; simpl; intro; unfold Rminus; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; - fold (r - IZR (up r)) in |- *; fold (r - IZR (up r) - -1) in |- *; + fold (r - IZR (up r)); fold (r - IZR (up r) - -1); apply Rge_minus; auto with zarith real. rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r); auto with zarith real. (*inf a 1*) cut (r - IZR (up r) < 0). - rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite <- Z_R_minus; simpl; intro; unfold Rminus; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; - fold (r - IZR (up r)) in |- *; rewrite Ropp_involutive; - elim (Rplus_ne 1); intros a b; pattern 1 at 2 in |- *; + fold (r - IZR (up r)); rewrite Ropp_involutive; + elim (Rplus_ne 1); intros a b; pattern 1 at 2; rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1); apply Rplus_lt_compat_l; auto with zarith real. elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr; @@ -110,8 +110,8 @@ Qed. Lemma base_Int_part : forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1. Proof. - intro; unfold Int_part in |- *; elim (archimed r); intros. - split; rewrite <- (Z_R_minus (up r) 1); simpl in |- *. + intro; unfold Int_part; elim (archimed r); intros. + split; rewrite <- (Z_R_minus (up r) 1); simpl. generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1; rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1; rewrite (Rplus_comm (- r) (-1)) in H1; @@ -132,29 +132,29 @@ Qed. (**********) Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n. Proof. - intros n; unfold Int_part in |- *. + intros n; unfold Int_part. cut (up (INR n) = (Z.of_nat n + Z.of_nat 1)%Z). - intros H'; rewrite H'; simpl in |- *; ring. + intros H'; rewrite H'; simpl; ring. symmetry; apply tech_up; auto. replace (Z.of_nat n + Z.of_nat 1)%Z with (Z.of_nat (S n)). repeat rewrite <- INR_IZR_INZ. apply lt_INR; auto. - rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl in |- *; auto. - rewrite plus_IZR; simpl in |- *; auto with real. + rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl; auto. + rewrite plus_IZR; simpl; auto with real. repeat rewrite <- INR_IZR_INZ; auto with real. Qed. (**********) Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z, r = IZR c. Proof. - unfold frac_part in |- *; intros; split with (Int_part r); + unfold frac_part; intros; split with (Int_part r); apply Rminus_diag_uniq; auto with zarith real. Qed. (**********) Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r. Proof. - red in |- *; intros; rewrite <- H0 in H; generalize fp_R0; intro; + red; intros; rewrite <- H0 in H; generalize fp_R0; intro; auto with zarith real. Qed. @@ -243,7 +243,7 @@ Proof. intro; rewrite H1 in H; clear H1; rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H; generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H); - intros; clear H H0; unfold Int_part at 1 in |- *; + intros; clear H H0; unfold Int_part at 1; omega. Qed. @@ -336,7 +336,7 @@ Proof. generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H); intro; clear H; generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0); - intros; clear H0 H1; unfold Int_part at 1 in |- *; + intros; clear H0 H1; unfold Int_part at 1; omega. Qed. @@ -346,9 +346,9 @@ Lemma Rminus_fp1 : frac_part r1 >= frac_part r2 -> frac_part (r1 - r2) = frac_part r1 - frac_part r2. Proof. - intros; unfold frac_part in |- *; generalize (Rminus_Int_part1 r1 r2 H); + intros; unfold frac_part; generalize (Rminus_Int_part1 r1 r2 H); intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2)); - unfold Rminus in |- *; + unfold Rminus; rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2))); rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))); rewrite (Ropp_involutive (IZR (Int_part r2))); @@ -366,17 +366,17 @@ Lemma Rminus_fp2 : frac_part r1 < frac_part r2 -> frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1. Proof. - intros; unfold frac_part in |- *; generalize (Rminus_Int_part2 r1 r2 H); + intros; unfold frac_part; generalize (Rminus_Int_part2 r1 r2 H); intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1); rewrite <- (Z_R_minus (Int_part r1) (Int_part r2)); - unfold Rminus in |- *; + unfold Rminus; rewrite (Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1)) ; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))); rewrite (Ropp_involutive (IZR 1)); rewrite (Ropp_involutive (IZR (Int_part r2))); rewrite (Ropp_plus_distr (IZR (Int_part r1))); - rewrite (Ropp_involutive (IZR (Int_part r2))); simpl in |- *; + rewrite (Ropp_involutive (IZR (Int_part r2))); simpl; rewrite <- (Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1) ; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))); @@ -451,7 +451,7 @@ Proof. rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0; rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0; generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0); - intro; clear H H0; unfold Int_part at 1 in |- *; omega. + intro; clear H H0; unfold Int_part at 1; omega. Qed. (**********) @@ -514,7 +514,7 @@ Proof. rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1; rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1; generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1); - intro; clear H0 H1; unfold Int_part at 1 in |- *; + intro; clear H0 H1; unfold Int_part at 1; omega. Qed. @@ -524,17 +524,17 @@ Lemma plus_frac_part1 : frac_part r1 + frac_part r2 >= 1 -> frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1. Proof. - intros; unfold frac_part in |- *; generalize (plus_Int_part1 r1 r2 H); intro; + intros; unfold frac_part; generalize (plus_Int_part1 r1 r2 H); intro; rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1); - rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl in |- *; - unfold Rminus at 3 4 in |- *; + rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl; + unfold Rminus at 3 4; rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))); rewrite (Rplus_comm r2 (- IZR (Int_part r2))); rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2); rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2); rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))); rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))); - unfold Rminus in |- *; + unfold Rminus; rewrite (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1)) ; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1); @@ -547,14 +547,14 @@ Lemma plus_frac_part2 : frac_part r1 + frac_part r2 < 1 -> frac_part (r1 + r2) = frac_part r1 + frac_part r2. Proof. - intros; unfold frac_part in |- *; generalize (plus_Int_part2 r1 r2 H); intro; + intros; unfold frac_part; generalize (plus_Int_part2 r1 r2 H); intro; rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2)); - unfold Rminus at 2 3 in |- *; + unfold Rminus at 2 3; rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))); rewrite (Rplus_comm r2 (- IZR (Int_part r2))); rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2); rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2); rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))); rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))); - unfold Rminus in |- *; trivial with zarith real. + unfold Rminus; trivial with zarith real. Qed. diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v index 868b8617f..8bea7618c 100644 --- a/theories/Reals/R_sqr.v +++ b/theories/Reals/R_sqr.v @@ -14,7 +14,7 @@ Local Open Scope R_scope. (** Rsqr : some results *) (****************************************************) -Ltac ring_Rsqr := unfold Rsqr in |- *; ring. +Ltac ring_Rsqr := unfold Rsqr; ring. Lemma Rsqr_neg : forall x:R, Rsqr x = Rsqr (- x). Proof. @@ -48,25 +48,25 @@ Qed. Lemma Rsqr_gt_0_0 : forall x:R, 0 < Rsqr x -> x <> 0. Proof. - intros; red in |- *; intro; rewrite H0 in H; rewrite Rsqr_0 in H; + intros; red; intro; rewrite H0 in H; rewrite Rsqr_0 in H; elim (Rlt_irrefl 0 H). Qed. Lemma Rsqr_pos_lt : forall x:R, x <> 0 -> 0 < Rsqr x. Proof. intros; case (Rtotal_order 0 x); intro; - [ unfold Rsqr in |- *; apply Rmult_lt_0_compat; assumption + [ unfold Rsqr; apply Rmult_lt_0_compat; assumption | elim H0; intro; - [ elim H; symmetry in |- *; exact H1 + [ elim H; symmetry ; exact H1 | rewrite Rsqr_neg; generalize (Ropp_lt_gt_contravar x 0 H1); - rewrite Ropp_0; intro; unfold Rsqr in |- *; + rewrite Ropp_0; intro; unfold Rsqr; apply Rmult_lt_0_compat; assumption ] ]. Qed. Lemma Rsqr_div : forall x y:R, y <> 0 -> Rsqr (x / y) = Rsqr x / Rsqr y. Proof. - intros; unfold Rsqr in |- *. - unfold Rdiv in |- *. + intros; unfold Rsqr. + unfold Rdiv. rewrite Rinv_mult_distr. repeat rewrite Rmult_assoc. apply Rmult_eq_compat_l. @@ -80,7 +80,7 @@ Qed. Lemma Rsqr_eq_0 : forall x:R, Rsqr x = 0 -> x = 0. Proof. - unfold Rsqr in |- *; intros; generalize (Rmult_integral x x H); intro; + unfold Rsqr; intros; generalize (Rmult_integral x x H); intro; elim H0; intro; assumption. Qed. @@ -122,7 +122,7 @@ Qed. Lemma Rsqr_incr_1 : forall x y:R, x <= y -> 0 <= x -> 0 <= y -> Rsqr x <= Rsqr y. Proof. - intros; unfold Rsqr in |- *; apply Rmult_le_compat; assumption. + intros; unfold Rsqr; apply Rmult_le_compat; assumption. Qed. Lemma Rsqr_incrst_0 : @@ -140,7 +140,7 @@ Qed. Lemma Rsqr_incrst_1 : forall x y:R, x < y -> 0 <= x -> 0 <= y -> Rsqr x < Rsqr y. Proof. - intros; unfold Rsqr in |- *; apply Rmult_le_0_lt_compat; assumption. + intros; unfold Rsqr; apply Rmult_le_0_lt_compat; assumption. Qed. Lemma Rsqr_neg_pos_le_0 : @@ -183,7 +183,7 @@ Qed. Lemma Rsqr_abs : forall x:R, Rsqr x = Rsqr (Rabs x). Proof. - intro; unfold Rabs in |- *; case (Rcase_abs x); intro; + intro; unfold Rabs; case (Rcase_abs x); intro; [ apply Rsqr_neg | reflexivity ]. Qed. @@ -220,7 +220,7 @@ Qed. Lemma Rsqr_eq_abs_0 : forall x y:R, Rsqr x = Rsqr y -> Rabs x = Rabs y. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs x); case (Rcase_abs y); intros. + intros; unfold Rabs; case (Rcase_abs x); case (Rcase_abs y); intros. rewrite (Rsqr_neg x) in H; rewrite (Rsqr_neg y) in H; generalize (Ropp_lt_gt_contravar y 0 r); generalize (Ropp_lt_gt_contravar x 0 r0); rewrite Ropp_0; @@ -288,7 +288,7 @@ Qed. Lemma Rsqr_inv : forall x:R, x <> 0 -> Rsqr (/ x) = / Rsqr x. Proof. - intros; unfold Rsqr in |- *. + intros; unfold Rsqr. rewrite Rinv_mult_distr; try reflexivity || assumption. Qed. @@ -302,7 +302,7 @@ Proof. repeat rewrite Rmult_plus_distr_l. repeat rewrite Rplus_assoc. apply Rplus_eq_compat_l. - unfold Rdiv, Rminus in |- *. + unfold Rdiv, Rminus. replace (2 * 1 + 2 * 1) with 4; [ idtac | ring ]. rewrite (Rmult_plus_distr_r (4 * a * c) (- Rsqr b) (/ (4 * a))). rewrite Rsqr_mult. @@ -332,7 +332,7 @@ Proof. rewrite (Rmult_comm x). apply Rplus_eq_compat_l. rewrite (Rmult_comm (/ a)). - unfold Rsqr in |- *; repeat rewrite Rmult_assoc. + unfold Rsqr; repeat rewrite Rmult_assoc. rewrite <- Rinv_l_sym. rewrite Rmult_1_r. ring. @@ -357,7 +357,7 @@ Proof. rewrite Rplus_opp_l; replace (- (y * y) + x * x) with ((x - y) * (x + y)). intro; generalize (Rmult_integral (x - y) (x + y) H0); intro; elim H1; intros. left; apply Rminus_diag_uniq; assumption. - right; apply Rminus_diag_uniq; unfold Rminus in |- *; rewrite Ropp_involutive; + right; apply Rminus_diag_uniq; unfold Rminus; rewrite Ropp_involutive; assumption. ring. Qed. diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index 47b9e6cf9..28db95b38 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -36,7 +36,7 @@ Qed. Lemma sqrt_sqrt : forall x:R, 0 <= x -> sqrt x * sqrt x = x. Proof. intros. - unfold sqrt in |- *. + unfold sqrt. case (Rcase_abs x); intro. elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). rewrite Rsqrt_Rsqrt; reflexivity. @@ -44,7 +44,7 @@ Qed. Lemma sqrt_0 : sqrt 0 = 0. Proof. - apply Rsqr_eq_0; unfold Rsqr in |- *; apply sqrt_sqrt; right; reflexivity. + apply Rsqr_eq_0; unfold Rsqr; apply sqrt_sqrt; right; reflexivity. Qed. Lemma sqrt_1 : sqrt 1 = 1. @@ -52,7 +52,7 @@ Proof. apply (Rsqr_inj (sqrt 1) 1); [ apply sqrt_positivity; left | left - | unfold Rsqr in |- *; rewrite sqrt_sqrt; [ ring | left ] ]; + | unfold Rsqr; rewrite sqrt_sqrt; [ ring | left ] ]; apply Rlt_0_1. Qed. @@ -73,7 +73,7 @@ Proof. intros; apply Rsqr_inj; [ apply (sqrt_positivity x H) | assumption - | unfold Rsqr in |- *; rewrite H1; apply (sqrt_sqrt x H) ]. + | unfold Rsqr; rewrite H1; apply (sqrt_sqrt x H) ]. Qed. Lemma sqrt_def : forall x:R, 0 <= x -> sqrt x * sqrt x = x. @@ -86,12 +86,12 @@ Proof. intros; apply (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (Rle_0_sqr x)) H); - unfold Rsqr in |- *; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)). + unfold Rsqr; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)). Qed. Lemma sqrt_Rsqr : forall x:R, 0 <= x -> sqrt (Rsqr x) = x. Proof. - intros; unfold Rsqr in |- *; apply sqrt_square; assumption. + intros; unfold Rsqr; apply sqrt_square; assumption. Qed. Lemma sqrt_Rsqr_abs : forall x:R, sqrt (Rsqr x) = Rabs x. @@ -101,7 +101,7 @@ Qed. Lemma Rsqr_sqrt : forall x:R, 0 <= x -> Rsqr (sqrt x) = x. Proof. - intros x H1; unfold Rsqr in |- *; apply (sqrt_sqrt x H1). + intros x H1; unfold Rsqr; apply (sqrt_sqrt x H1). Qed. Lemma sqrt_mult_alt : @@ -300,7 +300,7 @@ Proof. intros x H1 H2; generalize (sqrt_lt_1 x 1 (Rlt_le 0 x H1) (Rlt_le 0 1 Rlt_0_1) H2); intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x)); - intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1 in |- *; + intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1; rewrite <- (sqrt_def x (Rlt_le 0 x H1)); apply (Rmult_lt_compat_l (sqrt x) (sqrt x) 1 (sqrt_lt_R0 x H1) H3). Qed. @@ -310,7 +310,7 @@ Lemma sqrt_cauchy : a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d). Proof. intros a b c d; apply Rsqr_incr_0_var; - [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr in |- *; + [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr; [ replace ((a * c + b * d) * (a * c + b * d)) with (a * a * c * c + b * b * d * d + 2 * a * b * c * d); [ replace ((a * a + b * b) * (c * c + d * d)) with @@ -319,11 +319,11 @@ Proof. replace (a * a * d * d + b * b * c * c) with (2 * a * b * c * d + (a * a * d * d + b * b * c * c - 2 * a * b * c * d)); - [ pattern (2 * a * b * c * d) at 1 in |- *; rewrite <- Rplus_0_r; + [ pattern (2 * a * b * c * d) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; replace (a * a * d * d + b * b * c * c - 2 * a * b * c * d) with (Rsqr (a * d - b * c)); - [ apply Rle_0_sqr | unfold Rsqr in |- *; ring ] + [ apply Rle_0_sqr | unfold Rsqr; ring ] | ring ] | ring ] | ring ] @@ -355,16 +355,16 @@ Lemma Rsqr_sol_eq_0_1 : x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0. Proof. intros; elim H0; intro. - unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv; repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg; rewrite Rsqr_sqrt. rewrite Rsqr_inv. - unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr. + unfold Rsqr; repeat rewrite Rinv_mult_distr. repeat rewrite Rmult_assoc; rewrite (Rmult_comm a). repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite Rmult_plus_distr_r. repeat rewrite Rmult_assoc. - pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). + pattern 2 at 2; rewrite (Rmult_comm 2). repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r. rewrite @@ -376,7 +376,7 @@ Proof. (b * (- b * (/ 2 * / a)) + (b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + c))) with (b * (- b * (/ 2 * / a)) + c). - unfold Rminus in |- *; repeat rewrite <- Rplus_assoc. + unfold Rminus; repeat rewrite <- Rplus_assoc. replace (b * b + b * b) with (2 * (b * b)). rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc. rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc. @@ -407,17 +407,17 @@ Proof. apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. assumption. - unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv; repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg; rewrite Rsqr_sqrt. rewrite Rsqr_inv. - unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr; + unfold Rsqr; repeat rewrite Rinv_mult_distr; repeat rewrite Rmult_assoc. rewrite (Rmult_comm a); repeat rewrite Rmult_assoc. rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; unfold Rminus in |- *; rewrite Rmult_plus_distr_r. + rewrite Rmult_1_r; unfold Rminus; rewrite Rmult_plus_distr_r. rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; - pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). + pattern 2 at 2; rewrite (Rmult_comm 2). repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite @@ -480,23 +480,23 @@ Proof. intro; generalize (Rsqr_eq (x + b / (2 * a)) (sqrt (Delta a b c) / (2 * a)) H3); intro; elim H4; intro. - left; unfold sol_x1 in |- *; + left; unfold sol_x1; generalize (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) (sqrt (Delta a b c) / (2 * a)) H5); replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. - intro; rewrite H6; unfold Rdiv in |- *; ring. + intro; rewrite H6; unfold Rdiv; ring. ring. - right; unfold sol_x2 in |- *; + right; unfold sol_x2; generalize (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) (- (sqrt (Delta a b c) / (2 * a))) H5); replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. - intro; rewrite H6; unfold Rdiv in |- *; ring. + intro; rewrite H6; unfold Rdiv; ring. ring. rewrite Rsqr_div. rewrite Rsqr_sqrt. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rmult_assoc. rewrite (Rmult_comm (/ a)). rewrite Rmult_assoc. @@ -510,9 +510,9 @@ Proof. assumption. apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. - symmetry in |- *; apply Rmult_1_l. + symmetry ; apply Rmult_1_l. apply (cond_nonzero a). - unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. + unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse. rewrite Ropp_minus_distr. reflexivity. reflexivity. diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v index 804bfe114..4ca39d37c 100644 --- a/theories/Reals/Ranalysis1.v +++ b/theories/Reals/Ranalysis1.v @@ -82,14 +82,14 @@ Lemma continuity_pt_plus : forall f1 f2 (x0:R), continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0. Proof. - unfold continuity_pt, plus_fct in |- *; unfold continue_in in |- *; intros; + unfold continuity_pt, plus_fct; unfold continue_in; intros; apply limit_plus; assumption. Qed. Lemma continuity_pt_opp : forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0. Proof. - unfold continuity_pt, opp_fct in |- *; unfold continue_in in |- *; intros; + unfold continuity_pt, opp_fct; unfold continue_in; intros; apply limit_Ropp; assumption. Qed. @@ -97,7 +97,7 @@ Lemma continuity_pt_minus : forall f1 f2 (x0:R), continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0. Proof. - unfold continuity_pt, minus_fct in |- *; unfold continue_in in |- *; intros; + unfold continuity_pt, minus_fct; unfold continue_in; intros; apply limit_minus; assumption. Qed. @@ -105,17 +105,17 @@ Lemma continuity_pt_mult : forall f1 f2 (x0:R), continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0. Proof. - unfold continuity_pt, mult_fct in |- *; unfold continue_in in |- *; intros; + unfold continuity_pt, mult_fct; unfold continue_in; intros; apply limit_mul; assumption. Qed. Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0. Proof. - unfold constant, continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; + unfold constant, continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; intros; exists 1; split; [ apply Rlt_0_1 - | intros; generalize (H x x0); intro; rewrite H2; simpl in |- *; + | intros; generalize (H x x0); intro; rewrite H2; simpl; rewrite R_dist_eq; assumption ]. Qed. @@ -123,9 +123,9 @@ Lemma continuity_pt_scal : forall f (a x0:R), continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0. Proof. - unfold continuity_pt, mult_real_fct in |- *; unfold continue_in in |- *; + unfold continuity_pt, mult_real_fct; unfold continue_in; intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0). - unfold limit1_in in |- *; unfold limit_in in |- *; intros; exists 1; split. + unfold limit1_in; unfold limit_in; intros; exists 1; split. apply Rlt_0_1. intros; rewrite R_dist_eq; assumption. assumption. @@ -136,9 +136,9 @@ Lemma continuity_pt_inv : Proof. intros. replace (/ f)%F with (fun x:R => / f x). - unfold continuity_pt in |- *; unfold continue_in in |- *; intros; + unfold continuity_pt; unfold continue_in; intros; apply limit_inv; assumption. - unfold inv_fct in |- *; reflexivity. + unfold inv_fct; reflexivity. Qed. Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F. @@ -159,8 +159,8 @@ Lemma continuity_pt_comp : forall f1 f2 (x:R), continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x. Proof. - unfold continuity_pt in |- *; unfold continue_in in |- *; intros; - unfold comp in |- *. + unfold continuity_pt; unfold continue_in; intros; + unfold comp. cut (limit1_in (fun x0:R => f2 (f1 x0)) (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) ( @@ -170,23 +170,23 @@ Proof. eapply limit_comp. apply H. apply H0. - unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold limit1_in; unfold limit_in; unfold dist; + simpl; unfold R_dist; intros. assert (H3 := H1 eps H2). elim H3; intros. exists x0. split. elim H4; intros; assumption. intros; case (Req_dec (f1 x) (f1 x1)); intro. - rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. elim H4; intros; apply H8. split. - unfold Dgf, D_x, no_cond in |- *. + unfold Dgf, D_x, no_cond. split. split. trivial. - elim H5; unfold D_x, no_cond in |- *; intros. + elim H5; unfold D_x, no_cond; intros. elim H9; intros; assumption. split. trivial. @@ -198,44 +198,44 @@ Qed. Lemma continuity_plus : forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2). Proof. - unfold continuity in |- *; intros; + unfold continuity; intros; apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_opp : forall f, continuity f -> continuity (- f). Proof. - unfold continuity in |- *; intros; apply (continuity_pt_opp f x (H x)). + unfold continuity; intros; apply (continuity_pt_opp f x (H x)). Qed. Lemma continuity_minus : forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2). Proof. - unfold continuity in |- *; intros; + unfold continuity; intros; apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_mult : forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2). Proof. - unfold continuity in |- *; intros; + unfold continuity; intros; apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_const : forall f, constant f -> continuity f. Proof. - unfold continuity in |- *; intros; apply (continuity_pt_const f x H). + unfold continuity; intros; apply (continuity_pt_const f x H). Qed. Lemma continuity_scal : forall f (a:R), continuity f -> continuity (mult_real_fct a f). Proof. - unfold continuity in |- *; intros; apply (continuity_pt_scal f a x (H x)). + unfold continuity; intros; apply (continuity_pt_scal f a x (H x)). Qed. Lemma continuity_inv : forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f). Proof. - unfold continuity in |- *; intros; apply (continuity_pt_inv f x (H x) (H0 x)). + unfold continuity; intros; apply (continuity_pt_inv f x (H x) (H0 x)). Qed. Lemma continuity_div : @@ -243,14 +243,14 @@ Lemma continuity_div : continuity f1 -> continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2). Proof. - unfold continuity in |- *; intros; + unfold continuity; intros; apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). Qed. Lemma continuity_comp : forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1). Proof. - unfold continuity in |- *; intros. + unfold continuity; intros. apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). Qed. @@ -307,23 +307,23 @@ Proof. apply (single_limit (fun h:R => (f (x + h) - f x) / h) ( fun h:R => h <> 0) l1 l2 0); try assumption. - unfold adhDa in |- *; intros; exists (alp / 2). + unfold adhDa; intros; exists (alp / 2). split. - unfold Rdiv in |- *; apply prod_neq_R0. - red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). + unfold Rdiv; apply prod_neq_R0. + red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). apply Rinv_neq_0_compat; discrR. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; - rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite Rabs_mult. + unfold R_dist; unfold Rminus; rewrite Ropp_0; + rewrite Rplus_0_r; unfold Rdiv; rewrite Rabs_mult. replace (Rabs (/ 2)) with (/ 2). replace (Rabs alp) with alp. apply Rmult_lt_reg_l with 2. prove_sup0. rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double; - pattern alp at 1 in |- *; replace alp with (alp + 0); + pattern alp at 1; replace alp with (alp + 0); [ idtac | ring ]; apply Rplus_lt_compat_l; assumption. - symmetry in |- *; apply Rabs_right; left; assumption. - symmetry in |- *; apply Rabs_right; left; change (0 < / 2) in |- *; + symmetry ; apply Rabs_right; left; assumption. + symmetry ; apply Rabs_right; left; change (0 < / 2); apply Rinv_0_lt_compat; prove_sup0. Qed. @@ -332,14 +332,14 @@ Lemma uniqueness_step2 : derivable_pt_lim f x l -> limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0. Proof. - unfold derivable_pt_lim in |- *; intros; unfold limit1_in in |- *; - unfold limit_in in |- *; intros. + unfold derivable_pt_lim; intros; unfold limit1_in; + unfold limit_in; intros. assert (H1 := H eps H0). elim H1; intros. exists (pos x0). split. apply (cond_pos x0). - simpl in |- *; unfold R_dist in |- *; intros. + simpl; unfold R_dist; intros. elim H3; intros. apply H2; [ assumption @@ -352,15 +352,15 @@ Lemma uniqueness_step3 : limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 -> derivable_pt_lim f x l. Proof. - unfold limit1_in, derivable_pt_lim in |- *; unfold limit_in in |- *; - unfold dist in |- *; simpl in |- *; intros. + unfold limit1_in, derivable_pt_lim; unfold limit_in; + unfold dist; simpl; intros. elim (H eps H0). intros; elim H1; intros. exists (mkposreal x0 H2). - simpl in |- *; intros; unfold R_dist in H3; apply (H3 h). + simpl; intros; unfold R_dist in H3; apply (H3 h). split; [ assumption - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; assumption ]. + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; assumption ]. Qed. Lemma uniqueness_limite : @@ -383,8 +383,8 @@ Proof. assumption. intro; assert (H1 := proj2_sig pr); unfold derivable_pt_abs in H1. assert (H2 := uniqueness_limite _ _ _ _ H H1). - unfold derive_pt in |- *; unfold derivable_pt_abs in |- *. - symmetry in |- *; assumption. + unfold derive_pt; unfold derivable_pt_abs. + symmetry ; assumption. Qed. (**********) @@ -414,25 +414,25 @@ Lemma derive_pt_D_in : D_in f df no_cond x <-> derive_pt f x pr = df x. Proof. intros; split. - unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold D_in; unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros. apply derive_pt_eq_0. - unfold derivable_pt_lim in |- *. + unfold derivable_pt_lim. intros; elim (H eps H0); intros alpha H1; elim H1; intros; exists (mkposreal alpha H2); intros; generalize (H3 (x + h)); intro; cut (x + h - x = h); [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha); [ intro; generalize (H6 H8); rewrite H7; intro; assumption | split; - [ unfold D_x in |- *; split; - [ unfold no_cond in |- *; trivial + [ unfold D_x; split; + [ unfold no_cond; trivial | apply Rminus_not_eq_right; rewrite H7; assumption ] | rewrite H7; assumption ] ] | ring ]. intro. assert (H0 := derive_pt_eq_1 f x (df x) pr H). - unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; - unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + unfold D_in; unfold limit1_in; unfold limit_in; + unfold dist; simpl; unfold R_dist; intros. elim (H0 eps H1); intros alpha H2; exists (pos alpha); split. apply (cond_pos alpha). @@ -448,24 +448,24 @@ Lemma derivable_pt_lim_D_in : D_in f df no_cond x <-> derivable_pt_lim f x (df x). Proof. intros; split. - unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros. - unfold derivable_pt_lim in |- *. + unfold D_in; unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros. + unfold derivable_pt_lim. intros; elim (H eps H0); intros alpha H1; elim H1; intros; exists (mkposreal alpha H2); intros; generalize (H3 (x + h)); intro; cut (x + h - x = h); [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha); [ intro; generalize (H6 H8); rewrite H7; intro; assumption | split; - [ unfold D_x in |- *; split; - [ unfold no_cond in |- *; trivial + [ unfold D_x; split; + [ unfold no_cond; trivial | apply Rminus_not_eq_right; rewrite H7; assumption ] | rewrite H7; assumption ] ] | ring ]. intro. unfold derivable_pt_lim in H. - unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; - unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + unfold D_in; unfold limit1_in; unfold limit_in; + unfold dist; simpl; unfold R_dist; intros. elim (H eps H0); intros alpha H2; exists (pos alpha); split. apply (cond_pos alpha). @@ -486,7 +486,7 @@ Lemma derivable_derive : forall f (x:R) (pr:derivable_pt f x), exists l : R, derive_pt f x pr = l. Proof. intros; exists (proj1_sig pr). - unfold derive_pt in |- *; reflexivity. + unfold derive_pt; reflexivity. Qed. Theorem derivable_continuous_pt : @@ -501,14 +501,14 @@ Proof. generalize (derive_pt_D_in f (fct_cte l) x); intro. elim (H2 X); intros. generalize (H4 H1); intro. - unfold continuity_pt in |- *. + unfold continuity_pt. apply (cont_deriv f (fct_cte l) no_cond x H5). - unfold fct_cte in |- *; reflexivity. + unfold fct_cte; reflexivity. Qed. Theorem derivable_continuous : forall f, derivable f -> continuity f. Proof. - unfold derivable, continuity in |- *; intros f X x. + unfold derivable, continuity; intros f X x. apply (derivable_continuous_pt f x (X x)). Qed. @@ -524,7 +524,7 @@ Lemma derivable_pt_lim_plus : apply uniqueness_step3. assert (H1 := uniqueness_step2 _ _ _ H). assert (H2 := uniqueness_step2 _ _ _ H0). - unfold plus_fct in |- *. + unfold plus_fct. cut (forall h:R, (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h = @@ -533,15 +533,15 @@ Lemma derivable_pt_lim_plus : generalize (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h') (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). - unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold limit1_in; unfold limit_in; unfold dist; + simpl; unfold R_dist; intros. elim (H4 eps H5); intros. exists x0. elim H6; intros. split. assumption. intros; rewrite H3; apply H8; assumption. - intro; unfold Rdiv in |- *; ring. + intro; unfold Rdiv; ring. Qed. Lemma derivable_pt_lim_opp : @@ -550,20 +550,20 @@ Proof. intros. apply uniqueness_step3. assert (H1 := uniqueness_step2 _ _ _ H). - unfold opp_fct in |- *. + unfold opp_fct. cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)). intro. generalize (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1). - unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold limit1_in; unfold limit_in; unfold dist; + simpl; unfold R_dist; intros. elim (H2 eps H3); intros. exists x0. elim H4; intros. split. assumption. intros; rewrite H0; apply H6; assumption. - intro; unfold Rdiv in |- *; ring. + intro; unfold Rdiv; ring. Qed. Lemma derivable_pt_lim_minus : @@ -575,7 +575,7 @@ Proof. apply uniqueness_step3. assert (H1 := uniqueness_step2 _ _ _ H). assert (H2 := uniqueness_step2 _ _ _ H0). - unfold minus_fct in |- *. + unfold minus_fct. cut (forall h:R, (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h = @@ -584,15 +584,15 @@ Proof. generalize (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h') (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). - unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold limit1_in; unfold limit_in; unfold dist; + simpl; unfold R_dist; intros. elim (H4 eps H5); intros. exists x0. elim H6; intros. split. assumption. intros; rewrite <- H3; apply H8; assumption. - intro; unfold Rdiv in |- *; ring. + intro; unfold Rdiv; ring. Qed. Lemma derivable_pt_lim_mult : @@ -615,15 +615,15 @@ Proof. elim H1; intros. clear H1 H3. apply H2. - unfold mult_fct in |- *. + unfold mult_fct. apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption. Qed. Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0. Proof. - intros; unfold fct_cte, derivable_pt_lim in |- *. - intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus in |- *; - rewrite Rplus_opp_r; unfold Rdiv in |- *; rewrite Rmult_0_l; + intros; unfold fct_cte, derivable_pt_lim. + intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus; + rewrite Rplus_opp_r; unfold Rdiv; rewrite Rmult_0_l; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. Qed. @@ -636,34 +636,34 @@ Proof. replace (mult_real_fct a f) with (fct_cte a * f)%F. replace (a * l) with (0 * f x + a * l); [ idtac | ring ]. apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption. - unfold mult_real_fct, mult_fct, fct_cte in |- *; reflexivity. + unfold mult_real_fct, mult_fct, fct_cte; reflexivity. Qed. Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1. Proof. - intro; unfold derivable_pt_lim in |- *. + intro; unfold derivable_pt_lim. intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2; - unfold id in |- *; replace ((x + h - x) / h - 1) with 0. + unfold id; replace ((x + h - x) / h - 1) with 0. rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h). apply Rabs_pos. assumption. - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite (Rplus_comm x); + unfold Rminus; rewrite Rplus_assoc; rewrite (Rplus_comm x); rewrite Rplus_assoc. - rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv; rewrite <- Rinv_r_sym. - symmetry in |- *; apply Rplus_opp_r. + symmetry ; apply Rplus_opp_r. assumption. Qed. Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x). Proof. - intro; unfold derivable_pt_lim in |- *. - unfold Rsqr in |- *; intros eps Heps; exists (mkposreal eps Heps); + intro; unfold derivable_pt_lim. + unfold Rsqr; intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h. assumption. replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h); [ idtac | ring ]. - unfold Rdiv in |- *; rewrite Rmult_plus_distr_r. + unfold Rdiv; rewrite Rmult_plus_distr_r. repeat rewrite Rmult_assoc. repeat rewrite <- Rinv_r_sym; [ idtac | assumption ]. ring. @@ -684,7 +684,7 @@ Proof. assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x). elim H1; intros. clear H1 H3; apply H2. - unfold comp in |- *; + unfold comp; cut (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) (Dgf no_cond no_cond f1) x -> @@ -693,14 +693,14 @@ Proof. rewrite Rmult_comm; apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption. - unfold Dgf, D_in, no_cond in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; unfold dist in |- *; simpl in |- *; - unfold R_dist in |- *; intros. + unfold Dgf, D_in, no_cond; unfold limit1_in; + unfold limit_in; unfold dist; simpl; + unfold R_dist; intros. elim (H1 eps H3); intros. exists x0; intros; split. elim H5; intros; assumption. intros; elim H5; intros; apply H9; split. - unfold D_x in |- *; split. + unfold D_x; split. split; trivial. elim H6; intros; unfold D_x in H10; elim H10; intros; assumption. elim H6; intros; assumption. @@ -710,7 +710,7 @@ Lemma derivable_pt_plus : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x. Proof. - unfold derivable_pt in |- *; intros f1 f2 x X X0. + unfold derivable_pt; intros f1 f2 x X X0. elim X; intros. elim X0; intros. exists (x0 + x1). @@ -720,7 +720,7 @@ Qed. Lemma derivable_pt_opp : forall f (x:R), derivable_pt f x -> derivable_pt (- f) x. Proof. - unfold derivable_pt in |- *; intros f x X. + unfold derivable_pt; intros f x X. elim X; intros. exists (- x0). apply derivable_pt_lim_opp; assumption. @@ -730,7 +730,7 @@ Lemma derivable_pt_minus : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x. Proof. - unfold derivable_pt in |- *; intros f1 f2 x X X0. + unfold derivable_pt; intros f1 f2 x X X0. elim X; intros. elim X0; intros. exists (x0 - x1). @@ -741,7 +741,7 @@ Lemma derivable_pt_mult : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x. Proof. - unfold derivable_pt in |- *; intros f1 f2 x X X0. + unfold derivable_pt; intros f1 f2 x X X0. elim X; intros. elim X0; intros. exists (x0 * f2 x + f1 x * x1). @@ -750,7 +750,7 @@ Qed. Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x. Proof. - intros; unfold derivable_pt in |- *. + intros; unfold derivable_pt. exists 0. apply derivable_pt_lim_const. Qed. @@ -758,7 +758,7 @@ Qed. Lemma derivable_pt_scal : forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x. Proof. - unfold derivable_pt in |- *; intros f1 a x X. + unfold derivable_pt; intros f1 a x X. elim X; intros. exists (a * x0). apply derivable_pt_lim_scal; assumption. @@ -766,14 +766,14 @@ Qed. Lemma derivable_pt_id : forall x:R, derivable_pt id x. Proof. - unfold derivable_pt in |- *; intro. + unfold derivable_pt; intro. exists 1. apply derivable_pt_lim_id. Qed. Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x. Proof. - unfold derivable_pt in |- *; intro; exists (2 * x). + unfold derivable_pt; intro; exists (2 * x). apply derivable_pt_lim_Rsqr. Qed. @@ -781,7 +781,7 @@ Lemma derivable_pt_comp : forall f1 f2 (x:R), derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x. Proof. - unfold derivable_pt in |- *; intros f1 f2 x X X0. + unfold derivable_pt; intros f1 f2 x X X0. elim X; intros. elim X0; intros. exists (x1 * x0). @@ -791,57 +791,57 @@ Qed. Lemma derivable_plus : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2). Proof. - unfold derivable in |- *; intros f1 f2 X X0 x. + unfold derivable; intros f1 f2 X X0 x. apply (derivable_pt_plus _ _ x (X _) (X0 _)). Qed. Lemma derivable_opp : forall f, derivable f -> derivable (- f). Proof. - unfold derivable in |- *; intros f X x. + unfold derivable; intros f X x. apply (derivable_pt_opp _ x (X _)). Qed. Lemma derivable_minus : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2). Proof. - unfold derivable in |- *; intros f1 f2 X X0 x. + unfold derivable; intros f1 f2 X X0 x. apply (derivable_pt_minus _ _ x (X _) (X0 _)). Qed. Lemma derivable_mult : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2). Proof. - unfold derivable in |- *; intros f1 f2 X X0 x. + unfold derivable; intros f1 f2 X X0 x. apply (derivable_pt_mult _ _ x (X _) (X0 _)). Qed. Lemma derivable_const : forall a:R, derivable (fct_cte a). Proof. - unfold derivable in |- *; intros. + unfold derivable; intros. apply derivable_pt_const. Qed. Lemma derivable_scal : forall f (a:R), derivable f -> derivable (mult_real_fct a f). Proof. - unfold derivable in |- *; intros f a X x. + unfold derivable; intros f a X x. apply (derivable_pt_scal _ a x (X _)). Qed. Lemma derivable_id : derivable id. Proof. - unfold derivable in |- *; intro; apply derivable_pt_id. + unfold derivable; intro; apply derivable_pt_id. Qed. Lemma derivable_Rsqr : derivable Rsqr. Proof. - unfold derivable in |- *; intro; apply derivable_pt_Rsqr. + unfold derivable; intro; apply derivable_pt_Rsqr. Qed. Lemma derivable_comp : forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1). Proof. - unfold derivable in |- *; intros f1 f2 X X0 x. + unfold derivable; intros f1 f2 X X0 x. apply (derivable_pt_comp _ _ x (X _) (X0 _)). Qed. @@ -996,13 +996,13 @@ Proof. elim (lt_irrefl _ H). cut (n = 0%nat \/ (0 < n)%nat). intro; elim H0; intro. - rewrite H1; simpl in |- *. + rewrite H1; simpl. replace (fun y:R => y * 1) with (id * fct_cte 1)%F. replace (1 * 1) with (1 * fct_cte 1 x + id x * 0). apply derivable_pt_lim_mult. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte, id in |- *; ring. + unfold fct_cte, id; ring. reflexivity. replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n). replace (pred (S n)) with n; [ idtac | reflexivity ]. @@ -1011,13 +1011,13 @@ Proof. replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)). apply derivable_pt_lim_mult. apply derivable_pt_lim_id. - unfold f in |- *; apply Hrecn; assumption. - unfold f in |- *. - pattern n at 1 5 in |- *; replace n with (S (pred n)). - unfold id in |- *; rewrite S_INR; simpl in |- *. + unfold f; apply Hrecn; assumption. + unfold f. + pattern n at 1 5; replace n with (S (pred n)). + unfold id; rewrite S_INR; simpl. ring. - symmetry in |- *; apply S_pred with 0%nat; assumption. - unfold mult_fct, id in |- *; reflexivity. + symmetry ; apply S_pred with 0%nat; assumption. + unfold mult_fct, id; reflexivity. reflexivity. inversion H. left; reflexivity. @@ -1033,7 +1033,7 @@ Lemma derivable_pt_lim_pow : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *. + simpl. rewrite Rmult_0_l. replace (fun _:R => 1) with (fct_cte 1); [ apply derivable_pt_lim_const | reflexivity ]. @@ -1044,14 +1044,14 @@ Qed. Lemma derivable_pt_pow : forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x. Proof. - intros; unfold derivable_pt in |- *. + intros; unfold derivable_pt. exists (INR n * x ^ pred n). apply derivable_pt_lim_pow. Qed. Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n). Proof. - intro; unfold derivable in |- *; intro; apply derivable_pt_pow. + intro; unfold derivable; intro; apply derivable_pt_pow. Qed. Lemma derive_pt_pow : @@ -1073,7 +1073,7 @@ Proof. elim pr2; intros. unfold derivable_pt_abs in p. unfold derivable_pt_abs in p0. - simpl in |- *. + simpl. apply (uniqueness_limite f x x0 x1 p p0). Qed. @@ -1094,7 +1094,7 @@ Proof. assert (H5 := derive_pt_eq_1 f c l pr H4). cut (0 < l / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H5 (l / 2) H6); intros delta H7. cut (0 < (b - c) / 2). @@ -1119,7 +1119,7 @@ Proof. (Rabs ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2). - unfold Rabs in |- *; + unfold Rabs; case (Rcase_abs ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / @@ -1157,7 +1157,7 @@ Proof. (Rlt_le_trans 0 ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)). - pattern l at 2 in |- *; rewrite double_var. + pattern l at 2; rewrite double_var. ring. ring. intro. @@ -1183,7 +1183,7 @@ Proof. l + - ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) / - Rmin (delta / 2) ((b + - c) / 2))) in |- *; apply Rplus_lt_le_0_compat; + Rmin (delta / 2) ((b + - c) / 2))); apply Rplus_lt_le_0_compat; [ assumption | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ]. unfold Rminus; ring. @@ -1195,13 +1195,13 @@ Proof. ((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) / Rmin (delta / 2) ((b - c) / 2))). rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; - unfold Rdiv in |- *; apply Rmult_le_pos; + unfold Rdiv; apply Rmult_le_pos; [ generalize (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2))) (f (c + Rmin (delta * / 2) ((b - c) * / 2))) ( f c) H15); rewrite Rplus_opp_r; intro; assumption | left; apply Rinv_0_lt_compat; assumption ]. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Ropp_mult_distr_l_reverse. repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))). apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)). @@ -1209,9 +1209,9 @@ Proof. rewrite <- Rinv_r_sym. repeat rewrite Rmult_1_l. ring. - red in |- *; intro. + red; intro. unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). - red in |- *; intro. + red; intro. unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)). assert @@ -1225,7 +1225,7 @@ Proof. replace (2 * b) with (b + b). apply Rplus_lt_compat_r; assumption. ring. - unfold Rdiv in |- *; rewrite Rmult_plus_distr_l. + unfold Rdiv; rewrite Rmult_plus_distr_l. repeat rewrite (Rmult_comm 2). rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r. @@ -1233,51 +1233,51 @@ Proof. discrR. apply Rlt_trans with c. assumption. - pattern c at 1 in |- *; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l; + pattern c at 1; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l; assumption. cut (0 < delta / 2). intro; apply (Rmin_stable_in_posreal (mkposreal (delta / 2) H12) (mkposreal ((b - c) / 2) H8)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. - unfold Rabs in |- *; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))). + unfold Rabs; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))). intro. cut (0 < delta / 2). intro. generalize (Rmin_stable_in_posreal (mkposreal (delta / 2) H10) - (mkposreal ((b - c) / 2) H8)); simpl in |- *; intro; + (mkposreal ((b - c) / 2) H8)); simpl; intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 r)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. intro; apply Rle_lt_trans with (delta / 2). apply Rmin_l. - unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. + unfold Rdiv; apply Rmult_lt_reg_l with 2. prove_sup0. rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. rewrite Rmult_1_l. replace (2 * delta) with (delta + delta). - pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + pattern delta at 2; rewrite <- (Rplus_0_r delta); apply Rplus_lt_compat_l. rewrite Rplus_0_r; apply (cond_pos delta). - symmetry in |- *; apply double. + symmetry ; apply double. discrR. cut (0 < delta / 2). intro; generalize (Rmin_stable_in_posreal (mkposreal (delta / 2) H9) - (mkposreal ((b - c) / 2) H8)); simpl in |- *; - intro; red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + (mkposreal ((b - c) / 2) H8)); simpl; + intro; red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro; assumption. apply Rinv_0_lt_compat; prove_sup0. elim H2; intro. - symmetry in |- *; assumption. + symmetry ; assumption. generalize (derivable_derive f c pr); intro; elim H4; intros l H5. rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro; cut (0 < - (l / 2)). @@ -1307,7 +1307,7 @@ Proof. ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / Rmax (- (delta / 2)) ((a + - c) / 2) + - l) < - (l / 2)). - unfold Rabs in |- *; + unfold Rabs; case (Rcase_abs ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / @@ -1339,12 +1339,12 @@ Proof. Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)). rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. - pattern l at 3 in |- *; rewrite double_var. + pattern l at 3; rewrite double_var. ring. assumption. apply Rplus_le_lt_0_compat; assumption. rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. - unfold Rdiv in |- *; + unfold Rdiv; replace ((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) * / Rmax (- (delta * / 2)) ((a - c) * / 2)) with @@ -1361,7 +1361,7 @@ Proof. ring. left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Ropp_inv_permute. rewrite Rmult_opp_opp. reflexivity. @@ -1380,7 +1380,7 @@ Proof. apply Rplus_lt_compat_l; assumption. field; discrR. assumption. - unfold Rabs in |- *; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))). + unfold Rabs; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))). intro; generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro; generalize (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2)) @@ -1390,10 +1390,10 @@ Proof. assumption. 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. - pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + pattern delta at 2; rewrite <- (Rplus_0_r delta); apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta). discrR. cut (- (delta / 2) < 0). @@ -1401,7 +1401,7 @@ Proof. intros; generalize (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13) - (mknegreal ((a - c) / 2) H12)); simpl in |- *; + (mknegreal ((a - c) / 2) H12)); simpl; intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 r); intro; elim @@ -1410,41 +1410,41 @@ Proof. rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). assumption. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Ropp_mult_distr_l_reverse. rewrite (Ropp_minus_distr a c). reflexivity. - rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. - red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). + red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). cut ((a - c) / 2 < 0). intro; cut (- (delta / 2) < 0). intro; apply (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11) (mknegreal ((a - c) / 2) H10)). - rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). assumption. - unfold Rdiv in |- *. + unfold Rdiv. rewrite <- Ropp_mult_distr_l_reverse. rewrite (Ropp_minus_distr a c). reflexivity. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro; assumption | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. replace (- (l / 2)) with (- l / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ]. - unfold Rdiv in |- *; apply Ropp_mult_distr_l_reverse. + unfold Rdiv; apply Ropp_mult_distr_l_reverse. Qed. Theorem deriv_minimum : @@ -1460,7 +1460,7 @@ Proof. cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c). intro. apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2). - intros; unfold opp_fct in |- *; apply Ropp_ge_le_contravar; apply Rle_ge. + intros; unfold opp_fct; apply Ropp_ge_le_contravar; apply Rle_ge. apply (H1 x H2 H3). Qed. @@ -1493,7 +1493,7 @@ Proof. intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11); cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)). intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l). - intro; unfold Rabs in |- *; + intro; unfold Rabs; case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)). intro; elim @@ -1502,7 +1502,7 @@ Proof. intros; generalize (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l) - (- (l / 2)) H13); unfold Rminus in |- *; + (- (l / 2)) H13); unfold Rminus; replace (- (l / 2) + l) with (l / 2). rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro; generalize @@ -1512,50 +1512,50 @@ Proof. rewrite <- Ropp_0 in H5; generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5); repeat rewrite Ropp_involutive; intro; assumption. - pattern l at 3 in |- *; rewrite double_var. + pattern l at 3; rewrite double_var. ring. - unfold Rminus in |- *; apply Rplus_le_le_0_compat. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rminus; apply Rplus_le_le_0_compat. + unfold Rdiv; apply Rmult_le_pos. cut (x <= x + delta * / 2). intro; generalize (H x (x + delta * / 2) H12); intro; generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13); rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. - pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; left; assumption. left; apply Rinv_0_lt_compat; assumption. left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. cut (x <= x + delta * / 2). intro; generalize (H x (x + delta * / 2) H9); intro; generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12); rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. - pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; left; assumption. left; apply Rinv_0_lt_compat; assumption. split. - unfold Rdiv in |- *; apply prod_neq_R0. - generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H7; + unfold Rdiv; apply prod_neq_R0. + generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H7; elim (Rlt_irrefl 0 H7). apply Rinv_neq_0_compat; discrR. split. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. replace (Rabs (delta / 2)) with (delta / 2). - unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. + unfold Rdiv; apply Rmult_lt_reg_l with 2. prove_sup0. rewrite (Rmult_comm 2). rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. rewrite Rmult_1_r. rewrite double. - pattern (pos delta) at 1 in |- *; rewrite <- Rplus_0_r. + pattern (pos delta) at 1; rewrite <- Rplus_0_r. apply Rplus_lt_compat_l; apply (cond_pos delta). - symmetry in |- *; apply Rabs_right. - left; change (0 < delta / 2) in |- *; unfold Rdiv in |- *; + symmetry ; apply Rabs_right. + left; change (0 < delta / 2); unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. - unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; + unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_lt_0_compat. apply Rplus_lt_reg_r with l. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption. apply Rinv_0_lt_compat; prove_sup0. Qed. diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v index 218f2a38f..971983987 100644 --- a/theories/Reals/Ranalysis2.v +++ b/theories/Reals/Ranalysis2.v @@ -24,7 +24,7 @@ Lemma formule : f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) + l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x). Proof. - intros; unfold Rdiv, Rminus, Rsqr in |- *. + intros; unfold Rdiv, Rminus, Rsqr. repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; repeat rewrite Rinv_mult_distr; try assumption. replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x)); @@ -81,10 +81,10 @@ Proof. rewrite Rabs_Rinv; [ left; exact H7 | assumption ]. apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)). apply Rmult_lt_compat_l. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. exact H8. - right; unfold Rdiv in |- *. + right; unfold Rdiv. repeat rewrite Rabs_mult. rewrite Rabs_Rinv; discrR. replace (Rabs 8) with 8. @@ -96,8 +96,8 @@ Proof. replace (Rabs eps) with eps. repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). ring. - symmetry in |- *; apply Rabs_right; left; assumption. - symmetry in |- *; apply Rabs_right; left; prove_sup. + symmetry ; apply Rabs_right; left; assumption. + symmetry ; apply Rabs_right; left; prove_sup. Qed. Lemma maj_term2 : @@ -129,11 +129,11 @@ Proof. (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))). apply Rmult_lt_compat_r. apply Rabs_pos_lt. - unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0; try assumption || discrR. - red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). + red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rinv_mult_distr; try assumption. repeat rewrite Rabs_mult. replace (Rabs 2) with 2. @@ -147,9 +147,9 @@ Proof. repeat rewrite Rabs_Rinv; try assumption. rewrite <- (Rmult_comm 2). unfold Rdiv in H8; exact H8. - symmetry in |- *; apply Rabs_right; left; prove_sup0. + symmetry ; apply Rabs_right; left; prove_sup0. right. - unfold Rsqr, Rdiv in |- *. + unfold Rsqr, Rdiv. do 1 rewrite Rinv_mult_distr; try assumption || discrR. do 1 rewrite Rinv_mult_distr; try assumption || discrR. repeat rewrite Rabs_mult. @@ -166,9 +166,9 @@ Proof. (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ]. repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR. ring. - symmetry in |- *; apply Rabs_right; left; prove_sup0. - symmetry in |- *; apply Rabs_right; left; prove_sup. - symmetry in |- *; apply Rabs_right; left; assumption. + symmetry ; apply Rabs_right; left; prove_sup0. + symmetry ; apply Rabs_right; left; prove_sup. + symmetry ; apply Rabs_right; left; assumption. Qed. Lemma maj_term3 : @@ -204,11 +204,11 @@ Proof. (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))). apply Rmult_lt_compat_r. apply Rabs_pos_lt. - unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0; try assumption. - red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). + red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rinv_mult_distr; try assumption. repeat rewrite Rabs_mult. replace (Rabs 2) with 2. @@ -222,9 +222,9 @@ Proof. repeat rewrite Rabs_Rinv; assumption || idtac. rewrite <- (Rmult_comm 2). unfold Rdiv in H9; exact H9. - symmetry in |- *; apply Rabs_right; left; prove_sup0. + symmetry ; apply Rabs_right; left; prove_sup0. right. - unfold Rsqr, Rdiv in |- *. + unfold Rsqr, Rdiv. rewrite Rinv_mult_distr; try assumption || discrR. rewrite Rinv_mult_distr; try assumption || discrR. repeat rewrite Rabs_mult. @@ -241,9 +241,9 @@ Proof. (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ]. repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). ring. - symmetry in |- *; apply Rabs_right; left; prove_sup0. - symmetry in |- *; apply Rabs_right; left; prove_sup. - symmetry in |- *; apply Rabs_right; left; assumption. + symmetry ; apply Rabs_right; left; prove_sup0. + symmetry ; apply Rabs_right; left; prove_sup. + symmetry ; apply Rabs_right; left; assumption. Qed. Lemma maj_term4 : @@ -281,17 +281,17 @@ Proof. Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). apply Rmult_lt_compat_r. apply Rabs_pos_lt. - unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0; assumption || idtac. - red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H). + red; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H). apply Rinv_neq_0_compat; apply prod_neq_R0. apply prod_neq_R0. discrR. assumption. assumption. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rinv_mult_distr; - try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption). + try assumption || (unfold Rsqr; apply prod_neq_R0; assumption). repeat rewrite Rabs_mult. replace (Rabs 2) with 2. replace @@ -305,13 +305,13 @@ Proof. repeat apply Rmult_lt_compat_l. apply Rabs_pos_lt; assumption. apply Rabs_pos_lt; assumption. - apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *; + apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr; apply prod_neq_R0; assumption. repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ]. rewrite <- (Rmult_comm 2). unfold Rdiv in H10; exact H10. - symmetry in |- *; apply Rabs_right; left; prove_sup0. - right; unfold Rsqr, Rdiv in |- *. + symmetry ; apply Rabs_right; left; prove_sup0. + right; unfold Rsqr, Rdiv. rewrite Rinv_mult_distr; try assumption || discrR. rewrite Rinv_mult_distr; try assumption || discrR. rewrite Rinv_mult_distr; try assumption || discrR. @@ -333,9 +333,9 @@ Proof. (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ]. repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). ring. - symmetry in |- *; apply Rabs_right; left; prove_sup0. - symmetry in |- *; apply Rabs_right; left; prove_sup. - symmetry in |- *; apply Rabs_right; left; assumption. + symmetry ; apply Rabs_right; left; prove_sup0. + symmetry ; apply Rabs_right; left; prove_sup. + symmetry ; apply Rabs_right; left; assumption. apply prod_neq_R0; assumption || discrR. apply prod_neq_R0; assumption. Qed. @@ -343,11 +343,11 @@ Qed. Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a). Proof. intros. - unfold D_x, no_cond in |- *. + unfold D_x, no_cond. split. trivial. apply Rminus_not_eq. - unfold Rminus in |- *. + unfold Rminus. rewrite Ropp_plus_distr. rewrite <- Rplus_assoc. rewrite Rplus_opp_r. @@ -394,7 +394,7 @@ Qed. Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4. Proof. intro; rewrite <- quadruple. - unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR. + unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR. reflexivity. Qed. @@ -413,10 +413,10 @@ Proof. cut (dist R_met (x0 + h) x0 < x -> dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)). - unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + unfold dist; simpl; unfold R_dist; replace (x0 + h - x0) with h. intros; assert (H7 := H6 H4). - red in |- *; intro. + red; intro. rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7; rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7; pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7. @@ -429,10 +429,10 @@ Proof. rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12; [ idtac | discrR ]. cut (IZR 1 < IZR 2). - unfold IZR in |- *; unfold INR, Pos.to_nat in |- *; simpl in |- *; intro; + unfold IZR; unfold INR, Pos.to_nat; simpl; intro; elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)). apply IZR_lt; omega. - unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro. + unfold Rabs; case (Rcase_abs (/ 2)); intro. assert (Hyp : 0 < 2). prove_sup0. assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11; @@ -442,18 +442,18 @@ Proof. apply (Rabs_pos_lt _ H0). ring. assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro. - intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *; - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + intro; rewrite <- H7; unfold dist, R_met; unfold R_dist; + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rabs_pos_lt. - unfold Rdiv in |- *; apply prod_neq_R0; + unfold Rdiv; apply prod_neq_R0; [ assumption | apply Rinv_neq_0_compat; discrR ]. intro; apply H5. split. - unfold D_x, no_cond in |- *. + unfold D_x, no_cond. split; trivial || assumption. assumption. - change (0 < Rabs (f x0 / 2)) in |- *. - apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0. + change (0 < Rabs (f x0 / 2)). + apply Rabs_pos_lt; unfold Rdiv; apply prod_neq_R0. assumption. apply Rinv_neq_0_compat; discrR. Qed. diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v index aa9e6bb35..82341fee3 100644 --- a/theories/Reals/Ranalysis3.v +++ b/theories/Reals/Ranalysis3.v @@ -22,17 +22,17 @@ Theorem derivable_pt_lim_div : Proof. intros f1 f2 x l1 l2 H H0 H1. cut (derivable_pt f2 x); - [ intro X | unfold derivable_pt in |- *; exists l2; exact H0 ]. + [ intro X | unfold derivable_pt; exists l2; exact H0 ]. assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1). elim H2; clear H2; intros eps_f2 H2. - unfold div_fct in |- *. + unfold div_fct. assert (H3 := derivable_continuous_pt _ _ X). unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3; unfold limit_in in H3; unfold dist in H3. simpl in H3; unfold R_dist in H3. elim (H3 (Rabs (f2 x) / 2)); [ idtac - | unfold Rdiv in |- *; change (0 < Rabs (f2 x) * / 2) in |- *; + | unfold Rdiv; change (0 < Rabs (f2 x) * / 2); apply Rmult_lt_0_compat; [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. clear H3; intros alp_f2 H3. @@ -46,12 +46,12 @@ Proof. (forall a:R, Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)). intro Maj. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. elim (H (Rabs (eps * f2 x / 8))); [ idtac - | unfold Rdiv in |- *; change (0 < Rabs (eps * f2 x * / 8)) in |- *; + | unfold Rdiv; change (0 < Rabs (eps * f2 x * / 8)); apply Rabs_pos_lt; repeat apply prod_neq_R0; - [ red in |- *; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6) + [ red; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6) | assumption | apply Rinv_neq_0_compat; discrR ] ]. intros alp_f1d H7. @@ -68,7 +68,7 @@ Proof. | elim H3; intros; assumption | apply (cond_pos alp_f1d) ] ]. exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10). - simpl in |- *; intros. + simpl; intros. assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)). assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)). assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)). @@ -80,7 +80,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -98,15 +98,15 @@ Proof. intros. apply Rlt_4; assumption. rewrite H8. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite H8. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite H9. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. @@ -114,7 +114,7 @@ Proof. try assumption || apply H2. apply H14. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. (***********************************) (* Second case *) (* (f1 x)=0 l1<>0 *) @@ -137,7 +137,7 @@ Proof. cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)). intro. exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12). - simpl in |- *. + simpl. intros. assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). @@ -152,7 +152,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -170,11 +170,11 @@ Proof. intros. apply Rlt_4; assumption. rewrite H8. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite H8. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. @@ -185,7 +185,7 @@ Proof. apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. apply H2; assumption. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. apply H2; assumption. repeat apply Rmin_pos. apply (cond_pos eps_f2). @@ -196,21 +196,21 @@ Proof. elim H10; intros. case (Req_dec a 0); intro. rewrite H14; rewrite Rplus_0_r. - unfold Rminus in |- *; rewrite Rplus_opp_r. + unfold Rminus; rewrite Rplus_opp_r. rewrite Rabs_R0. apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc. + unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc. repeat apply prod_neq_R0; try assumption. - red in |- *; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6). + red; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption. apply H13. split. apply D_x_no_cond; assumption. replace (x + a - x) with a; [ assumption | ring ]. - change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. - apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))). + apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc; repeat apply prod_neq_R0. - red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6). + red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6). assumption. assumption. apply Rinv_neq_0_compat; repeat apply prod_neq_R0; @@ -223,17 +223,17 @@ Proof. case (Req_dec l2 0); intro. elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); [ idtac - | apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + | apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc; repeat apply prod_neq_R0; [ assumption | assumption - | red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6) + | red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6) | apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption ] ]. intros alp_f2d H12. cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)). intro. exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11). - simpl in |- *. + simpl. intros. assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). @@ -248,7 +248,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -266,7 +266,7 @@ Proof. intros. apply Rlt_4; assumption. rewrite H10. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. @@ -274,14 +274,14 @@ Proof. apply H2; assumption. apply Rmin_2; assumption. rewrite H9. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac. apply H2; assumption. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. apply H2; assumption. repeat apply Rmin_pos. apply (cond_pos eps_f2). @@ -294,7 +294,7 @@ Proof. (***********************************) elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); [ idtac - | apply Rabs_pos_lt; unfold Rsqr, Rdiv in |- *; + | apply Rabs_pos_lt; unfold Rsqr, Rdiv; repeat rewrite Rinv_mult_distr; repeat apply prod_neq_R0; try assumption || discrR ]. intros alp_f2d H11. @@ -313,7 +313,7 @@ Proof. exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))) H14). - simpl in |- *; intros. + simpl; intros. assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). @@ -335,7 +335,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -361,24 +361,24 @@ Proof. apply H2; assumption. apply Rmin_2; assumption. rewrite H9. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. apply H2; assumption. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. apply H2; assumption. intros. case (Req_dec a 0); intro. rewrite H17; rewrite Rplus_0_r. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0. apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *. + unfold Rdiv, Rsqr. repeat rewrite Rinv_mult_distr; try assumption. repeat apply prod_neq_R0; try assumption. - red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6). + red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. @@ -401,19 +401,19 @@ Proof. apply (cond_pos alp_f1d). apply (cond_pos alp_f2d). elim H13; intros; assumption. - change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *. + change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). apply Rabs_pos_lt. - unfold Rsqr, Rdiv in |- *. + unfold Rsqr, Rdiv. repeat rewrite Rinv_mult_distr; try assumption || discrR. repeat apply prod_neq_R0; try assumption. - red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6). + red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; assumption. apply Rinv_neq_0_compat; assumption. apply prod_neq_R0; [ discrR | assumption ]. - red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6). + red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6). apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. apply Rinv_neq_0_compat; discrR. @@ -440,7 +440,7 @@ Proof. exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13). - simpl in |- *. + simpl. intros. cut (forall a:R, @@ -462,7 +462,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -480,7 +480,7 @@ Proof. intros. apply Rlt_4; assumption. rewrite H10. - unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. + unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. rewrite Rabs_R0; rewrite Rmult_0_l. apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. rewrite <- Rabs_mult. @@ -495,20 +495,20 @@ Proof. apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. apply H2; assumption. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. apply H2; assumption. intros. case (Req_dec a 0); intro. - rewrite H17; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite H17; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0. apply Rabs_pos_lt. - unfold Rdiv in |- *; rewrite Rinv_mult_distr; try discrR || assumption. - unfold Rsqr in |- *. + unfold Rdiv; rewrite Rinv_mult_distr; try discrR || assumption. + unfold Rsqr. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)). elim H11; intros. apply H19. split. @@ -521,20 +521,20 @@ Proof. apply (cond_pos alp_f2d). elim H11; intros; assumption. apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). - change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. + (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). + change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))). apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). (***********************************) (* Sixth case *) (* (f1 x)<>0 l1<>0 l2<>0 *) @@ -562,7 +562,7 @@ Proof. (mkposreal (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) (Rmin alp_f2c alp_f2t2)) H15). - simpl in |- *. + simpl. intros. assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). @@ -591,7 +591,7 @@ Proof. Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). - unfold Rminus in |- *. + unfold Rminus. rewrite <- (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) . @@ -624,18 +624,18 @@ Proof. apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. apply H2; assumption. apply Rmin_2; assumption. - right; symmetry in |- *; apply quadruple_var. + right; symmetry ; apply quadruple_var. apply H2; assumption. intros. case (Req_dec a 0); intro. - rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). apply prod_neq_R0; [ discrR | assumption ]. apply prod_neq_R0; [ discrR | assumption ]. assumption. @@ -646,20 +646,20 @@ Proof. replace (x + a - x) with a; [ assumption | ring ]. intros. case (Req_dec a 0); intro. - rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). discrR. assumption. elim H14; intros. apply H20. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. apply Rminus_not_eq_right. replace (x + a - x) with a; [ assumption | ring ]. @@ -671,34 +671,34 @@ Proof. apply (cond_pos alp_f2d). elim H13; intros; assumption. elim H14; intros; assumption. - change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *; apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. + change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))); apply Rabs_pos_lt. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)). - change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *; + (red; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)). + change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))); apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)). apply prod_neq_R0; [ discrR | assumption ]. apply prod_neq_R0; [ discrR | assumption ]. assumption. apply Rabs_pos_lt. - unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; + unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; [ idtac | discrR | assumption ]. repeat apply prod_neq_R0; assumption || (apply Rinv_neq_0_compat; assumption) || (apply Rinv_neq_0_compat; discrR) || - (red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)). + (red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)). intros. - unfold Rdiv in |- *. + unfold Rdiv. apply Rmult_lt_reg_l with (Rabs (f2 (x + a))). apply Rabs_pos_lt; apply H2. apply Rlt_le_trans with (Rmin eps_f2 alp_f2). @@ -739,13 +739,13 @@ Proof. unfold Rminus in H7; assumption. intros. case (Req_dec x x0); intro. - rewrite <- H5; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + rewrite <- H5; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim H3; intros. apply H7. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. assumption. assumption. @@ -756,7 +756,7 @@ Lemma derivable_pt_div : derivable_pt f1 x -> derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x. Proof. - unfold derivable_pt in |- *. + unfold derivable_pt. intros f1 f2 x X X0 H. elim X; intros. elim X0; intros. @@ -769,7 +769,7 @@ Lemma derivable_div : derivable f1 -> derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2). Proof. - unfold derivable in |- *; intros f1 f2 X X0 H x. + unfold derivable; intros f1 f2 X X0 H x. apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)). Qed. diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v index 83bc28318..2a2b1b9b8 100644 --- a/theories/Reals/Ranalysis4.v +++ b/theories/Reals/Ranalysis4.v @@ -26,12 +26,12 @@ Proof. apply derivable_pt_const. assumption. assumption. - unfold div_fct, inv_fct, fct_cte in |- *; intro X0; elim X0; intros; - unfold derivable_pt in |- *; exists x0; - unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *; + unfold div_fct, inv_fct, fct_cte; intro X0; elim X0; intros; + unfold derivable_pt; exists x0; + unfold derivable_pt_abs; unfold derivable_pt_lim; unfold derivable_pt_abs in p; unfold derivable_pt_lim in p; intros; elim (p eps H0); intros; exists x1; intros; - unfold Rdiv in H1; unfold Rdiv in |- *; rewrite <- (Rmult_1_l (/ f x)); + unfold Rdiv in H1; unfold Rdiv; rewrite <- (Rmult_1_l (/ f x)); rewrite <- (Rmult_1_l (/ f (x + h))). apply H1; assumption. Qed. @@ -41,10 +41,10 @@ Lemma pr_nu_var : forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x), f = g -> derive_pt f x pr1 = derive_pt g x pr2. Proof. - unfold derivable_pt, derive_pt in |- *; intros. + unfold derivable_pt, derive_pt; intros. elim pr1; intros. elim pr2; intros. - simpl in |- *. + simpl. rewrite H in p. apply uniqueness_limite with g x; assumption. Qed. @@ -54,17 +54,17 @@ Lemma pr_nu_var2 : forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x), (forall h:R, f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2. Proof. - unfold derivable_pt, derive_pt in |- *; intros. + unfold derivable_pt, derive_pt; intros. elim pr1; intros. elim pr2; intros. - simpl in |- *. + simpl. assert (H0 := uniqueness_step2 _ _ _ p). assert (H1 := uniqueness_step2 _ _ _ p0). cut (limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) x1 0). intro; assert (H3 := uniqueness_step1 _ _ _ _ H0 H2). assumption. - unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; - simpl in |- *; unfold R_dist in |- *; unfold limit1_in in H1; + unfold limit1_in; unfold limit_in; unfold dist; + simpl; unfold R_dist; unfold limit1_in in H1; unfold limit_in in H1; unfold dist in H1; simpl in H1; unfold R_dist in H1. intros; elim (H1 eps H2); intros. @@ -80,7 +80,7 @@ Lemma derivable_inv : forall f:R -> R, (forall x:R, f x <> 0) -> derivable f -> derivable (/ f). Proof. intros f H X. - unfold derivable in |- *; intro x. + unfold derivable; intro x. apply derivable_pt_inv. apply (H x). apply (X x). @@ -95,25 +95,25 @@ Proof. replace (derive_pt (/ f) x (derivable_pt_inv f x na pr)) with (derive_pt (fct_cte 1 / f) x (derivable_pt_div (fct_cte 1) f x (derivable_pt_const 1 x) pr na)). - rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte in |- *; - rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte; + rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus; rewrite Rplus_0_l; reflexivity. apply pr_nu_var2. - intro; unfold div_fct, fct_cte, inv_fct in |- *. - unfold Rdiv in |- *; ring. + intro; unfold div_fct, fct_cte, inv_fct. + unfold Rdiv; ring. Qed. (** Rabsolu *) Lemma Rabs_derive_1 : forall x:R, 0 < x -> derivable_pt_lim Rabs x 1. Proof. intros. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. exists (mkposreal x H); intros. rewrite (Rabs_right x). rewrite (Rabs_right (x + h)). rewrite Rplus_comm. - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r. - rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r. + rewrite Rplus_0_r; unfold Rdiv; rewrite <- Rinv_r_sym. rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0. apply H1. apply Rle_ge. @@ -131,16 +131,16 @@ Qed. Lemma Rabs_derive_2 : forall x:R, x < 0 -> derivable_pt_lim Rabs x (-1). Proof. intros. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. cut (0 < - x). intro; exists (mkposreal (- x) H1); intros. rewrite (Rabs_left x). rewrite (Rabs_left (x + h)). rewrite Rplus_comm. rewrite Ropp_plus_distr. - unfold Rminus in |- *; rewrite Ropp_involutive; rewrite Rplus_assoc; + unfold Rminus; rewrite Ropp_involutive; rewrite Rplus_assoc; rewrite Rplus_opp_l. - rewrite Rplus_0_r; unfold Rdiv in |- *. + rewrite Rplus_0_r; unfold Rdiv. rewrite Ropp_mult_distr_l_reverse. rewrite <- Rinv_r_sym. rewrite Ropp_involutive; rewrite Rplus_opp_l; rewrite Rabs_R0; apply H0. @@ -163,24 +163,24 @@ Proof. intros. case (total_order_T x 0); intro. elim s; intro. - unfold derivable_pt in |- *; exists (-1). + unfold derivable_pt; exists (-1). apply (Rabs_derive_2 x a). elim H; exact b. - unfold derivable_pt in |- *; exists 1. + unfold derivable_pt; exists 1. apply (Rabs_derive_1 x r). Qed. (** Rabsolu is continuous for all x *) Lemma Rcontinuity_abs : continuity Rabs. Proof. - unfold continuity in |- *; intro. + unfold continuity; intro. case (Req_dec x 0); intro. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros; exists eps; + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros; exists eps; split. apply H0. - intros; rewrite H; rewrite Rabs_R0; unfold Rminus in |- *; rewrite Ropp_0; + intros; rewrite H; rewrite Rabs_R0; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; elim H1; intros; rewrite H in H3; unfold Rminus in H3; rewrite Ropp_0 in H3; rewrite Rplus_0_r in H3; apply H3. @@ -192,11 +192,11 @@ Lemma continuity_finite_sum : forall (An:nat -> R) (N:nat), continuity (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N). Proof. - intros; unfold continuity in |- *; intro. + intros; unfold continuity; intro. induction N as [| N HrecN]. - simpl in |- *. + simpl. apply continuity_pt_const. - unfold constant in |- *; intros; reflexivity. + unfold constant; intros; reflexivity. replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) + (fun y:R => (An (S N) * y ^ S N)%R))%F. @@ -222,7 +222,7 @@ Proof. cut (N = 0%nat \/ (0 < N)%nat). intro; elim H0; intro. rewrite H1. - simpl in |- *. + simpl. replace (fun y:R => An 0%nat * 1 + An 1%nat * (y * 1)) with (fct_cte (An 0%nat * 1) + mult_real_fct (An 1%nat) (id * fct_cte 1))%F. replace (1 * An 1%nat * 1) with (0 + An 1%nat * (1 * fct_cte 1 x + id x * 0)). @@ -232,7 +232,7 @@ Proof. apply derivable_pt_lim_mult. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte, id in |- *; ring. + unfold fct_cte, id; ring. reflexivity. replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) + @@ -248,7 +248,7 @@ Proof. (mult_real_fct (An (S N)) (fun y:R => y ^ S N)). apply derivable_pt_lim_scal. replace (pred (S N)) with N; [ idtac | reflexivity ]. - pattern N at 3 in |- *; replace N with (pred (S N)). + pattern N at 3; replace N with (pred (S N)). apply derivable_pt_lim_pow. reflexivity. reflexivity. @@ -259,10 +259,10 @@ Proof. rewrite <- H2. replace (pred (S N)) with N; [ idtac | reflexivity ]. ring. - simpl in |- *. + simpl. apply S_pred with 0%nat; assumption. - unfold plus_fct in |- *. - simpl in |- *; reflexivity. + unfold plus_fct. + simpl; reflexivity. inversion H. left; reflexivity. right; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. @@ -278,7 +278,7 @@ Lemma derivable_pt_lim_finite_sum : Proof. intros. induction N as [| N HrecN]. - simpl in |- *. + simpl. rewrite Rmult_1_r. replace (fun _:R => An 0%nat) with (fct_cte (An 0%nat)); [ apply derivable_pt_lim_const | reflexivity ]. @@ -290,7 +290,7 @@ Lemma derivable_pt_finite_sum : derivable_pt (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x. Proof. intros. - unfold derivable_pt in |- *. + unfold derivable_pt. assert (H := derivable_pt_lim_finite_sum An x N). induction N as [| N HrecN]. exists 0; apply H. @@ -303,14 +303,14 @@ Lemma derivable_finite_sum : forall (An:nat -> R) (N:nat), derivable (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N). Proof. - intros; unfold derivable in |- *; intro; apply derivable_pt_finite_sum. + intros; unfold derivable; intro; apply derivable_pt_finite_sum. Qed. (** Regularity of hyperbolic functions *) Lemma derivable_pt_lim_cosh : forall x:R, derivable_pt_lim cosh x (sinh x). Proof. intro. - unfold cosh, sinh in |- *; unfold Rdiv in |- *. + unfold cosh, sinh; unfold Rdiv. replace (fun x0:R => (exp x0 + exp (- x0)) * / 2) with ((exp + comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ]. replace ((exp x - exp (- x)) * / 2) with @@ -324,13 +324,13 @@ Proof. apply derivable_pt_lim_id. apply derivable_pt_lim_exp. apply derivable_pt_lim_const. - unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring. + unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring. Qed. Lemma derivable_pt_lim_sinh : forall x:R, derivable_pt_lim sinh x (cosh x). Proof. intro. - unfold cosh, sinh in |- *; unfold Rdiv in |- *. + unfold cosh, sinh; unfold Rdiv. replace (fun x0:R => (exp x0 - exp (- x0)) * / 2) with ((exp - comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ]. replace ((exp x + exp (- x)) * / 2) with @@ -344,13 +344,13 @@ Proof. apply derivable_pt_lim_id. apply derivable_pt_lim_exp. apply derivable_pt_lim_const. - unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring. + unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring. Qed. Lemma derivable_pt_exp : forall x:R, derivable_pt exp x. Proof. intro. - unfold derivable_pt in |- *. + unfold derivable_pt. exists (exp x). apply derivable_pt_lim_exp. Qed. @@ -358,7 +358,7 @@ Qed. Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x. Proof. intro. - unfold derivable_pt in |- *. + unfold derivable_pt. exists (sinh x). apply derivable_pt_lim_cosh. Qed. @@ -366,24 +366,24 @@ Qed. Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x. Proof. intro. - unfold derivable_pt in |- *. + unfold derivable_pt. exists (cosh x). apply derivable_pt_lim_sinh. Qed. Lemma derivable_exp : derivable exp. Proof. - unfold derivable in |- *; apply derivable_pt_exp. + unfold derivable; apply derivable_pt_exp. Qed. Lemma derivable_cosh : derivable cosh. Proof. - unfold derivable in |- *; apply derivable_pt_cosh. + unfold derivable; apply derivable_pt_cosh. Qed. Lemma derivable_sinh : derivable sinh. Proof. - unfold derivable in |- *; apply derivable_pt_sinh. + unfold derivable; apply derivable_pt_sinh. Qed. Lemma derive_pt_exp : diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v index ee9ea921c..15db598ad 100644 --- a/theories/Reals/Rbasic_fun.v +++ b/theories/Reals/Rbasic_fun.v @@ -45,10 +45,10 @@ Qed. (*********) Lemma Rmin_Rgt_l : forall r1 r2 r, Rmin r1 r2 > r -> r1 > r /\ r2 > r. Proof. - intros r1 r2 r; unfold Rmin in |- *; case (Rle_dec r1 r2); intros. + intros r1 r2 r; unfold Rmin; case (Rle_dec r1 r2); intros. split. assumption. - unfold Rgt in |- *; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0). + unfold Rgt; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0). split. generalize (Rnot_le_lt r1 r2 n); intro; exact (Rgt_trans r1 r2 r H0 H). assumption. @@ -57,7 +57,7 @@ Qed. (*********) Lemma Rmin_Rgt_r : forall r1 r2 r, r1 > r /\ r2 > r -> Rmin r1 r2 > r. Proof. - intros; unfold Rmin in |- *; case (Rle_dec r1 r2); elim H; clear H; intros; + intros; unfold Rmin; case (Rle_dec r1 r2); elim H; clear H; intros; assumption. Qed. @@ -72,14 +72,14 @@ Qed. (*********) Lemma Rmin_l : forall x y:R, Rmin x y <= x. Proof. - intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1; + intros; unfold Rmin; case (Rle_dec x y); intro H1; [ right; reflexivity | auto with real ]. Qed. (*********) Lemma Rmin_r : forall x y:R, Rmin x y <= y. Proof. - intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1; + intros; unfold Rmin; case (Rle_dec x y); intro H1; [ assumption | auto with real ]. Qed. @@ -123,20 +123,20 @@ Qed. (*********) Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y. Proof. - intros; unfold Rmin in |- *. + intros; unfold Rmin. case (Rle_dec x y); intro; assumption. Qed. (*********) Lemma Rmin_glb : forall x y z:R, z <= x -> z <= y -> z <= Rmin x y. Proof. - intros; unfold Rmin in |- *; case (Rle_dec x y); intro; assumption. + intros; unfold Rmin; case (Rle_dec x y); intro; assumption. Qed. (*********) Lemma Rmin_glb_lt : forall x y z:R, z < x -> z < y -> z < Rmin x y. Proof. - intros; unfold Rmin in |- *; case (Rle_dec x y); intro; assumption. + intros; unfold Rmin; case (Rle_dec x y); intro; assumption. Qed. (*******************************) @@ -167,8 +167,8 @@ Qed. Lemma Rmax_Rle : forall r1 r2 r, r <= Rmax r1 r2 <-> r <= r1 \/ r <= r2. Proof. intros; split. - unfold Rmax in |- *; case (Rle_dec r1 r2); intros; auto. - intro; unfold Rmax in |- *; case (Rle_dec r1 r2); elim H; clear H; intros; + unfold Rmax; case (Rle_dec r1 r2); intros; auto. + intro; unfold Rmax; case (Rle_dec r1 r2); elim H; clear H; intros; auto. apply (Rle_trans r r1 r2); auto. generalize (Rnot_le_lt r1 r2 n); clear n; intro; unfold Rgt in H0; @@ -177,7 +177,7 @@ Qed. Lemma Rmax_comm : forall x y:R, Rmax x y = Rmax y x. Proof. - intros p q; unfold Rmax in |- *; case (Rle_dec p q); case (Rle_dec q p); auto; + intros p q; unfold Rmax; case (Rle_dec p q); case (Rle_dec q p); auto; intros H1 H2; apply Rle_antisym; auto with real. Qed. @@ -188,14 +188,14 @@ Notation RmaxSym := Rmax_comm (only parsing). (*********) Lemma Rmax_l : forall x y:R, x <= Rmax x y. Proof. - intros; unfold Rmax in |- *; case (Rle_dec x y); intro H1; + intros; unfold Rmax; case (Rle_dec x y); intro H1; [ assumption | auto with real ]. Qed. (*********) Lemma Rmax_r : forall x y:R, y <= Rmax x y. Proof. - intros; unfold Rmax in |- *; case (Rle_dec x y); intro H1; + intros; unfold Rmax; case (Rle_dec x y); intro H1; [ right; reflexivity | auto with real ]. Qed. @@ -232,7 +232,7 @@ Qed. Lemma RmaxRmult : forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q. Proof. - intros p q r H; unfold Rmax in |- *. + intros p q r H; unfold Rmax. case (Rle_dec p q); case (Rle_dec (r * p) (r * q)); auto; intros H1 H2; auto. case H; intros E1. case H1; auto with real. @@ -246,7 +246,7 @@ Qed. (*********) Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0. Proof. - intros; unfold Rmax in |- *; case (Rle_dec x y); intro; + intros; unfold Rmax; case (Rle_dec x y); intro; [ apply (cond_neg y) | apply (cond_neg x) ]. Qed. @@ -265,7 +265,7 @@ Qed. (*********) Lemma Rmax_neg : forall x y:R, x < 0 -> y < 0 -> Rmax x y < 0. Proof. - intros; unfold Rmax in |- *. + intros; unfold Rmax. case (Rle_dec x y); intro; assumption. Qed. @@ -278,7 +278,7 @@ Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}. Proof. intro; generalize (Rle_dec 0 r); intro X; elim X; intro; clear X. right; apply (Rle_ge 0 r a). - left; fold (0 > r) in |- *; apply (Rnot_le_lt 0 r b). + left; fold (0 > r); apply (Rnot_le_lt 0 r b). Qed. (*********) @@ -291,27 +291,27 @@ Definition Rabs r : R := (*********) Lemma Rabs_R0 : Rabs 0 = 0. Proof. - unfold Rabs in |- *; case (Rcase_abs 0); auto; intro. + unfold Rabs; case (Rcase_abs 0); auto; intro. generalize (Rlt_irrefl 0); intro; exfalso; auto. Qed. Lemma Rabs_R1 : Rabs 1 = 1. Proof. -unfold Rabs in |- *; case (Rcase_abs 1); auto with real. +unfold Rabs; case (Rcase_abs 1); auto with real. intros H; absurd (1 < 0); auto with real. Qed. (*********) Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs r); intro; auto. + intros; unfold Rabs; case (Rcase_abs r); intro; auto. apply Ropp_neq_0_compat; auto. Qed. (*********) Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs r); trivial; intro; + intros; unfold Rabs; case (Rcase_abs r); trivial; intro; absurd (r >= 0). exact (Rlt_not_ge r 0 H). assumption. @@ -320,7 +320,7 @@ Qed. (*********) Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs r); intro. + intros; unfold Rabs; case (Rcase_abs r); intro. absurd (r >= 0). exact (Rlt_not_ge r 0 r0). assumption. @@ -331,21 +331,21 @@ Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a. Proof. intros a H; case H; intros H1. apply Rabs_left; auto. - rewrite H1; simpl in |- *; rewrite Rabs_right; auto with real. + rewrite H1; simpl; rewrite Rabs_right; auto with real. Qed. (*********) Lemma Rabs_pos : forall x:R, 0 <= Rabs x. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs x); intro. + intros; unfold Rabs; case (Rcase_abs x); intro. generalize (Ropp_lt_gt_contravar x 0 r); intro; unfold Rgt in H; - rewrite Ropp_0 in H; unfold Rle in |- *; left; assumption. + rewrite Ropp_0 in H; unfold Rle; left; assumption. apply Rge_le; assumption. Qed. Lemma Rle_abs : forall x:R, x <= Rabs x. Proof. - intro; unfold Rabs in |- *; case (Rcase_abs x); intros; fourier. + intro; unfold Rabs; case (Rcase_abs x); intros; fourier. Qed. Definition RRle_abs := Rle_abs. @@ -353,7 +353,7 @@ Definition RRle_abs := Rle_abs. (*********) Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs x); intro; + intros; unfold Rabs; case (Rcase_abs x); intro; [ generalize (Rgt_not_le 0 x r); intro; exfalso; auto | trivial ]. Qed. @@ -368,7 +368,7 @@ Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x. Proof. intros; generalize (Rabs_pos x); intro; unfold Rle in H0; elim H0; intro; auto. - exfalso; clear H0; elim H; clear H; generalize H1; unfold Rabs in |- *; + exfalso; clear H0; elim H; clear H; generalize H1; unfold Rabs; case (Rcase_abs x); intros; auto. clear r H1; generalize (Rplus_eq_compat_l x 0 (- x) H0); rewrite (let (H1, H2) := Rplus_ne x in H1); rewrite (Rplus_opp_r x); @@ -378,7 +378,7 @@ Qed. (*********) Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x). Proof. - intros; unfold Rabs in |- *; case (Rcase_abs (x - y)); + intros; unfold Rabs; case (Rcase_abs (x - y)); case (Rcase_abs (y - x)); intros. generalize (Rminus_lt y x r); generalize (Rminus_lt x y r0); intros; generalize (Rlt_asym x y H); intro; exfalso; @@ -397,7 +397,7 @@ Qed. (*********) Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y. Proof. - intros; unfold Rabs in |- *; case (Rcase_abs (x * y)); case (Rcase_abs x); + intros; unfold Rabs; case (Rcase_abs (x * y)); case (Rcase_abs x); case (Rcase_abs y); intros; auto. generalize (Rmult_lt_gt_compat_neg_l y x 0 r r0); intro; rewrite (Rmult_0_r y) in H; generalize (Rlt_asym (x * y) 0 r1); @@ -448,7 +448,7 @@ Qed. (*********) Lemma Rabs_Rinv : forall r, r <> 0 -> Rabs (/ r) = / Rabs r. Proof. - intro; unfold Rabs in |- *; case (Rcase_abs r); case (Rcase_abs (/ r)); auto; + intro; unfold Rabs; case (Rcase_abs r); case (Rcase_abs (/ r)); auto; intros. apply Ropp_inv_permute; auto. generalize (Rinv_lt_0_compat r r1); intro; unfold Rge in r0; elim r0; intros. @@ -470,7 +470,7 @@ Proof. cut (Rabs (-1) = 1). intros; rewrite H0. ring. - unfold Rabs in |- *; case (Rcase_abs (-1)). + unfold Rabs; case (Rcase_abs (-1)). intro; ring. intro H0; generalize (Rge_le (-1) 0 H0); intros. generalize (Ropp_le_ge_contravar 0 (-1) H1). @@ -483,13 +483,13 @@ Qed. (*********) Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b. Proof. - intros a b; unfold Rabs in |- *; case (Rcase_abs (a + b)); case (Rcase_abs a); + intros a b; unfold Rabs; case (Rcase_abs (a + b)); case (Rcase_abs a); case (Rcase_abs b); intros. apply (Req_le (- (a + b)) (- a + - b)); rewrite (Ropp_plus_distr a b); reflexivity. (**) rewrite (Ropp_plus_distr a b); apply (Rplus_le_compat_l (- a) (- b) b); - unfold Rle in |- *; unfold Rge in r; elim r; intro. + unfold Rle; unfold Rge in r; elim r; intro. left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- b) 0 b H); intro; elim (Rplus_ne (- b)); intros v w; rewrite v in H0; clear v w; rewrite (Rplus_opp_l b) in H0; apply (Rlt_trans (- b) 0 b H0 H). @@ -497,7 +497,7 @@ Proof. (**) rewrite (Ropp_plus_distr a b); rewrite (Rplus_comm (- a) (- b)); rewrite (Rplus_comm a (- b)); apply (Rplus_le_compat_l (- b) (- a) a); - unfold Rle in |- *; unfold Rge in r0; elim r0; intro. + unfold Rle; unfold Rge in r0; elim r0; intro. left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- a) 0 a H); intro; elim (Rplus_ne (- a)); intros v w; rewrite v in H0; clear v w; rewrite (Rplus_opp_l a) in H0; apply (Rlt_trans (- a) 0 a H0 H). @@ -521,27 +521,27 @@ Proof. (**) rewrite (Rplus_comm a b); rewrite (Rplus_comm (- a) b); apply (Rplus_le_compat_l b a (- a)); apply (Rminus_le a (- a)); - unfold Rminus in |- *; rewrite (Ropp_involutive a); + unfold Rminus; rewrite (Ropp_involutive a); generalize (Rplus_lt_compat_l a a 0 r0); clear r r1; intro; elim (Rplus_ne a); intros v w; rewrite v in H; clear v w; generalize (Rlt_trans (a + a) a 0 H r0); intro; apply (Rlt_le (a + a) 0 H0). (**) apply (Rplus_le_compat_l a b (- b)); apply (Rminus_le b (- b)); - unfold Rminus in |- *; rewrite (Ropp_involutive b); + unfold Rminus; rewrite (Ropp_involutive b); generalize (Rplus_lt_compat_l b b 0 r); clear r0 r1; intro; elim (Rplus_ne b); intros v w; rewrite v in H; clear v w; generalize (Rlt_trans (b + b) b 0 H r); intro; apply (Rlt_le (b + b) 0 H0). (**) - unfold Rle in |- *; right; reflexivity. + unfold Rle; right; reflexivity. Qed. (*********) Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b). Proof. intros; apply (Rplus_le_reg_l (Rabs b) (Rabs a - Rabs b) (Rabs (a - b))); - unfold Rminus in |- *; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b)); + unfold Rminus; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b)); rewrite (Rplus_comm (Rabs b) (Rabs a)); rewrite (Rplus_assoc (Rabs a) (Rabs b) (- Rabs b)); rewrite (Rplus_opp_r (Rabs b)); rewrite (proj1 (Rplus_ne (Rabs a))); @@ -561,7 +561,7 @@ Proof. rewrite <- (Rabs_Ropp (Rabs a - Rabs b)); rewrite <- (Rabs_Ropp (a - b)); do 2 rewrite Ropp_minus_distr. apply H; left; assumption. - rewrite Heq; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite Heq; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rabs_pos. apply H; left; assumption. intros; replace (Rabs (Rabs a - Rabs b)) with (Rabs a - Rabs b). @@ -576,8 +576,8 @@ Qed. (*********) Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a. Proof. - unfold Rabs in |- *; intros; case (Rcase_abs x); intro. - generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt in |- *; + unfold Rabs; intros; case (Rcase_abs x); intro. + generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt; rewrite Ropp_involutive; intro; assumption. assumption. Qed. @@ -585,15 +585,15 @@ Qed. (*********) Lemma Rabs_def2 : forall x a:R, Rabs x < a -> x < a /\ - a < x. Proof. - unfold Rabs in |- *; intro x; case (Rcase_abs x); intros. - generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt in |- *; intro; + unfold Rabs; intro x; case (Rcase_abs x); intros. + generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt; intro; generalize (Rlt_trans 0 (- x) a H0 H); intro; split. apply (Rlt_trans x 0 a r H1). generalize (Ropp_lt_gt_contravar (- x) a H); rewrite (Ropp_involutive x); - unfold Rgt in |- *; trivial. + unfold Rgt; trivial. fold (a > x) in H; generalize (Rgt_ge_trans a x 0 H r); intro; - generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a) in |- *; - generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt in |- *; + generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a); + generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt; intro; split; assumption. Qed. @@ -625,7 +625,7 @@ Qed. Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Z.abs z). Proof. - intros z; case z; simpl in |- *; auto with real. + intros z; case z; simpl; auto with real. apply Rabs_right; auto with real. intros p0; apply Rabs_right; auto with real zarith. intros p0; rewrite Rabs_Ropp. diff --git a/theories/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v index 2841ac8e3..88747ee03 100644 --- a/theories/Reals/Rcomplete.v +++ b/theories/Reals/Rcomplete.v @@ -37,7 +37,7 @@ Proof. intros. exists x. rewrite <- H2 in p0. - unfold Un_cv in |- *. + unfold Un_cv. intros. unfold Un_cv in p; unfold Un_cv in p0. cut (0 < eps / 3). @@ -46,7 +46,7 @@ Proof. elim (p0 (eps / 3) H4); intros. exists (max x1 x2). intros. - unfold R_dist in |- *. + unfold R_dist. apply Rle_lt_trans with (Rabs (Un n - Vn n) + Rabs (Vn n - x)). replace (Un n - x) with (Un n - Vn n + (Vn n - x)); [ apply Rabs_triang | ring ]. @@ -54,14 +54,14 @@ Proof. do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))). apply Rplus_le_compat_l. repeat rewrite Rabs_right. - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- Vn n)); + unfold Rminus; do 2 rewrite <- (Rplus_comm (- Vn n)); apply Rplus_le_compat_l. assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). fold Vn Wn in H8. elim (H8 n); intros. assumption. apply Rle_ge. - unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n). + unfold Rminus; apply Rplus_le_reg_l with (Vn n). rewrite Rplus_0_r. replace (Vn n + (Wn n + - Vn n)) with (Wn n); [ idtac | ring ]. assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). @@ -69,7 +69,7 @@ Proof. elim (H8 n); intros. apply Rle_trans with (Un n); assumption. apply Rle_ge. - unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n). + unfold Rminus; apply Rplus_le_reg_l with (Vn n). rewrite Rplus_0_r. replace (Vn n + (Un n + - Vn n)) with (Un n); [ idtac | ring ]. assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). @@ -85,26 +85,26 @@ Proof. repeat apply Rplus_lt_compat. unfold R_dist in H5. apply H5. - unfold ge in |- *; apply le_trans with (max x1 x2). + unfold ge; apply le_trans with (max x1 x2). apply le_max_l. assumption. rewrite <- Rabs_Ropp. replace (- (x - Vn n)) with (Vn n - x); [ idtac | ring ]. unfold R_dist in H6. apply H6. - unfold ge in |- *; apply le_trans with (max x1 x2). + unfold ge; apply le_trans with (max x1 x2). apply le_max_r. assumption. unfold R_dist in H6. apply H6. - unfold ge in |- *; apply le_trans with (max x1 x2). + unfold ge; apply le_trans with (max x1 x2). apply le_max_r. assumption. right. - pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)). + pattern eps at 4; replace eps with (3 * (eps / 3)). ring. - unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR. + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. apply cond_eq. intros. @@ -130,10 +130,10 @@ Proof. repeat apply Rplus_lt_compat. rewrite <- Rabs_Ropp. replace (- (x - Wn N)) with (Wn N - x); [ apply H4 | ring ]. - unfold ge, N in |- *. + unfold ge, N. apply le_trans with (max N1 N2); apply le_max_l. - unfold Wn, Vn in |- *. - unfold sequence_majorant, sequence_minorant in |- *. + unfold Wn, Vn. + unfold sequence_majorant, sequence_minorant. assert (H7 := approx_maj (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))). @@ -169,13 +169,13 @@ Proof. [ repeat apply Rplus_lt_compat | ring ]. assumption. apply H6. - unfold ge in |- *. + unfold ge. apply le_trans with N. - unfold N in |- *; apply le_max_r. + unfold N; apply le_max_r. apply le_plus_l. - unfold ge in |- *. + unfold ge. apply le_trans with N. - unfold N in |- *; apply le_max_r. + unfold N; apply le_max_r. apply le_plus_l. rewrite <- Rabs_Ropp. replace (- (Un (N + k1)%nat - Vn N)) with (Vn N - Un (N + k1)%nat); @@ -183,14 +183,14 @@ Proof. reflexivity. reflexivity. apply H5. - unfold ge in |- *; apply le_trans with (max N1 N2). + unfold ge; apply le_trans with (max N1 N2). apply le_max_r. - unfold N in |- *; apply le_max_l. - pattern eps at 4 in |- *; replace eps with (5 * (eps / 5)). + unfold N; apply le_max_l. + pattern eps at 4; replace eps with (5 * (eps / 5)). ring. - unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. + unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. discrR. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat. prove_sup0; try apply lt_O_Sn. diff --git a/theories/Reals/Rderiv.v b/theories/Reals/Rderiv.v index 0a6d728f5..1808acea8 100644 --- a/theories/Reals/Rderiv.v +++ b/theories/Reals/Rderiv.v @@ -34,18 +34,18 @@ Lemma cont_deriv : forall (f d:R -> R) (D:R -> Prop) (x0:R), D_in f d D x0 -> continue_in f D x0. Proof. - unfold continue_in in |- *; unfold D_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; unfold Rdiv in |- *; simpl in |- *; + unfold continue_in; unfold D_in; unfold limit1_in; + unfold limit_in; unfold Rdiv; simpl; intros; elim (H eps H0); clear H; intros; elim H; clear H; intros; elim (Req_dec (d x0) 0); intro. split with (Rmin 1 x); split. elim (Rmin_Rgt 1 x 0); intros a b; apply (b (conj Rlt_0_1 H)). intros; elim H3; clear H3; intros; generalize (let (H1, H2) := Rmin_Rgt 1 x (R_dist x1 x0) in H1); - unfold Rgt in |- *; intro; elim (H5 H4); clear H5; + unfold Rgt; intro; elim (H5 H4); clear H5; intros; generalize (H1 x1 (conj H3 H6)); clear H1; intro; unfold D_x in H3; elim H3; intros. - rewrite H2 in H1; unfold R_dist in |- *; unfold R_dist in H1; + rewrite H2 in H1; unfold R_dist; unfold R_dist in H1; cut (Rabs (f x1 - f x0) < eps * Rabs (x1 - x0)). intro; unfold R_dist in H5; generalize (Rmult_lt_compat_l eps (Rabs (x1 - x0)) 1 H0 H5); @@ -68,7 +68,7 @@ Proof. intros; elim (Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) 0); intros a b; apply (b (conj H4 H3)). apply Rmult_gt_0_compat; auto. - unfold Rgt in |- *; apply Rinv_0_lt_compat; apply Rabs_pos_lt; + unfold Rgt; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply Rmult_integral_contrapositive; split. discrR. assumption. @@ -80,17 +80,17 @@ Proof. generalize (let (H1, H2) := Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) (R_dist x1 x0) in - H1); unfold Rgt in |- *; intro; elim (H5 H4); clear H5; + H1); unfold Rgt; intro; elim (H5 H4); clear H5; intros; generalize (let (H1, H2) := Rmin_Rgt (/ 2) x (R_dist x1 x0) in H1); - unfold Rgt in |- *; intro; elim (H7 H5); clear H7; + unfold Rgt; intro; elim (H7 H5); clear H7; intros; clear H4 H5; generalize (H1 x1 (conj H3 H8)); clear H1; intro; unfold D_x in H3; elim H3; intros; generalize (not_eq_sym H5); clear H5; intro H5; generalize (Rminus_eq_contra x1 x0 H5); intro; generalize H1; - pattern (d x0) at 1 in |- *; + pattern (d x0) at 1; rewrite <- (let (H1, H2) := Rmult_ne (d x0) in H2); - rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist in |- *; - unfold Rminus at 1 in |- *; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))); + rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist; + unfold Rminus at 1; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))); rewrite (Rmult_comm (/ (x1 - x0) * (x1 - x0)) (d x0)); rewrite <- (Ropp_mult_distr_l_reverse (d x0) (/ (x1 - x0) * (x1 - x0))); rewrite (Rmult_comm (- d x0) (/ (x1 - x0) * (x1 - x0))); @@ -113,7 +113,7 @@ Proof. ; generalize (Rabs_triang_inv (f x1 - f x0) ((x1 - x0) * d x0)); intro; rewrite (Rmult_comm (x1 - x0) (- d x0)); rewrite (Ropp_mult_distr_l_reverse (d x0) (x1 - x0)); - fold (f x1 - f x0 - d x0 * (x1 - x0)) in |- *; + fold (f x1 - f x0 - d x0 * (x1 - x0)); rewrite (Rmult_comm (x1 - x0) (d x0)) in H10; clear H1; intro; generalize @@ -123,7 +123,7 @@ Proof. generalize (Rplus_lt_compat_l (Rabs (d x0 * (x1 - x0))) (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) ( - Rabs (x1 - x0) * eps) H1); unfold Rminus at 2 in |- *; + Rabs (x1 - x0) * eps) H1); unfold Rminus at 2; rewrite (Rplus_comm (Rabs (f x1 - f x0)) (- Rabs (d x0 * (x1 - x0)))); rewrite <- (Rplus_assoc (Rabs (d x0 * (x1 - x0))) (- Rabs (d x0 * (x1 - x0))) @@ -162,7 +162,7 @@ Proof. (Rplus_lt_compat (Rabs (d x0 * (x1 - x0))) (eps * / 2) (Rabs (x1 - x0) * eps) (eps * / 2) H5 H3); intro; rewrite eps2 in H10; assumption. - unfold Rabs in |- *; case (Rcase_abs 2); auto. + unfold Rabs; case (Rcase_abs 2); auto. intro; cut (0 < 2). intro ; elim (Rlt_asym 0 2 H7 r). fourier. @@ -174,14 +174,14 @@ Qed. Lemma Dconst : forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0. Proof. - unfold D_in in |- *; intros; unfold limit1_in in |- *; - unfold limit_in in |- *; unfold Rdiv in |- *; intros; - simpl in |- *; split with eps; split; auto. + unfold D_in; intros; unfold limit1_in; + unfold limit_in; unfold Rdiv; intros; + simpl; split with eps; split; auto. intros; rewrite (Rminus_diag_eq y y (eq_refl y)); rewrite Rmult_0_l; - unfold R_dist in |- *; rewrite (Rminus_diag_eq 0 0 (eq_refl 0)); - unfold Rabs in |- *; case (Rcase_abs 0); intro. + unfold R_dist; rewrite (Rminus_diag_eq 0 0 (eq_refl 0)); + unfold Rabs; case (Rcase_abs 0); intro. absurd (0 < 0); auto. - red in |- *; intro; apply (Rlt_irrefl 0 H1). + red; intro; apply (Rlt_irrefl 0 H1). unfold Rgt in H0; assumption. Qed. @@ -189,15 +189,15 @@ Qed. Lemma Dx : forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0. Proof. - unfold D_in in |- *; unfold Rdiv in |- *; intros; unfold limit1_in in |- *; - unfold limit_in in |- *; intros; simpl in |- *; split with eps; + unfold D_in; unfold Rdiv; intros; unfold limit1_in; + unfold limit_in; intros; simpl; split with eps; split; auto. intros; elim H0; clear H0; intros; unfold D_x in H0; elim H0; intros; rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (not_eq_sym H3))); - unfold R_dist in |- *; rewrite (Rminus_diag_eq 1 1 (eq_refl 1)); - unfold Rabs in |- *; case (Rcase_abs 0); intro. + unfold R_dist; rewrite (Rminus_diag_eq 1 1 (eq_refl 1)); + unfold Rabs; case (Rcase_abs 0); intro. absurd (0 < 0); auto. - red in |- *; intro; apply (Rlt_irrefl 0 r). + red; intro; apply (Rlt_irrefl 0 r). unfold Rgt in H; assumption. Qed. @@ -208,12 +208,12 @@ Lemma Dadd : D_in g dg D x0 -> D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0. Proof. - unfold D_in in |- *; intros; + unfold D_in; intros; generalize (limit_plus (fun x:R => (f x - f x0) * / (x - x0)) (fun x:R => (g x - g x0) * / (x - x0)) (D_x D x0) ( - df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; intros; elim (H eps H0); + df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in; + unfold limit_in; simpl; intros; elim (H eps H0); clear H; intros; elim H; clear H; intros; split with x; split; auto; intros; generalize (H1 x1 H2); clear H1; intro; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1; @@ -233,8 +233,8 @@ Lemma Dmult : D_in g dg D x0 -> D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0. Proof. - intros; unfold D_in in |- *; generalize H H0; intros; unfold D_in in H, H0; - generalize (cont_deriv f df D x0 H1); unfold continue_in in |- *; + intros; unfold D_in; generalize H H0; intros; unfold D_in in H, H0; + generalize (cont_deriv f df D x0 H1); unfold continue_in; intro; generalize (limit_mul (fun x:R => (g x - g x0) * / (x - x0)) ( @@ -250,8 +250,8 @@ Proof. (fun x:R => (g x - g x0) * / (x - x0) * f x) ( D_x D x0) (df x0 * g x0) (dg x0 * f x0) x0 H H4); clear H4 H; intro; unfold limit1_in in H; unfold limit_in in H; - simpl in H; unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; intros; elim (H eps H0); clear H; intros; + simpl in H; unfold limit1_in; unfold limit_in; + simpl; intros; elim (H eps H0); clear H; intros; elim H; clear H; intros; split with x; split; auto; intros; generalize (H1 x1 H2); clear H1; intro; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1; @@ -268,7 +268,7 @@ Proof. ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1 = f x1 * g x1 - f x0 * g x0). intro; rewrite H3 in H1; assumption. ring. - unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + unfold limit1_in; unfold limit_in; simpl; intros; split with eps; split; auto; intros; elim (R_dist_refl (g x0) (g x0)); intros a b; rewrite (b (eq_refl (g x0))); unfold Rgt in H; assumption. @@ -281,7 +281,7 @@ Lemma Dmult_const : Proof. intros; generalize (Dmult D (fun _:R => 0) df (fun _:R => a) f x0 (Dconst D a x0) H); - unfold D_in in |- *; intros; rewrite (Rmult_0_l (f x0)) in H0; + unfold D_in; intros; rewrite (Rmult_0_l (f x0)) in H0; rewrite (let (H1, H2) := Rplus_ne (a * df x0) in H2) in H0; assumption. Qed. @@ -291,10 +291,10 @@ Lemma Dopp : forall (D:R -> Prop) (f df:R -> R) (x0:R), D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0. Proof. - intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; + intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in; + unfold limit1_in; unfold limit_in; intros; generalize (H0 eps H1); clear H0; intro; elim H0; - clear H0; intros; elim H0; clear H0; simpl in |- *; + clear H0; intros; elim H0; clear H0; simpl; intros; split with x; split; auto. intros; generalize (H2 x1 H3); clear H2; intro; rewrite Ropp_mult_distr_l_reverse in H2; @@ -313,7 +313,7 @@ Lemma Dminus : D_in g dg D x0 -> D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0. Proof. - unfold Rminus in |- *; intros; generalize (Dopp D g dg x0 H0); intro; + unfold Rminus; intros; generalize (Dopp D g dg x0 H0); intro; apply (Dadd D df (fun x:R => - dg x) f (fun x:R => - g x) x0); assumption. Qed. @@ -324,14 +324,14 @@ Lemma Dx_pow_n : D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0. Proof. simple induction n; intros. - simpl in |- *; rewrite Rmult_0_l; apply Dconst. + simpl; rewrite Rmult_0_l; apply Dconst. intros; cut (n0 = (S n0 - 1)%nat); - [ intro a; rewrite <- a; clear a | simpl in |- *; apply minus_n_O ]. + [ intro a; rewrite <- a; clear a | simpl; apply minus_n_O ]. generalize (Dmult D (fun _:R => 1) (fun x:R => INR n0 * x ^ (n0 - 1)) ( fun x:R => x) (fun x:R => x ^ n0) x0 (Dx D x0) ( - H D x0)); unfold D_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; intros; elim (H0 eps H1); + H D x0)); unfold D_in; unfold limit1_in; + unfold limit_in; simpl; intros; elim (H0 eps H1); clear H0; intros; elim H0; clear H0; intros; split with x; split; auto. intros; generalize (H2 x1 H3); clear H2 H3; intro; @@ -340,7 +340,7 @@ Proof. rewrite (Rmult_comm (INR n0) (x0 ^ (n0 - 1))) in H2; rewrite <- (Rmult_assoc x0 (x0 ^ (n0 - 1)) (INR n0)) in H2; rewrite (tech_pow_Rmult x0 (n0 - 1)) in H2; elim (Peano_dec.eq_nat_dec n0 0) ; intros cond. - rewrite cond in H2; rewrite cond; simpl in H2; simpl in |- *; + rewrite cond in H2; rewrite cond; simpl in H2; simpl; cut (1 + x0 * 1 * 0 = 1 * 1); [ intro A; rewrite A in H2; assumption | ring ]. cut (n0 <> 0%nat -> S (n0 - 1) = n0); [ intro | omega ]; @@ -355,8 +355,8 @@ Lemma Dcomp : D_in g dg Dg (f x0) -> D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0. Proof. - intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in in |- *; - unfold Rdiv in |- *; intros; + intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in; + unfold Rdiv; intros; generalize (limit_comp f (fun x:R => (g x - g (f x0)) * / (x - f x0)) ( D_x Df x0) (D_x Dg (f x0)) (f x0) (dg (f x0)) x0); @@ -376,8 +376,8 @@ Proof. (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) ( fun x:R => dg (f x0)) (D_x Df x0) (df x0) (dg (f x0)) x0 H1 (limit_free (fun x:R => dg (f x0)) (D_x Df x0) x0 x0)); - intro; unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold limit1_in in H5, H7; unfold limit_in in H5, H7; + intro; unfold limit1_in; unfold limit_in; + simpl; unfold limit1_in in H5, H7; unfold limit_in in H5, H7; simpl in H5, H7; intros; elim (H5 eps H8); elim (H7 eps H8); clear H5 H7; intros; elim H5; elim H7; clear H5 H7; intros; split with (Rmin x x1); split. @@ -405,8 +405,8 @@ Proof. in H15; rewrite (Rinv_l (f x2 - f x0) H16) in H15; rewrite (let (H1, H2) := Rmult_ne (/ (x2 - x0)) in H2) in H15; rewrite (Rmult_comm (df x0) (dg (f x0))); assumption. - clear H5 H3 H4 H2; unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold limit1_in in H1; unfold limit_in in H1; + clear H5 H3 H4 H2; unfold limit1_in; unfold limit_in; + simpl; unfold limit1_in in H1; unfold limit_in in H1; simpl in H1; intros; elim (H1 eps H2); clear H1; intros; elim H1; clear H1; intros; split with x; split; auto; intros; unfold D_x, Dgf in H4, H3; elim H4; clear H4; @@ -425,8 +425,8 @@ Proof. generalize (Dcomp D D dexpr (fun x:R => INR n * x ^ (n - 1)) expr ( fun x:R => x ^ n) x0 H (Dx_pow_n n D (expr x0))); - intro; unfold D_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; intros; unfold D_in in H0; + intro; unfold D_in; unfold limit1_in; + unfold limit_in; simpl; intros; unfold D_in in H0; unfold limit1_in in H0; unfold limit_in in H0; simpl in H0; elim (H0 eps H1); clear H0; intros; elim H0; clear H0; intros; split with x; split; intros; auto. diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v index c00917b6b..c5363be5f 100644 --- a/theories/Reals/Rgeom.v +++ b/theories/Reals/Rgeom.v @@ -20,23 +20,23 @@ Definition dist_euc (x0 y0 x1 y1:R) : R := Lemma distance_refl : forall x0 y0:R, dist_euc x0 y0 x0 y0 = 0. Proof. - intros x0 y0; unfold dist_euc in |- *; apply Rsqr_inj; + intros x0 y0; unfold dist_euc; apply Rsqr_inj; [ apply sqrt_positivity; apply Rplus_le_le_0_compat; [ apply Rle_0_sqr | apply Rle_0_sqr ] | right; reflexivity | rewrite Rsqr_0; rewrite Rsqr_sqrt; - [ unfold Rsqr in |- *; ring + [ unfold Rsqr; ring | apply Rplus_le_le_0_compat; [ apply Rle_0_sqr | apply Rle_0_sqr ] ] ]. Qed. Lemma distance_symm : forall x0 y0 x1 y1:R, dist_euc x0 y0 x1 y1 = dist_euc x1 y1 x0 y0. Proof. - intros x0 y0 x1 y1; unfold dist_euc in |- *; apply Rsqr_inj; + intros x0 y0 x1 y1; unfold dist_euc; apply Rsqr_inj; [ apply sqrt_positivity; apply Rplus_le_le_0_compat | apply sqrt_positivity; apply Rplus_le_le_0_compat | repeat rewrite Rsqr_sqrt; - [ unfold Rsqr in |- *; ring + [ unfold Rsqr; ring | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr. Qed. @@ -49,8 +49,8 @@ Lemma law_cosines : a * c * cos ac = (x0 - x1) * (x2 - x1) + (y0 - y1) * (y2 - y1) -> Rsqr b = Rsqr c + Rsqr a - 2 * (a * c * cos ac). Proof. - unfold dist_euc in |- *; intros; repeat rewrite Rsqr_sqrt; - [ rewrite H; unfold Rsqr in |- *; ring + unfold dist_euc; intros; repeat rewrite Rsqr_sqrt; + [ rewrite H; unfold Rsqr; ring | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat ]; apply Rle_0_sqr. @@ -60,7 +60,7 @@ Lemma triangle : forall x0 y0 x1 y1 x2 y2:R, dist_euc x0 y0 x1 y1 <= dist_euc x0 y0 x2 y2 + dist_euc x2 y2 x1 y1. Proof. - intros; unfold dist_euc in |- *; apply Rsqr_incr_0; + intros; unfold dist_euc; apply Rsqr_incr_0; [ rewrite Rsqr_plus; repeat rewrite Rsqr_sqrt; [ replace (Rsqr (x0 - x1)) with (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1)); @@ -112,7 +112,7 @@ Definition yt (y ty:R) : R := y + ty. Lemma translation_0 : forall x y:R, xt x 0 = x /\ yt y 0 = y. Proof. - intros x y; split; [ unfold xt in |- * | unfold yt in |- * ]; ring. + intros x y; split; [ unfold xt | unfold yt ]; ring. Qed. Lemma isometric_translation : @@ -120,7 +120,7 @@ Lemma isometric_translation : Rsqr (x1 - x2) + Rsqr (y1 - y2) = Rsqr (xt x1 tx - xt x2 tx) + Rsqr (yt y1 ty - yt y2 ty). Proof. - intros; unfold Rsqr, xt, yt in |- *; ring. + intros; unfold Rsqr, xt, yt; ring. Qed. (******************************************************************) @@ -132,13 +132,13 @@ Definition yr (x y theta:R) : R := - x * sin theta + y * cos theta. Lemma rotation_0 : forall x y:R, xr x y 0 = x /\ yr x y 0 = y. Proof. - intros x y; unfold xr, yr in |- *; split; rewrite cos_0; rewrite sin_0; ring. + intros x y; unfold xr, yr; split; rewrite cos_0; rewrite sin_0; ring. Qed. Lemma rotation_PI2 : forall x y:R, xr x y (PI / 2) = y /\ yr x y (PI / 2) = - x. Proof. - intros x y; unfold xr, yr in |- *; split; rewrite cos_PI2; rewrite sin_PI2; + intros x y; unfold xr, yr; split; rewrite cos_PI2; rewrite sin_PI2; ring. Qed. @@ -148,7 +148,7 @@ Lemma isometric_rotation_0 : Rsqr (xr x1 y1 theta - xr x2 y2 theta) + Rsqr (yr x1 y1 theta - yr x2 y2 theta). Proof. - intros; unfold xr, yr in |- *; + intros; unfold xr, yr; replace (x1 * cos theta + y1 * sin theta - (x2 * cos theta + y2 * sin theta)) with (cos theta * (x1 - x2) + sin theta * (y1 - y2)); @@ -168,7 +168,7 @@ Lemma isometric_rotation : dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta) (yr x2 y2 theta). Proof. - unfold dist_euc in |- *; intros; apply Rsqr_inj; + unfold dist_euc; intros; apply Rsqr_inj; [ apply sqrt_positivity; apply Rplus_le_le_0_compat | apply sqrt_positivity; apply Rplus_le_le_0_compat | repeat rewrite Rsqr_sqrt; diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index 9e4a5e4b1..83e059f53 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.v @@ -51,19 +51,19 @@ Lemma RiemannInt_P1 : forall (f:R -> R) (a b:R), Riemann_integrable f a b -> Riemann_integrable f b a. Proof. - unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; intros; + unfold Riemann_integrable; intros; elim (X eps); clear X; intros; elim p; clear p; intros; exists (mkStepFun (StepFun_P6 (pre x))); exists (mkStepFun (StepFun_P6 (pre x0))); elim p; clear p; intros; split. intros; apply (H t); elim H1; clear H1; intros; split; [ apply Rle_trans with (Rmin b a); try assumption; right; - unfold Rmin in |- * + unfold Rmin | apply Rle_trans with (Rmax b a); try assumption; right; - unfold Rmax in |- * ]; + unfold Rmax ]; (case (Rle_dec a b); case (Rle_dec b a); intros; try reflexivity || apply Rle_antisym; [ assumption | assumption | auto with real | auto with real ]). - generalize H0; unfold RiemannInt_SF in |- *; case (Rle_dec a b); + generalize H0; unfold RiemannInt_SF; case (Rle_dec a b); case (Rle_dec b a); intros; (replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0)))) @@ -89,11 +89,11 @@ Lemma RiemannInt_P2 : Rabs (RiemannInt_SF (wn n)) < un n) -> { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }. Proof. - intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit in |- *; + intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit; intros; assert (H3 : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist in |- *; + elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist; unfold R_dist in H4; elim (H1 n); elim (H1 m); intros; replace (RiemannInt_SF (vn n) - RiemannInt_SF (vn m)) with (RiemannInt_SF (vn n) + -1 * RiemannInt_SF (vn m)); @@ -105,15 +105,15 @@ Proof. apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 1 (wn n) (wn m)))). apply StepFun_P37; try assumption. - intros; simpl in |- *; + intros; simpl; apply Rle_trans with (Rabs (vn n x - f x) + Rabs (f x - vn m x)). replace (vn n x + -1 * vn m x) with (vn n x - f x + (f x - vn m x)); [ apply Rabs_triang | ring ]. assert (H12 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H13 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. rewrite <- H12 in H11; pattern b at 2 in H11; rewrite <- H13 in H11; rewrite Rmult_1_l; apply Rplus_le_compat. @@ -156,14 +156,14 @@ Proof. intro; elim (H0 n0); intros; split. intros; apply (H2 t); elim H4; clear H4; intros; split; [ apply Rle_trans with (Rmin b a); try assumption; right; - unfold Rmin in |- * + unfold Rmin | apply Rle_trans with (Rmax b a); try assumption; right; - unfold Rmax in |- * ]; + unfold Rmax ]; (case (Rle_dec a b); case (Rle_dec b a); intros; try reflexivity || apply Rle_antisym; [ assumption | assumption | auto with real | auto with real ]). - generalize H3; unfold RiemannInt_SF in |- *; case (Rle_dec a b); - case (Rle_dec b a); unfold wn' in |- *; intros; + generalize H3; unfold RiemannInt_SF; case (Rle_dec a b); + case (Rle_dec b a); unfold wn'; intros; (replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with @@ -178,19 +178,19 @@ Proof. rewrite Rabs_Ropp in H4; apply H4. apply H4. assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros; - exists (- x); unfold Un_cv in |- *; unfold Un_cv in p; + exists (- x); unfold Un_cv; unfold Un_cv in p; intros; elim (p _ H4); intros; exists x0; intros; - generalize (H5 _ H6); unfold R_dist, RiemannInt_SF in |- *; + generalize (H5 _ H6); unfold R_dist, RiemannInt_SF; case (Rle_dec b a); case (Rle_dec a b); intros. elim n; assumption. unfold vn' in H7; replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (vn n0)))))); - [ unfold Rminus in |- *; rewrite Ropp_involutive; rewrite <- Rabs_Ropp; + [ unfold Rminus; rewrite Ropp_involutive; rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive; apply H7 - | symmetry in |- *; apply StepFun_P17 with (fe (vn n0)) a b; + | symmetry ; apply StepFun_P17 with (fe (vn n0)) a b; [ apply StepFun_P1 | apply StepFun_P2; apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0))))) ] ]. @@ -218,9 +218,9 @@ Lemma RiemannInt_P4 : Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr1 N)) l -> Un_cv (fun N:nat => RiemannInt_SF (phi_sequence vn pr2 N)) l. Proof. - unfold Un_cv in |- *; unfold R_dist in |- *; intros f; intros; + unfold Un_cv; unfold R_dist; intros f; intros; assert (H3 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H _ H3); clear H; intros N0 H; elim (H0 _ H3); clear H0; intros N1 H0; elim (H1 _ H3); clear H1; intros N2 H1; set (N := max (max N0 N1) N2); @@ -255,7 +255,7 @@ Proof. apply StepFun_P34; assumption. apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 1 psi_un psi_vn))). - apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l; + apply StepFun_P37; try assumption; intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (phi_sequence vn pr2 n x - f x) + Rabs (f x - phi_sequence un pr1 n x)). @@ -263,10 +263,10 @@ Proof. (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x)); [ apply Rabs_triang | ring ]. assert (H10 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H11 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. rewrite (Rplus_comm (psi_un x)); apply Rplus_le_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8. @@ -279,20 +279,20 @@ Proof. apply RRle_abs. assumption. replace (pos (un n)) with (Rabs (un n - 0)); - [ apply H; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_trans with (max N0 N1); + [ apply H; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_trans with (max N0 N1); apply le_max_l - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (un n)) ]. apply Rlt_trans with (pos (vn n)). elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)). apply RRle_abs; assumption. assumption. replace (pos (vn n)) with (Rabs (vn n - 0)); - [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_trans with (max N0 N1); + [ apply H0; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_trans with (max N0 N1); [ apply le_max_r | apply le_max_l ] - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (vn n)) ]. rewrite StepFun_P39; rewrite Rabs_Ropp; apply Rle_lt_trans with @@ -311,7 +311,7 @@ Proof. (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))). apply StepFun_P37. auto with real. - intros; simpl in |- *; rewrite Rmult_1_l; + intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (phi_sequence vn pr2 n x - f x) + Rabs (f x - phi_sequence un pr1 n x)). @@ -319,10 +319,10 @@ Proof. (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x)); [ apply Rabs_triang | ring ]. assert (H10 : Rmin a b = b). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ elim n0; assumption | reflexivity ]. assert (H11 : Rmax a b = a). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ elim n0; assumption | reflexivity ]. apply Rplus_le_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8. @@ -341,10 +341,10 @@ Proof. rewrite <- Rabs_Ropp; apply RRle_abs. assumption. replace (pos (vn n)) with (Rabs (vn n - 0)); - [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_trans with (max N0 N1); + [ apply H0; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_trans with (max N0 N1); [ apply le_max_r | apply le_max_l ] - | unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + | unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (vn n)) ]. apply Rlt_trans with (pos (un n)). @@ -352,15 +352,15 @@ Proof. rewrite <- Rabs_Ropp; apply RRle_abs; assumption. assumption. replace (pos (un n)) with (Rabs (un n - 0)); - [ apply H; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_trans with (max N0 N1); + [ apply H; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_trans with (max N0 N1); apply le_max_l - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (un n)) ]. - apply H1; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_max_r. + apply H1; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_max_r. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + [ unfold Rdiv; rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -376,17 +376,17 @@ Definition RinvN (N:nat) : posreal := mkposreal _ (RinvN_pos N). Lemma RinvN_cv : Un_cv RinvN 0. Proof. - unfold Un_cv in |- *; intros; assert (H0 := archimed (/ eps)); elim H0; + unfold Un_cv; intros; assert (H0 := archimed (/ eps)); elim H0; clear H0; intros; assert (H2 : (0 <= up (/ eps))%Z). apply le_IZR; left; apply Rlt_trans with (/ eps); [ apply Rinv_0_lt_compat; assumption | assumption ]. - elim (IZN _ H2); intros; exists x; intros; unfold R_dist in |- *; - simpl in |- *; unfold Rminus in |- *; rewrite Ropp_0; + elim (IZN _ H2); intros; exists x; intros; unfold R_dist; + simpl; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; assert (H5 : 0 < INR n + 1). apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. rewrite Rabs_right; [ idtac - | left; change (0 < / (INR n + 1)) in |- *; apply Rinv_0_lt_compat; + | left; change (0 < / (INR n + 1)); apply Rinv_0_lt_compat; assumption ]; apply Rle_lt_trans with (/ (INR x + 1)). apply Rle_Rinv. apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. @@ -400,9 +400,9 @@ Proof. apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. apply Rlt_trans with (INR x); [ rewrite INR_IZR_INZ; rewrite <- H3; apply H0 - | pattern (INR x) at 1 in |- *; rewrite <- Rplus_0_r; + | pattern (INR x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1 ]. - red in |- *; intro; rewrite H6 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H6 in H; elim (Rlt_irrefl _ H). Qed. (**********) @@ -413,7 +413,7 @@ Lemma RiemannInt_P5 : forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b), RiemannInt pr1 = RiemannInt pr2. Proof. - intros; unfold RiemannInt in |- *; + intros; unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; eapply UL_sequence; @@ -431,7 +431,7 @@ Lemma maxN : Proof. intros; set (I := fun n:nat => a + INR n * del < b); assert (H0 : exists n : nat, I n). - exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; + exists 0%nat; unfold I; rewrite Rmult_0_l; rewrite Rplus_0_r; assumption. cut (Nbound I). intro; assert (H2 := Nzorn H0 H1); elim H2; intros; exists x; elim p; intros; @@ -440,27 +440,27 @@ Proof. case (total_order_T (a + INR (S x) * del) b); intro. elim s; intro. assert (H5 := H4 (S x) a0); elim (le_Sn_n _ H5). - right; symmetry in |- *; assumption. + right; symmetry ; assumption. left; apply r. assert (H1 : 0 <= (b - a) / del). - unfold Rdiv in |- *; apply Rmult_le_pos; + unfold Rdiv; apply Rmult_le_pos; [ apply Rge_le; apply Rge_minus; apply Rle_ge; left; apply H | left; apply Rinv_0_lt_compat; apply (cond_pos del) ]. elim (archimed ((b - a) / del)); intros; assert (H4 : (0 <= up ((b - a) / del))%Z). - apply le_IZR; simpl in |- *; left; apply Rle_lt_trans with ((b - a) / del); + apply le_IZR; simpl; left; apply Rle_lt_trans with ((b - a) / del); assumption. assert (H5 := IZN _ H4); elim H5; clear H5; intros N H5; - unfold Nbound in |- *; exists N; intros; unfold I in H6; + unfold Nbound; exists N; intros; unfold I in H6; apply INR_le; rewrite H5 in H2; rewrite <- INR_IZR_INZ in H2; left; apply Rle_lt_trans with ((b - a) / del); try assumption; apply Rmult_le_reg_l with (pos del); [ apply (cond_pos del) - | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ del)); + | unfold Rdiv; rewrite <- (Rmult_comm (/ del)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ rewrite Rmult_1_l; rewrite Rmult_comm; apply Rplus_le_reg_l with a; replace (a + (b - a)) with b; [ left; assumption | ring ] - | assert (H7 := cond_pos del); red in |- *; intro; rewrite H8 in H7; + | assert (H7 := cond_pos del); red; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7) ] ]. Qed. @@ -496,15 +496,15 @@ Proof. a <= x <= b -> a <= y <= b -> Rabs (x - y) < l -> Rabs (f x - f y) < eps)); assert (H1 : bound E). - unfold bound in |- *; exists (b - a); unfold is_upper_bound in |- *; intros; + unfold bound; exists (b - a); unfold is_upper_bound; intros; unfold E in H1; elim H1; clear H1; intros H1 _; elim H1; intros; assumption. assert (H2 : exists x : R, E x). assert (H2 := Heine f (fun x:R => a <= x <= b) (compact_P3 a b) H0 eps); - elim H2; intros; exists (Rmin x (b - a)); unfold E in |- *; + elim H2; intros; exists (Rmin x (b - a)); unfold E; split; [ split; - [ unfold Rmin in |- *; case (Rle_dec x (b - a)); intro; + [ unfold Rmin; case (Rle_dec x (b - a)); intro; [ apply (cond_pos x) | apply Rlt_Rminus; assumption ] | apply Rmin_r ] | intros; apply H3; try assumption; apply Rlt_le_trans with (Rmin x (b - a)); @@ -519,7 +519,7 @@ Proof. intros; apply H15; assumption. assert (H12 := not_ex_all_not _ (fun y:R => D < y /\ E y) H11); assert (H13 : is_upper_bound E D). - unfold is_upper_bound in |- *; intros; assert (H14 := H12 x1); + unfold is_upper_bound; intros; assert (H14 := H12 x1); elim (not_and_or (D < x1) (E x1) H14); intro. case (Rle_dec x1 D); intro. assumption. @@ -551,7 +551,7 @@ Proof. exists (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y); [ elim H0; elim H1; intros; rewrite b0 in H3; rewrite b0 in H5; apply Rle_antisym; apply Rle_trans with b; assumption - | rewrite H3; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + | rewrite H3; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos eps) ]. exists (mkposreal _ Rlt_0_1); intros; elim H0; intros; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H3 H4) r)). @@ -560,14 +560,14 @@ Qed. Lemma SubEqui_P1 : forall (a b:R) (del:posreal) (h:a < b), pos_Rl (SubEqui del h) 0 = a. Proof. - intros; unfold SubEqui in |- *; case (maxN del h); intros; reflexivity. + intros; unfold SubEqui; case (maxN del h); intros; reflexivity. Qed. Lemma SubEqui_P2 : forall (a b:R) (del:posreal) (h:a < b), pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))) = b. Proof. - intros; unfold SubEqui in |- *; case (maxN del h); intros; clear a0; + intros; unfold SubEqui; case (maxN del h); intros; clear a0; cut (forall (x:nat) (a:R) (del:posreal), pos_Rl (SubEquiN (S x) a b del) @@ -579,14 +579,14 @@ Proof. change (pos_Rl (SubEquiN (S n) (a0 + del0) b del0) (pred (Rlength (SubEquiN (S n) (a0 + del0) b del0))) = b) - in |- *; apply H ] ]. + ; apply H ] ]. Qed. Lemma SubEqui_P3 : forall (N:nat) (a b:R) (del:posreal), Rlength (SubEquiN N a b del) = S N. Proof. simple induction N; intros; - [ reflexivity | simpl in |- *; rewrite H; reflexivity ]. + [ reflexivity | simpl; rewrite H; reflexivity ]. Qed. Lemma SubEqui_P4 : @@ -594,36 +594,36 @@ Lemma SubEqui_P4 : (i < S N)%nat -> pos_Rl (SubEquiN (S N) a b del) i = a + INR i * del. Proof. simple induction N; - [ intros; inversion H; [ simpl in |- *; ring | elim (le_Sn_O _ H1) ] + [ intros; inversion H; [ simpl; ring | elim (le_Sn_O _ H1) ] | intros; induction i as [| i Hreci]; - [ simpl in |- *; ring + [ simpl; ring | change (pos_Rl (SubEquiN (S n) (a + del) b del) i = a + INR (S i) * del) - in |- *; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ]. + ; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ]. Qed. Lemma SubEqui_P5 : forall (a b:R) (del:posreal) (h:a < b), Rlength (SubEqui del h) = S (S (max_N del h)). Proof. - intros; unfold SubEqui in |- *; apply SubEqui_P3. + intros; unfold SubEqui; apply SubEqui_P3. Qed. Lemma SubEqui_P6 : forall (a b:R) (del:posreal) (h:a < b) (i:nat), (i < S (max_N del h))%nat -> pos_Rl (SubEqui del h) i = a + INR i * del. Proof. - intros; unfold SubEqui in |- *; apply SubEqui_P4; assumption. + intros; unfold SubEqui; apply SubEqui_P4; assumption. Qed. Lemma SubEqui_P7 : forall (a b:R) (del:posreal) (h:a < b), ordered_Rlist (SubEqui del h). Proof. - intros; unfold ordered_Rlist in |- *; intros; rewrite SubEqui_P5 in H; + intros; unfold ordered_Rlist; intros; rewrite SubEqui_P5 in H; simpl in H; inversion H. rewrite (SubEqui_P6 del h (i:=(max_N del h))). replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))). - rewrite SubEqui_P2; unfold max_N in |- *; case (maxN del h); intros; left; + rewrite SubEqui_P2; unfold max_N; case (maxN del h); intros; left; elim a0; intros; assumption. rewrite SubEqui_P5; reflexivity. apply lt_n_Sn. @@ -631,7 +631,7 @@ Proof. 3: assumption. 2: apply le_lt_n_Sm; assumption. apply Rplus_le_compat_l; rewrite S_INR; rewrite Rmult_plus_distr_r; - pattern (INR i * del) at 1 in |- *; rewrite <- Rplus_0_r; + pattern (INR i * del) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite Rmult_1_l; left; apply (cond_pos del). Qed. @@ -641,11 +641,11 @@ Lemma SubEqui_P8 : (i < Rlength (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b. Proof. intros; split. - pattern a at 1 in |- *; rewrite <- (SubEqui_P1 del h); apply RList_P5. + pattern a at 1; rewrite <- (SubEqui_P1 del h); apply RList_P5. apply SubEqui_P7. elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; apply H1; exists i; split; [ reflexivity | assumption ]. - pattern b at 2 in |- *; rewrite <- (SubEqui_P2 del h); apply RList_P7; + pattern b at 2; rewrite <- (SubEqui_P2 del h); apply RList_P7; [ apply SubEqui_P7 | elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; apply H1; exists i; split; [ reflexivity | assumption ] ]. @@ -671,42 +671,42 @@ Lemma RiemannInt_P6 : a < b -> (forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b. Proof. - intros; unfold Riemann_integrable in |- *; intro; + intros; unfold Riemann_integrable; intro; assert (H1 : 0 < eps / (2 * (b - a))). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rlt_Rminus; assumption ] ]. assert (H2 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; left; assumption ]. assert (H3 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; left; assumption ]. elim (Heine_cor2 H0 (mkposreal _ H1)); intros del H4; elim (SubEqui_P9 del f H); intros phi [H5 H6]; split with phi; split with (mkStepFun (StepFun_P4 a b (eps / (2 * (b - a))))); split. - 2: rewrite StepFun_P18; unfold Rdiv in |- *; rewrite Rinv_mult_distr. + 2: rewrite StepFun_P18; unfold Rdiv; rewrite Rinv_mult_distr. 2: do 2 rewrite Rmult_assoc; rewrite <- Rinv_l_sym. 2: rewrite Rmult_1_r; rewrite Rabs_right. 2: apply Rmult_lt_reg_l with 2. 2: prove_sup0. 2: rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. - 2: rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r; + 2: rewrite Rmult_1_l; pattern (pos eps) at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps). 2: discrR. 2: apply Rle_ge; left; apply Rmult_lt_0_compat. 2: apply (cond_pos eps). 2: apply Rinv_0_lt_compat; prove_sup0. - 2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H; + 2: apply Rminus_eq_contra; red; intro; clear H6; rewrite H7 in H; elim (Rlt_irrefl _ H). 2: discrR. - 2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H; + 2: apply Rminus_eq_contra; red; intro; clear H6; rewrite H7 in H; elim (Rlt_irrefl _ H). - intros; rewrite H2 in H7; rewrite H3 in H7; simpl in |- *; - unfold fct_cte in |- *; + intros; rewrite H2 in H7; rewrite H3 in H7; simpl; + unfold fct_cte; cut (forall t:R, a <= t <= b -> @@ -716,14 +716,14 @@ Proof. co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i)) t)). intro; elim (H8 _ H7); intro. - rewrite H9; rewrite H5; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite H9; rewrite H5; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left; assumption. elim H9; clear H9; intros I [H9 H10]; assert (H11 := H6 I H9 t H10); rewrite H11; left; apply H4. assumption. apply SubEqui_P8; apply lt_trans with (pred (Rlength (SubEqui del H))). assumption. - apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H9; + apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H9; elim (lt_n_O _ H9). unfold co_interval in H10; elim H10; clear H10; intros; rewrite Rabs_right. rewrite SubEqui_P5 in H9; simpl in H9; inversion H9. @@ -738,7 +738,7 @@ Proof. rewrite H13 in H12; rewrite SubEqui_P2 in H12; apply H12. rewrite SubEqui_P6. 2: apply lt_n_Sn. - unfold max_N in |- *; case (maxN del H); intros; elim a0; clear a0; + unfold max_N; case (maxN del H); intros; elim a0; clear a0; intros _ H13; replace (a + INR x * del + del) with (a + INR (S x) * del); [ assumption | rewrite S_INR; ring ]. apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) I); @@ -755,10 +755,10 @@ Proof. left; assumption. right; set (I := fun j:nat => a + INR j * del <= t0); assert (H1 : exists n : nat, I n). - exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8; + exists 0%nat; unfold I; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8; intros; assumption. assert (H4 : Nbound I). - unfold Nbound in |- *; exists (S (max_N del H)); intros; unfold max_N in |- *; + unfold Nbound; exists (S (max_N del H)); intros; unfold max_N; case (maxN del H); intros; elim a0; clear a0; intros _ H5; apply INR_le; apply Rmult_le_reg_l with (pos del). apply (cond_pos del). @@ -767,7 +767,7 @@ Proof. apply Rle_trans with b; try assumption; elim H8; intros; assumption. elim (Nzorn H1 H4); intros N [H5 H6]; assert (H7 : (N < S (max_N del H))%nat). - unfold max_N in |- *; case (maxN del H); intros; apply INR_lt; + unfold max_N; case (maxN del H); intros; apply INR_lt; apply Rmult_lt_reg_l with (pos del). apply (cond_pos del). apply Rplus_lt_reg_r with a; do 2 rewrite (Rmult_comm del); @@ -778,8 +778,8 @@ Proof. assumption. elim H0; assumption. exists N; split. - rewrite SubEqui_P5; simpl in |- *; assumption. - unfold co_interval in |- *; split. + rewrite SubEqui_P5; simpl; assumption. + unfold co_interval; split. rewrite SubEqui_P6. apply H5. assumption. @@ -799,13 +799,13 @@ Qed. Lemma RiemannInt_P7 : forall (f:R -> R) (a:R), Riemann_integrable f a a. Proof. - unfold Riemann_integrable in |- *; intro f; intros; + unfold Riemann_integrable; intro f; intros; split with (mkStepFun (StepFun_P4 a a (f a))); split with (mkStepFun (StepFun_P4 a a 0)); split. - intros; simpl in |- *; unfold fct_cte in |- *; replace t with a. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; right; + intros; simpl; unfold fct_cte; replace t with a. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; right; reflexivity. - generalize H; unfold Rmin, Rmax in |- *; case (Rle_dec a a); intros; elim H0; + generalize H; unfold Rmin, Rmax; case (Rle_dec a a); intros; elim H0; intros; apply Rle_antisym; assumption. rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos eps). Qed. @@ -826,9 +826,9 @@ Lemma RiemannInt_P8 : (pr2:Riemann_integrable f b a), RiemannInt pr1 = - RiemannInt pr2. Proof. intro f; intros; eapply UL_sequence. - unfold RiemannInt in |- *; case (RiemannInt_exists pr1 RinvN RinvN_cv); + unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); intros; apply u. - unfold RiemannInt in |- *; case (RiemannInt_exists pr2 RinvN RinvN_cv); + unfold RiemannInt; case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; cut (exists psi1 : nat -> StepFun a b, @@ -845,9 +845,9 @@ Proof. Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\ Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). intros; elim H; clear H; intros psi2 H; elim H0; clear H0; intros psi1 H0; - assert (H1 := RinvN_cv); unfold Un_cv in |- *; intros; + assert (H1 := RinvN_cv); unfold Un_cv; intros; assert (H3 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. unfold Un_cv in H1; elim (H1 _ H3); clear H1; intros N0 H1; unfold R_dist in H1; simpl in H1; @@ -855,10 +855,10 @@ Proof. intros; assert (H5 := H1 _ H4); replace (pos (RinvN n)) with (Rabs (/ (INR n + 1) - 0)); [ assumption - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; left; apply (cond_pos (RinvN n)) ]. clear H1; unfold Un_cv in u; elim (u _ H3); clear u; intros N1 H1; - exists (max N0 N1); intros; unfold R_dist in |- *; + exists (max N0 N1); intros; unfold R_dist; apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) + @@ -895,7 +895,7 @@ Proof. (mkStepFun (StepFun_P28 1 (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))). apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l; + intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (phi_sequence RinvN pr1 n x0 - f x0) + Rabs (f x0 - phi_sequence RinvN pr2 n x0)). @@ -903,10 +903,10 @@ Proof. (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0)); [ apply Rabs_triang | ring ]. assert (H7 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H8 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. apply Rplus_le_compat. elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9; @@ -919,7 +919,7 @@ Proof. [ apply RRle_abs | apply Rlt_trans with (pos (RinvN n)); [ assumption - | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + | apply H4; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_l | assumption ] ] ]. elim (H n); intros; rewrite <- @@ -929,7 +929,7 @@ Proof. [ rewrite <- Rabs_Ropp; apply RRle_abs | apply Rlt_trans with (pos (RinvN n)); [ assumption - | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + | apply H4; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_l | assumption ] ] ]. assert (Hyp : b <= a). auto with real. @@ -948,7 +948,7 @@ Proof. (mkStepFun (StepFun_P28 1 (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))). apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l; + intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (phi_sequence RinvN pr1 n x0 - f x0) + Rabs (f x0 - phi_sequence RinvN pr2 n x0)). @@ -956,10 +956,10 @@ Proof. (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0)); [ apply Rabs_triang | ring ]. assert (H7 : Rmin a b = b). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ elim n0; assumption | reflexivity ]. assert (H8 : Rmax a b = a). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ elim n0; assumption | reflexivity ]. apply Rplus_le_compat. elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9; @@ -976,18 +976,18 @@ Proof. [ rewrite <- Rabs_Ropp; apply RRle_abs | apply Rlt_trans with (pos (RinvN n)); [ assumption - | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + | apply H4; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_l | assumption ] ] ]. elim (H n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))); [ apply RRle_abs | apply Rlt_trans with (pos (RinvN n)); [ assumption - | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + | apply H4; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_l | assumption ] ] ]. - unfold R_dist in H1; apply H1; unfold ge in |- *; + unfold R_dist in H1; apply H1; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_r | assumption ]. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + [ unfold Rdiv; rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -1002,7 +1002,7 @@ Lemma RiemannInt_P9 : forall (f:R -> R) (a:R) (pr:Riemann_integrable f a a), RiemannInt pr = 0. Proof. intros; assert (H := RiemannInt_P8 pr pr); apply Rmult_eq_reg_l with 2; - [ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2 in |- *; + [ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2; rewrite H; apply Rplus_opp_r | discrR ]. Qed. @@ -1011,9 +1011,9 @@ Lemma Req_EM_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}. Proof. intros; elim (total_order_T r1 r2); intros; [ elim a; intro; - [ right; red in |- *; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0) + [ right; red; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0) | left; assumption ] - | right; red in |- *; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ]. + | right; red; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ]. Qed. (* L1([a,b]) is a vectorial space *) @@ -1023,16 +1023,16 @@ Lemma RiemannInt_P10 : Riemann_integrable g a b -> Riemann_integrable (fun x:R => f x + l * g x) a b. Proof. - unfold Riemann_integrable in |- *; intros f g; intros; case (Req_EM_T l 0); + unfold Riemann_integrable; intros f g; intros; case (Req_EM_T l 0); intro. elim (X eps); intros; split with x; elim p; intros; split with x0; elim p0; intros; split; try assumption; rewrite e; intros; rewrite Rmult_0_l; rewrite Rplus_0_r; apply H; assumption. assert (H : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. assert (H0 : 0 < eps / (2 * Rabs l)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rabs_pos_lt; assumption ] ]. @@ -1040,7 +1040,7 @@ Proof. split with (mkStepFun (StepFun_P28 l x x0)); elim p0; elim p; intros; split with (mkStepFun (StepFun_P28 (Rabs l) x1 x2)); elim p1; elim p2; clear p1 p2 p0 p X X0; intros; split. - intros; simpl in |- *; + intros; simpl; apply Rle_trans with (Rabs (f t - x t) + Rabs (l * (g t - x0 t))). replace (f t + l * g t - (x t + l * x0 t)) with (f t - x t + l * (g t - x0 t)); [ apply Rabs_triang | ring ]. @@ -1060,7 +1060,7 @@ Proof. [ rewrite Rmult_1_l; replace (/ Rabs l * (eps / 2)) with (eps / (2 * Rabs l)); [ apply H2 - | unfold Rdiv in |- *; rewrite Rinv_mult_distr; + | unfold Rdiv; rewrite Rinv_mult_distr; [ ring | discrR | apply Rabs_no_R0; assumption ] ] | apply Rabs_no_R0; assumption ]. Qed. @@ -1080,14 +1080,14 @@ Lemma RiemannInt_P11 : Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) l -> Un_cv (fun N:nat => RiemannInt_SF (phi2 N)) l. Proof. - unfold Un_cv in |- *; intro f; intros; intros. + unfold Un_cv; intro f; intros; intros. case (Rle_dec a b); intro Hyp. assert (H4 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H _ H4); clear H; intros N0 H. elim (H2 _ H4); clear H2; intros N1 H2. - set (N := max N0 N1); exists N; intros; unfold R_dist in |- *. + set (N := max N0 N1); exists N; intros; unfold R_dist. apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) + Rabs (RiemannInt_SF (phi1 n) - l)). @@ -1106,24 +1106,24 @@ Proof. apply StepFun_P34; assumption. apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n)))). - apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l. + apply StepFun_P37; try assumption; intros; simpl; rewrite Rmult_1_l. apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)). replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x)); [ apply Rabs_triang | ring ]. rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7. assert (H10 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H11 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H11 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat. @@ -1132,9 +1132,9 @@ Proof. apply RRle_abs. assumption. replace (pos (un n)) with (R_dist (un n) 0). - apply H; unfold ge in |- *; apply le_trans with N; try assumption. - unfold N in |- *; apply le_max_l. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + apply H; unfold ge; apply le_trans with N; try assumption. + unfold N; apply le_max_l. + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right. apply Rle_ge; left; apply (cond_pos (un n)). apply Rlt_trans with (pos (un n)). @@ -1142,24 +1142,24 @@ Proof. apply RRle_abs; assumption. assumption. replace (pos (un n)) with (R_dist (un n) 0). - apply H; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_max_l. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + apply H; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_max_l. + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (un n)). - unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N; - try assumption; unfold N in |- *; apply le_max_r. + unfold R_dist in H2; apply H2; unfold ge; apply le_trans with N; + try assumption; unfold N; apply le_max_r. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + [ unfold Rdiv; rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. assert (H4 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H _ H4); clear H; intros N0 H. elim (H2 _ H4); clear H2; intros N1 H2. - set (N := max N0 N1); exists N; intros; unfold R_dist in |- *. + set (N := max N0 N1); exists N; intros; unfold R_dist. apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) + Rabs (RiemannInt_SF (phi1 n) - l)). @@ -1189,24 +1189,24 @@ Proof. (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n))))))). apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l. + intros; simpl; rewrite Rmult_1_l. apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)). replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x)); [ apply Rabs_triang | ring ]. rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7. assert (H10 : Rmin a b = b). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ elim Hyp; assumption | reflexivity ]. assert (H11 : Rmax a b = a). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ elim Hyp; assumption | reflexivity ]. rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = b). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ elim Hyp; assumption | reflexivity ]. assert (H11 : Rmax a b = a). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ elim Hyp; assumption | reflexivity ]. rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. rewrite <- @@ -1224,9 +1224,9 @@ Proof. rewrite <- Rabs_Ropp; apply RRle_abs. assumption. replace (pos (un n)) with (R_dist (un n) 0). - apply H; unfold ge in |- *; apply le_trans with N; try assumption. - unfold N in |- *; apply le_max_l. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + apply H; unfold ge; apply le_trans with N; try assumption. + unfold N; apply le_max_l. + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right. apply Rle_ge; left; apply (cond_pos (un n)). apply Rlt_trans with (pos (un n)). @@ -1234,15 +1234,15 @@ Proof. rewrite <- Rabs_Ropp; apply RRle_abs; assumption. assumption. replace (pos (un n)) with (R_dist (un n) 0). - apply H; unfold ge in |- *; apply le_trans with N; try assumption; - unfold N in |- *; apply le_max_l. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + apply H; unfold ge; apply le_trans with N; try assumption; + unfold N; apply le_max_l. + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (un n)). - unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N; - try assumption; unfold N in |- *; apply le_max_r. + unfold R_dist in H2; apply H2; unfold ge; apply le_trans with N; + try assumption; unfold N; apply le_max_r. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + [ unfold Rdiv; rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -1255,8 +1255,8 @@ Lemma RiemannInt_P12 : a <= b -> RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2. Proof. intro f; intros; case (Req_dec l 0); intro. - pattern l at 2 in |- *; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r; - unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv); + pattern l at 2; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r; + unfold RiemannInt; case (RiemannInt_exists pr3 RinvN RinvN_cv); case (RiemannInt_exists pr1 RinvN RinvN_cv); intros; eapply UL_sequence; [ apply u0 @@ -1278,18 +1278,18 @@ Proof. [ apply H2; assumption | rewrite H0; ring ] ] | assumption ] ]. eapply UL_sequence. - unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv); + unfold RiemannInt; case (RiemannInt_exists pr3 RinvN RinvN_cv); intros; apply u. - unfold Un_cv in |- *; intros; unfold RiemannInt in |- *; + unfold Un_cv; intros; unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); - case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv in |- *; + case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv; intros; assert (H2 : 0 < eps / 5). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (u0 _ H2); clear u0; intros N0 H3; assert (H4 := RinvN_cv); unfold Un_cv in H4; elim (H4 _ H2); clear H4 H2; intros N1 H4; assert (H5 : 0 < eps / (5 * Rabs l)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rabs_pos_lt; assumption ] ]. @@ -1298,17 +1298,17 @@ Proof. unfold R_dist in H3, H4, H5, H6; set (N := max (max N0 N1) (max N2 N3)). assert (H7 : forall n:nat, (n >= N1)%nat -> RinvN n < eps / 5). intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0)); - [ unfold RinvN in |- *; apply H4; assumption - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + [ unfold RinvN; apply H4; assumption + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; left; apply (cond_pos (RinvN n)) ]. clear H4; assert (H4 := H7); clear H7; assert (H7 : forall n:nat, (n >= N3)%nat -> RinvN n < eps / (5 * Rabs l)). intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0)); - [ unfold RinvN in |- *; apply H5; assumption - | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + [ unfold RinvN; apply H5; assumption + | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; left; apply (cond_pos (RinvN n)) ]. clear H5; assert (H5 := H7); clear H7; exists N; intros; - unfold R_dist in |- *. + unfold R_dist. apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi_sequence RinvN pr3 n) - @@ -1381,10 +1381,10 @@ Proof. (RiemannInt_SF (phi_sequence RinvN pr1 n) + l * RiemannInt_SF (phi_sequence RinvN pr2 n))); [ idtac | ring ]; do 2 rewrite <- StepFun_P30; assert (H10 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. assert (H11 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. rewrite H10 in H7; rewrite H10 in H8; rewrite H10 in H9; rewrite H11 in H7; rewrite H11 in H8; rewrite H11 in H9; @@ -1404,7 +1404,7 @@ Proof. (StepFun_P28 1 (psi3 n) (mkStepFun (StepFun_P28 (Rabs l) (psi1 n) (psi2 n)))))). apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l. + intros; simpl; rewrite Rmult_1_l. apply Rle_trans with (Rabs (phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1)) + Rabs @@ -1444,16 +1444,16 @@ Proof. apply Rlt_trans with (pos (RinvN n)); [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n))); [ apply RRle_abs | elim (H9 n); intros; assumption ] - | apply H4; unfold ge in |- *; apply le_trans with N; + | apply H4; unfold ge; apply le_trans with N; [ apply le_trans with (max N0 N1); - [ apply le_max_r | unfold N in |- *; apply le_max_l ] + [ apply le_max_r | unfold N; apply le_max_l ] | assumption ] ]. apply Rlt_trans with (pos (RinvN n)); [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))); [ apply RRle_abs | elim (H7 n); intros; assumption ] - | apply H4; unfold ge in |- *; apply le_trans with N; + | apply H4; unfold ge; apply le_trans with N; [ apply le_trans with (max N0 N1); - [ apply le_max_r | unfold N in |- *; apply le_max_l ] + [ apply le_max_r | unfold N; apply le_max_l ] | assumption ] ]. apply Rmult_lt_reg_l with (/ Rabs l). apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. @@ -1462,28 +1462,28 @@ Proof. apply Rlt_trans with (pos (RinvN n)); [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))); [ apply RRle_abs | elim (H8 n); intros; assumption ] - | apply H5; unfold ge in |- *; apply le_trans with N; + | apply H5; unfold ge; apply le_trans with N; [ apply le_trans with (max N2 N3); - [ apply le_max_r | unfold N in |- *; apply le_max_r ] + [ apply le_max_r | unfold N; apply le_max_r ] | assumption ] ]. - unfold Rdiv in |- *; rewrite Rinv_mult_distr; + unfold Rdiv; rewrite Rinv_mult_distr; [ ring | discrR | apply Rabs_no_R0; assumption ]. apply Rabs_no_R0; assumption. - apply H3; unfold ge in |- *; apply le_trans with (max N0 N1); + apply H3; unfold ge; apply le_trans with (max N0 N1); [ apply le_max_l - | apply le_trans with N; [ unfold N in |- *; apply le_max_l | assumption ] ]. + | apply le_trans with N; [ unfold N; apply le_max_l | assumption ] ]. apply Rmult_lt_reg_l with (/ Rabs l). apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)). - apply H6; unfold ge in |- *; apply le_trans with (max N2 N3); + apply H6; unfold ge; apply le_trans with (max N2 N3); [ apply le_max_l - | apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ] ]. - unfold Rdiv in |- *; rewrite Rinv_mult_distr; + | apply le_trans with N; [ unfold N; apply le_max_r | assumption ] ]. + unfold Rdiv; rewrite Rinv_mult_distr; [ ring | discrR | apply Rabs_no_R0; assumption ]. apply Rabs_no_R0; assumption. apply Rmult_eq_reg_l with 5; - [ unfold Rdiv in |- *; do 2 rewrite Rmult_plus_distr_l; + [ unfold Rdiv; do 2 rewrite Rmult_plus_distr_l; do 3 rewrite (Rmult_comm 5); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -1500,11 +1500,11 @@ Proof. | assert (H : b <= a); [ auto with real | replace (RiemannInt pr3) with (- RiemannInt (RiemannInt_P1 pr3)); - [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + [ idtac | symmetry ; apply RiemannInt_P8 ]; replace (RiemannInt pr2) with (- RiemannInt (RiemannInt_P1 pr2)); - [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + [ idtac | symmetry ; apply RiemannInt_P8 ]; replace (RiemannInt pr1) with (- RiemannInt (RiemannInt_P1 pr1)); - [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + [ idtac | symmetry ; apply RiemannInt_P8 ]; rewrite (RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2) (RiemannInt_P1 pr3) H); ring ] ]. @@ -1512,11 +1512,11 @@ Qed. Lemma RiemannInt_P14 : forall a b c:R, Riemann_integrable (fct_cte c) a b. Proof. - unfold Riemann_integrable in |- *; intros; + unfold Riemann_integrable; intros; split with (mkStepFun (StepFun_P4 a b c)); split with (mkStepFun (StepFun_P4 a b 0)); split; - [ intros; simpl in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; - rewrite Rabs_R0; unfold fct_cte in |- *; right; + [ intros; simpl; unfold Rminus; rewrite Rplus_opp_r; + rewrite Rabs_R0; unfold fct_cte; right; reflexivity | rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos eps) ]. @@ -1526,11 +1526,11 @@ Lemma RiemannInt_P15 : forall (a b c:R) (pr:Riemann_integrable (fct_cte c) a b), RiemannInt pr = c * (b - a). Proof. - intros; unfold RiemannInt in |- *; case (RiemannInt_exists pr RinvN RinvN_cv); + intros; unfold RiemannInt; case (RiemannInt_exists pr RinvN RinvN_cv); intros; eapply UL_sequence. apply u. set (phi1 := fun N:nat => phi_sequence RinvN pr N); - change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a))) in |- *; + change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a))); set (f := fct_cte c); assert (H1 : @@ -1549,13 +1549,13 @@ Proof. try assumption. apply RinvN_cv. intro; split. - intros; unfold f in |- *; simpl in |- *; unfold Rminus in |- *; - rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte in |- *; + intros; unfold f; simpl; unfold Rminus; + rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte; right; reflexivity. - unfold psi2 in |- *; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; + unfold psi2; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos (RinvN n)). - unfold Un_cv in |- *; intros; split with 0%nat; intros; unfold R_dist in |- *; - unfold phi2 in |- *; rewrite StepFun_P18; unfold Rminus in |- *; + unfold Un_cv; intros; split with 0%nat; intros; unfold R_dist; + unfold phi2; rewrite StepFun_P18; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply H. Qed. @@ -1563,9 +1563,9 @@ Lemma RiemannInt_P16 : forall (f:R -> R) (a b:R), Riemann_integrable f a b -> Riemann_integrable (fun x:R => Rabs (f x)) a b. Proof. - unfold Riemann_integrable in |- *; intro f; intros; elim (X eps); clear X; + unfold Riemann_integrable; intro f; intros; elim (X eps); clear X; intros phi [psi [H H0]]; split with (mkStepFun (StepFun_P32 phi)); - split with psi; split; try assumption; intros; simpl in |- *; + split with psi; split; try assumption; intros; simpl; apply Rle_trans with (Rabs (f t - phi t)); [ apply Rabs_triang_inv2 | apply H; assumption ]. Qed. @@ -1579,9 +1579,9 @@ Proof. assert (H2 : l2 < l1). auto with real. clear n; assert (H3 : 0 < (l1 - l2) / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply Rlt_Rminus; assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist in |- *; intros; + elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist; intros; set (N := max x x0); cut (Vn N < Un N). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (H N) H4)). apply Rlt_trans with ((l1 + l2) / 2). @@ -1589,9 +1589,9 @@ Proof. replace (- l2 + (l1 + l2) / 2) with ((l1 - l2) / 2). rewrite Rplus_comm; apply Rle_lt_trans with (Rabs (Vn N - l2)). apply RRle_abs. - apply H1; unfold ge in |- *; unfold N in |- *; apply le_max_r. + apply H1; unfold ge; unfold N; apply le_max_r. apply Rmult_eq_reg_l with 2; - [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2); + [ unfold Rdiv; do 2 rewrite (Rmult_comm 2); rewrite (Rmult_plus_distr_r (- l2) ((l1 + l2) * / 2) 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] @@ -1600,9 +1600,9 @@ Proof. replace (l1 + - ((l1 + l2) / 2)) with ((l1 - l2) / 2). apply Rle_lt_trans with (Rabs (Un N - l1)). rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. - apply H0; unfold ge in |- *; unfold N in |- *; apply le_max_l. + apply H0; unfold ge; unfold N; apply le_max_l. apply Rmult_eq_reg_l with 2; - [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2); + [ unfold Rdiv; do 2 rewrite (Rmult_comm 2); rewrite (Rmult_plus_distr_r l1 (- ((l1 + l2) * / 2)) 2); rewrite <- Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] @@ -1614,7 +1614,7 @@ Lemma RiemannInt_P17 : (pr2:Riemann_integrable (fun x:R => Rabs (f x)) a b), a <= b -> Rabs (RiemannInt pr1) <= RiemannInt pr2. Proof. - intro f; intros; unfold RiemannInt in |- *; + intro f; intros; unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; set (phi1 := phi_sequence RinvN pr1) in u0; @@ -1622,7 +1622,7 @@ Proof. apply Rle_cv_lim with (fun N:nat => Rabs (RiemannInt_SF (phi1 N))) (fun N:nat => RiemannInt_SF (phi2 N)). - intro; unfold phi2 in |- *; apply StepFun_P34; assumption. + intro; unfold phi2; apply StepFun_P34; assumption. apply (continuity_seq Rabs (fun N:nat => RiemannInt_SF (phi1 N)) x0); try assumption. apply Rcontinuity_abs. @@ -1656,7 +1656,7 @@ Proof. apply (proj2_sig (phi_sequence_prop RinvN pr1 n)). elim H1; clear H1; intros psi2 H1; split with psi2; intros; elim (H1 n); clear H1; intros; split; try assumption. - intros; unfold phi2 in |- *; simpl in |- *; + intros; unfold phi2; simpl; apply Rle_trans with (Rabs (f t - phi1 n t)). apply Rabs_triang_inv2. apply H1; assumption. @@ -1671,13 +1671,13 @@ Lemma RiemannInt_P18 : a <= b -> (forall x:R, a < x < b -> f x = g x) -> RiemannInt pr1 = RiemannInt pr2. Proof. - intro f; intros; unfold RiemannInt in |- *; + intro f; intros; unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; eapply UL_sequence. apply u0. set (phi1 := fun N:nat => phi_sequence RinvN pr1 N); - change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x) in |- *; + change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x); assert (H1 : exists psi1 : nat -> StepFun a b, @@ -1717,45 +1717,45 @@ Proof. try assumption. apply RinvN_cv. intro; elim (H2 n); intros; split; try assumption. - intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + intros; unfold phi2_m; simpl; unfold phi2_aux; case (Req_EM_T t a); case (Req_EM_T t b); intros. - rewrite e0; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite e0; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rle_trans with (Rabs (g t - phi2 n t)). apply Rabs_pos. - pattern a at 3 in |- *; rewrite <- e0; apply H3; assumption. - rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + pattern a at 3; rewrite <- e0; apply H3; assumption. + rewrite e; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rle_trans with (Rabs (g t - phi2 n t)). apply Rabs_pos. - pattern a at 3 in |- *; rewrite <- e; apply H3; assumption. - rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + pattern a at 3; rewrite <- e; apply H3; assumption. + rewrite e; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply Rle_trans with (Rabs (g t - phi2 n t)). apply Rabs_pos. - pattern b at 3 in |- *; rewrite <- e; apply H3; assumption. + pattern b at 3; rewrite <- e; apply H3; assumption. replace (f t) with (g t). apply H3; assumption. - symmetry in |- *; apply H0; elim H5; clear H5; intros. + symmetry ; apply H0; elim H5; clear H5; intros. assert (H7 : Rmin a b = a). - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n2; assumption ]. assert (H8 : Rmax a b = b). - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n2; assumption ]. rewrite H7 in H5; rewrite H8 in H6; split. - elim H5; intro; [ assumption | elim n1; symmetry in |- *; assumption ]. + elim H5; intro; [ assumption | elim n1; symmetry ; assumption ]. elim H6; intro; [ assumption | elim n0; assumption ]. cut (forall N:nat, RiemannInt_SF (phi2_m N) = RiemannInt_SF (phi2 N)). - intro; unfold Un_cv in |- *; intros; elim (u _ H4); intros; exists x1; intros; + intro; unfold Un_cv; intros; elim (u _ H4); intros; exists x1; intros; rewrite (H3 n); apply H5; assumption. intro; apply Rle_antisym. apply StepFun_P37; try assumption. - intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + intros; unfold phi2_m; simpl; unfold phi2_aux; case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros. elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4). elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4). elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5). right; reflexivity. apply StepFun_P37; try assumption. - intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + intros; unfold phi2_m; simpl; unfold phi2_aux; case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros. elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4). elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4). @@ -1764,10 +1764,10 @@ Proof. intro; assert (H2 := pre (phi2 N)); unfold IsStepFun in H2; unfold is_subdivision in H2; elim H2; clear H2; intros l [lf H2]; split with l; split with lf; unfold adapted_couple in H2; - decompose [and] H2; clear H2; unfold adapted_couple in |- *; + decompose [and] H2; clear H2; unfold adapted_couple; repeat split; try assumption. intros; assert (H9 := H8 i H2); unfold constant_D_eq, open_interval in H9; - unfold constant_D_eq, open_interval in |- *; intros; + unfold constant_D_eq, open_interval; intros; rewrite <- (H9 x1 H7); assert (H10 : a <= pos_Rl l i). replace a with (Rmin a b). rewrite <- H5; elim (RList_P6 l); intros; apply H10. @@ -1775,7 +1775,7 @@ Proof. apply le_O_n. apply lt_trans with (pred (Rlength l)); [ assumption | apply lt_pred_n_n ]. apply neq_O_lt; intro; rewrite <- H12 in H6; discriminate. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. assert (H11 : pos_Rl l (S i) <= b). replace b with (Rmax a b). @@ -1783,9 +1783,9 @@ Proof. assumption. apply lt_le_S; assumption. apply lt_pred_n_n; apply neq_O_lt; intro; rewrite <- H13 in H6; discriminate. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. - elim H7; clear H7; intros; unfold phi2_aux in |- *; case (Req_EM_T x1 a); + elim H7; clear H7; intros; unfold phi2_aux; case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros. rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)). rewrite e in H7; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H7)). @@ -1852,12 +1852,12 @@ Proof. intros; replace (primitive h pr a) with 0. replace (RiemannInt pr0) with (primitive h pr b). ring. - unfold primitive in |- *; case (Rle_dec a b); case (Rle_dec b b); intros; + unfold primitive; case (Rle_dec a b); case (Rle_dec b b); intros; [ apply RiemannInt_P5 | elim n; right; reflexivity | elim n; assumption | elim n0; assumption ]. - symmetry in |- *; unfold primitive in |- *; case (Rle_dec a a); + symmetry ; unfold primitive; case (Rle_dec a a); case (Rle_dec a b); intros; [ apply RiemannInt_P9 | elim n; assumption @@ -1872,9 +1872,9 @@ Lemma RiemannInt_P21 : Riemann_integrable f a b -> Riemann_integrable f b c -> Riemann_integrable f a c. Proof. - unfold Riemann_integrable in |- *; intros f a b c Hyp1 Hyp2 X X0 eps. + unfold Riemann_integrable; intros f a b c Hyp1 Hyp2 X X0 eps. assert (H : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. elim (X (mkposreal _ H)); clear X; intros phi1 [psi1 H1]; elim (X0 (mkposreal _ H)); clear X0; intros phi2 [psi2 H2]. @@ -1904,35 +1904,35 @@ Proof. intro; cut (IsStepFun psi3 a b). intro; cut (IsStepFun psi3 b c). intro; cut (IsStepFun psi3 a c). - intro; split with (mkStepFun X); split with (mkStepFun X2); simpl in |- *; + intro; split with (mkStepFun X); split with (mkStepFun X2); simpl; split. - intros; unfold phi3, psi3 in |- *; case (Rle_dec t b); case (Rle_dec a t); + intros; unfold phi3, psi3; case (Rle_dec t b); case (Rle_dec a t); intros. elim H1; intros; apply H3. replace (Rmin a b) with a. replace (Rmax a b) with b. split; assumption. - 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 ]. elim n; replace a with (Rmin a c). elim H0; intros; assumption. - unfold Rmin in |- *; case (Rle_dec a c); intro; + unfold Rmin; case (Rle_dec a c); intro; [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. elim H2; intros; apply H3. replace (Rmax b c) with (Rmax a c). elim H0; intros; split; try assumption. replace (Rmin b c) with b. auto with real. - unfold Rmin in |- *; case (Rle_dec b c); intro; + unfold Rmin; case (Rle_dec b c); intro; [ reflexivity | elim n0; assumption ]. - unfold Rmax in |- *; case (Rle_dec a c); case (Rle_dec b c); intros; + unfold Rmax; case (Rle_dec a c); case (Rle_dec b c); intros; try (elim n0; assumption || elim n0; apply Rle_trans with b; assumption). reflexivity. elim n; replace a with (Rmin a c). elim H0; intros; assumption. - unfold Rmin in |- *; case (Rle_dec a c); intro; + unfold Rmin; case (Rle_dec a c); intro; [ reflexivity | elim n1; apply Rle_trans with b; assumption ]. rewrite <- (StepFun_P43 X0 X1 X2). apply Rle_lt_trans with @@ -1946,14 +1946,14 @@ Proof. elim H2; intros; assumption. apply Rle_antisym. apply StepFun_P37; try assumption. - simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + simpl; intros; unfold psi3; elim H0; clear H0; intros; case (Rle_dec a x); case (Rle_dec x b); intros; [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0)) | right; reflexivity | elim n; apply Rle_trans with b; [ assumption | left; assumption ] | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. apply StepFun_P37; try assumption. - simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + simpl; intros; unfold psi3; elim H0; clear H0; intros; case (Rle_dec a x); case (Rle_dec x b); intros; [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0)) | right; reflexivity @@ -1961,14 +1961,14 @@ Proof. | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. apply Rle_antisym. apply StepFun_P37; try assumption. - simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + simpl; intros; unfold psi3; elim H0; clear H0; intros; case (Rle_dec a x); case (Rle_dec x b); intros; [ right; reflexivity | elim n; left; assumption | elim n; left; assumption | elim n0; left; assumption ]. apply StepFun_P37; try assumption. - simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + simpl; intros; unfold psi3; elim H0; clear H0; intros; case (Rle_dec a x); case (Rle_dec x b); intros; [ right; reflexivity | elim n; left; assumption @@ -1978,19 +1978,19 @@ Proof. assert (H3 := pre psi2); unfold IsStepFun in H3; unfold is_subdivision in H3; elim H3; clear H3; intros l1 [lf1 H3]; split with l1; split with lf1; unfold adapted_couple in H3; decompose [and] H3; - clear H3; unfold adapted_couple in |- *; repeat split; + clear H3; unfold adapted_couple; repeat split; try assumption. - intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval; unfold constant_D_eq, open_interval in H9; intros; - rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x). + rewrite <- (H9 x H7); unfold psi3; assert (H10 : b < x). apply Rle_lt_trans with (pos_Rl l1 i). replace b with (Rmin b c). rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. apply le_O_n. apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; - apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. - unfold Rmin in |- *; case (Rle_dec b c); intro; + unfold Rmin; case (Rle_dec b c); intro; [ reflexivity | elim n; assumption ]. elim H7; intros; assumption. case (Rle_dec a x); case (Rle_dec x b); intros; @@ -2001,18 +2001,18 @@ Proof. assert (H3 := pre psi1); unfold IsStepFun in H3; unfold is_subdivision in H3; elim H3; clear H3; intros l1 [lf1 H3]; split with l1; split with lf1; unfold adapted_couple in H3; decompose [and] H3; - clear H3; unfold adapted_couple in |- *; repeat split; + clear H3; unfold adapted_couple; repeat split; try assumption. - intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval; unfold constant_D_eq, open_interval in H9; intros; - rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b). + rewrite <- (H9 x H7); unfold psi3; assert (H10 : x <= b). apply Rle_trans with (pos_Rl l1 (S i)). elim H7; intros; left; assumption. replace b with (Rmax a b). rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption. - apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. assert (H11 : a <= x). apply Rle_trans with (pos_Rl l1 i). @@ -2020,9 +2020,9 @@ Proof. rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. apply le_O_n. apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; - apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6; + apply neq_O_lt; red; intro; rewrite <- H13 in H6; discriminate. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. left; elim H7; intros; assumption. case (Rle_dec a x); case (Rle_dec x b); intros; reflexivity || elim n; @@ -2031,18 +2031,18 @@ Proof. assert (H3 := pre phi1); unfold IsStepFun in H3; unfold is_subdivision in H3; elim H3; clear H3; intros l1 [lf1 H3]; split with l1; split with lf1; unfold adapted_couple in H3; decompose [and] H3; - clear H3; unfold adapted_couple in |- *; repeat split; + clear H3; unfold adapted_couple; repeat split; try assumption. - intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval; unfold constant_D_eq, open_interval in H9; intros; - rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b). + rewrite <- (H9 x H7); unfold psi3; assert (H10 : x <= b). apply Rle_trans with (pos_Rl l1 (S i)). elim H7; intros; left; assumption. replace b with (Rmax a b). rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption. - apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. assert (H11 : a <= x). apply Rle_trans with (pos_Rl l1 i). @@ -2050,32 +2050,32 @@ Proof. rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. apply le_O_n. apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; - apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6; + apply neq_O_lt; red; intro; rewrite <- H13 in H6; discriminate. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; assumption ]. left; elim H7; intros; assumption. - unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros; + unfold phi3; case (Rle_dec a x); case (Rle_dec x b); intros; reflexivity || elim n; assumption. assert (H3 := pre phi2); unfold IsStepFun in H3; unfold is_subdivision in H3; elim H3; clear H3; intros l1 [lf1 H3]; split with l1; split with lf1; unfold adapted_couple in H3; decompose [and] H3; - clear H3; unfold adapted_couple in |- *; repeat split; + clear H3; unfold adapted_couple; repeat split; try assumption. - intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval; unfold constant_D_eq, open_interval in H9; intros; - rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x). + rewrite <- (H9 x H7); unfold psi3; assert (H10 : b < x). apply Rle_lt_trans with (pos_Rl l1 i). replace b with (Rmin b c). rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. apply le_O_n. apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; - apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + apply neq_O_lt; red; intro; rewrite <- H12 in H6; discriminate. - unfold Rmin in |- *; case (Rle_dec b c); intro; + unfold Rmin; case (Rle_dec b c); intro; [ reflexivity | elim n; assumption ]. elim H7; intros; assumption. - unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros; + unfold phi3; case (Rle_dec a x); case (Rle_dec x b); intros; [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)) | reflexivity | elim n; apply Rle_trans with b; [ assumption | left; assumption ] @@ -2086,7 +2086,7 @@ Lemma RiemannInt_P22 : forall (f:R -> R) (a b c:R), Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f a c. Proof. - unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; + unfold Riemann_integrable; intros; elim (X eps); clear X; intros phi [psi H0]; elim H; elim H0; clear H H0; intros; assert (H3 : IsStepFun phi a c). apply StepFun_P44 with b. @@ -2097,18 +2097,18 @@ Proof. apply (pre psi). split; assumption. split with (mkStepFun H3); split with (mkStepFun H4); split. - simpl in |- *; intros; apply H. + simpl; intros; apply H. replace (Rmin a b) with (Rmin a c). elim H5; intros; split; try assumption. apply Rle_trans with (Rmax a c); try assumption. replace (Rmax a b) with b. replace (Rmax a c) with c. assumption. - unfold Rmax in |- *; case (Rle_dec a c); intro; + unfold Rmax; case (Rle_dec a c); intro; [ reflexivity | elim n; assumption ]. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. - unfold Rmin in |- *; case (Rle_dec a c); case (Rle_dec a b); intros; + unfold Rmin; case (Rle_dec a c); case (Rle_dec a b); intros; [ reflexivity | elim n; apply Rle_trans with c; assumption | elim n; assumption @@ -2121,12 +2121,12 @@ Proof. replace (RiemannInt_SF (mkStepFun H4)) with (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)). apply Rle_lt_trans with (RiemannInt_SF psi). - unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *; + unfold Rminus; pattern (RiemannInt_SF psi) at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))). apply StepFun_P37; try assumption. - intros; simpl in |- *; unfold fct_cte in |- *; + intros; simpl; unfold fct_cte; apply Rle_trans with (Rabs (f x - phi x)). apply Rabs_pos. apply H. @@ -2135,9 +2135,9 @@ Proof. elim H6; intros; split; left. apply Rle_lt_trans with c; assumption. assumption. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. rewrite StepFun_P18; ring. apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)). @@ -2147,16 +2147,16 @@ Proof. apply (pre psi). replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). rewrite <- (StepFun_P43 H4 H5 H6); ring. - unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. + unfold RiemannInt_SF; case (Rle_dec a b); intro. eapply StepFun_P17. apply StepFun_P1. - simpl in |- *; apply StepFun_P1. + simpl; apply StepFun_P1. apply Ropp_eq_compat; eapply StepFun_P17. apply StepFun_P1. - simpl in |- *; apply StepFun_P1. + simpl; apply StepFun_P1. apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))). apply StepFun_P37; try assumption. - intros; simpl in |- *; unfold fct_cte in |- *; + intros; simpl; unfold fct_cte; apply Rle_trans with (Rabs (f x - phi x)). apply Rabs_pos. apply H. @@ -2165,9 +2165,9 @@ Proof. elim H5; intros; split; left. assumption. apply Rlt_le_trans with c; assumption. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. rewrite StepFun_P18; ring. Qed. @@ -2176,7 +2176,7 @@ Lemma RiemannInt_P23 : forall (f:R -> R) (a b c:R), Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f c b. Proof. - unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; + unfold Riemann_integrable; intros; elim (X eps); clear X; intros phi [psi H0]; elim H; elim H0; clear H H0; intros; assert (H3 : IsStepFun phi c b). apply StepFun_P45 with a. @@ -2187,18 +2187,18 @@ Proof. apply (pre psi). split; assumption. split with (mkStepFun H3); split with (mkStepFun H4); split. - simpl in |- *; intros; apply H. + simpl; intros; apply H. replace (Rmax a b) with (Rmax c b). elim H5; intros; split; try assumption. apply Rle_trans with (Rmin c b); try assumption. replace (Rmin a b) with a. replace (Rmin c b) with c. assumption. - unfold Rmin in |- *; case (Rle_dec c b); intro; + unfold Rmin; case (Rle_dec c 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; apply Rle_trans with c; assumption ]. - unfold Rmax in |- *; case (Rle_dec c b); case (Rle_dec a b); intros; + unfold Rmax; case (Rle_dec c b); case (Rle_dec a b); intros; [ reflexivity | elim n; apply Rle_trans with c; assumption | elim n; assumption @@ -2211,12 +2211,12 @@ Proof. replace (RiemannInt_SF (mkStepFun H4)) with (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)). apply Rle_lt_trans with (RiemannInt_SF psi). - unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *; + unfold Rminus; pattern (RiemannInt_SF psi) at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))). apply StepFun_P37; try assumption. - intros; simpl in |- *; unfold fct_cte in |- *; + intros; simpl; unfold fct_cte; apply Rle_trans with (Rabs (f x - phi x)). apply Rabs_pos. apply H. @@ -2225,9 +2225,9 @@ Proof. elim H6; intros; split; left. assumption. apply Rlt_le_trans with c; assumption. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. rewrite StepFun_P18; ring. apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)). @@ -2237,16 +2237,16 @@ Proof. apply (pre psi). replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). rewrite <- (StepFun_P43 H5 H4 H6); ring. - unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. + unfold RiemannInt_SF; case (Rle_dec a b); intro. eapply StepFun_P17. apply StepFun_P1. - simpl in |- *; apply StepFun_P1. + simpl; apply StepFun_P1. apply Ropp_eq_compat; eapply StepFun_P17. apply StepFun_P1. - simpl in |- *; apply StepFun_P1. + simpl; apply StepFun_P1. apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))). apply StepFun_P37; try assumption. - intros; simpl in |- *; unfold fct_cte in |- *; + intros; simpl; unfold fct_cte; apply Rle_trans with (Rabs (f x - phi x)). apply Rabs_pos. apply H. @@ -2255,9 +2255,9 @@ Proof. elim H5; intros; split; left. apply Rle_lt_trans with c; assumption. assumption. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n; apply Rle_trans with c; assumption ]. rewrite StepFun_P18; ring. Qed. @@ -2290,14 +2290,14 @@ Lemma RiemannInt_P25 : (pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c), a <= b -> b <= c -> RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3. Proof. - intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt in |- *; + intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv); case (RiemannInt_exists pr2 RinvN RinvN_cv); case (RiemannInt_exists pr3 RinvN RinvN_cv); intros; - symmetry in |- *; eapply UL_sequence. + symmetry ; eapply UL_sequence. apply u. - unfold Un_cv in |- *; intros; assert (H0 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Un_cv; intros; assert (H0 : 0 < eps / 3). + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (u1 _ H0); clear u1; intros N1 H1; elim (u0 _ H0); clear u0; intros N2 H2; @@ -2309,7 +2309,7 @@ Proof. RiemannInt_SF (phi_sequence RinvN pr2 n))) 0). intro; elim (H3 _ H0); clear H3; intros N3 H3; set (N0 := max (max N1 N2) N3); exists N0; intros; - unfold R_dist in |- *; + unfold R_dist; apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi_sequence RinvN pr3 n) - @@ -2330,8 +2330,8 @@ Proof. unfold R_dist in H3; cut (n >= N3)%nat. intro; assert (H6 := H3 _ H5); unfold Rminus in H6; rewrite Ropp_0 in H6; rewrite Rplus_0_r in H6; apply H6. - unfold ge in |- *; apply le_trans with N0; - [ unfold N0 in |- *; apply le_max_r | assumption ]. + unfold ge; apply le_trans with N0; + [ unfold N0; apply le_max_r | assumption ]. apply Rle_lt_trans with (Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x1) + Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x0)). @@ -2343,17 +2343,17 @@ Proof. [ apply Rabs_triang | ring ]. apply Rplus_lt_compat. unfold R_dist in H1; apply H1. - unfold ge in |- *; apply le_trans with N0; + unfold ge; apply le_trans with N0; [ apply le_trans with (max N1 N2); - [ apply le_max_l | unfold N0 in |- *; apply le_max_l ] + [ apply le_max_l | unfold N0; apply le_max_l ] | assumption ]. unfold R_dist in H2; apply H2. - unfold ge in |- *; apply le_trans with N0; + unfold ge; apply le_trans with N0; [ apply le_trans with (max N1 N2); - [ apply le_max_r | unfold N0 in |- *; apply le_max_l ] + [ apply le_max_r | unfold N0; apply le_max_l ] | assumption ]. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l; + [ unfold Rdiv; repeat rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -2390,8 +2390,8 @@ Proof. apply (proj2_sig (phi_sequence_prop RinvN pr3 n)). elim H1; clear H1; intros psi1 H1; elim H2; clear H2; intros psi2 H2; elim H3; clear H3; intros psi3 H3; assert (H := RinvN_cv); - unfold Un_cv in |- *; intros; assert (H4 : 0 < eps / 3). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Un_cv; intros; assert (H4 : 0 < eps / 3). + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H _ H4); clear H; intros N0 H; assert (H5 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3). @@ -2399,11 +2399,11 @@ Proof. replace (pos (RinvN n)) with (R_dist (mkposreal (/ (INR n + 1)) (RinvN_pos n)) 0). apply H; assumption. - unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; left; apply (cond_pos (RinvN n)). exists N0; intros; elim (H1 n); elim (H2 n); elim (H3 n); clear H1 H2 H3; - intros; unfold R_dist in |- *; unfold Rminus in |- *; + intros; unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; set (phi1 := phi_sequence RinvN pr1 n) in H8 |- *; set (phi2 := phi_sequence RinvN pr2 n) in H3 |- *; @@ -2469,7 +2469,7 @@ Proof. (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) + RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))). apply Rplus_le_compat_l; apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l; + intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi2 x)). rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr; replace (phi3 x + -1 * phi2 x) with (phi3 x - f x + (f x - phi2 x)); @@ -2480,28 +2480,28 @@ Proof. replace (Rmin a c) with a. apply Rle_trans with b; try assumption. left; assumption. - unfold Rmin in |- *; case (Rle_dec a c); intro; + unfold Rmin; case (Rle_dec a c); intro; [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. replace (Rmax a c) with c. left; assumption. - unfold Rmax in |- *; case (Rle_dec a c); intro; + unfold Rmax; case (Rle_dec a c); intro; [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. apply H3. elim H14; intros; split. replace (Rmin b c) with b. left; assumption. - unfold Rmin in |- *; case (Rle_dec b c); intro; + unfold Rmin; case (Rle_dec b c); intro; [ reflexivity | elim n0; assumption ]. replace (Rmax b c) with c. left; assumption. - unfold Rmax in |- *; case (Rle_dec b c); intro; + unfold Rmax; case (Rle_dec b c); intro; [ reflexivity | elim n0; assumption ]. do 2 rewrite <- (Rplus_comm (RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n))))) ; apply Rplus_le_compat_l; apply StepFun_P37; try assumption. - intros; simpl in |- *; rewrite Rmult_1_l; + intros; simpl; rewrite Rmult_1_l; apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi1 x)). rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr; replace (phi3 x + -1 * phi1 x) with (phi3 x - f x + (f x - phi1 x)); @@ -2511,23 +2511,23 @@ Proof. elim H14; intros; split. replace (Rmin a c) with a. left; assumption. - unfold Rmin in |- *; case (Rle_dec a c); intro; + unfold Rmin; case (Rle_dec a c); intro; [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. replace (Rmax a c) with c. apply Rle_trans with b. left; assumption. assumption. - unfold Rmax in |- *; case (Rle_dec a c); intro; + unfold Rmax; case (Rle_dec a c); intro; [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. apply H8. elim H14; intros; split. replace (Rmin a b) with a. left; assumption. - unfold Rmin in |- *; case (Rle_dec a b); intro; + unfold Rmin; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. replace (Rmax a b) with b. left; assumption. - unfold Rmax in |- *; case (Rle_dec a b); intro; + unfold Rmax; case (Rle_dec a b); intro; [ reflexivity | elim n0; assumption ]. do 2 rewrite StepFun_P30. do 2 rewrite Rmult_1_l; @@ -2553,7 +2553,7 @@ Proof. assumption. apply H5; assumption. apply Rmult_eq_reg_l with 3; - [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l; + [ unfold Rdiv; repeat rewrite Rmult_plus_distr_l; do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | discrR ] | discrR ]. @@ -2608,13 +2608,13 @@ Lemma RiemannInt_P27 : Proof. intro f; intros; elim H; clear H; intros; assert (H1 : continuity_pt f x). apply C0; split; left; assumption. - unfold derivable_pt_lim in |- *; intros; assert (Hyp : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold derivable_pt_lim; intros; assert (Hyp : 0 < eps / 2). + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - elim (H1 _ Hyp); unfold dist, D_x, no_cond in |- *; simpl in |- *; - unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin (b - x) (x - a))); + elim (H1 _ Hyp); unfold dist, D_x, no_cond; simpl; + unfold R_dist; intros; set (del := Rmin x0 (Rmin (b - x) (x - a))); assert (H4 : 0 < del). - unfold del in |- *; unfold Rmin in |- *; case (Rle_dec (b - x) (x - a)); + unfold del; unfold Rmin; case (Rle_dec (b - x) (x - a)); intro. case (Rle_dec x0 (b - x)); intro; [ elim H3; intros; assumption | apply Rlt_Rminus; assumption ]. @@ -2631,22 +2631,22 @@ Proof. left; apply Rlt_le_trans with (x + del). apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h0); [ apply RRle_abs | apply H6 ]. - unfold del in |- *; apply Rle_trans with (x + Rmin (b - x) (x - a)). + unfold del; apply Rle_trans with (x + Rmin (b - x) (x - a)). apply Rplus_le_compat_l; apply Rmin_r. - pattern b at 2 in |- *; replace b with (x + (b - x)); + pattern b at 2; replace b with (x + (b - x)); [ apply Rplus_le_compat_l; apply Rmin_l | ring ]. apply RiemannInt_P1; apply continuity_implies_RiemannInt; auto with real. intros; apply C0; elim H7; intros; split. apply Rle_trans with (x + h0). left; apply Rle_lt_trans with (x - del). - unfold del in |- *; apply Rle_trans with (x - Rmin (b - x) (x - a)). - pattern a at 1 in |- *; replace a with (x + (a - x)); [ idtac | ring ]. - unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. + unfold del; apply Rle_trans with (x - Rmin (b - x) (x - a)). + pattern a at 1; replace a with (x + (a - x)); [ idtac | ring ]. + unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel. rewrite Ropp_involutive; rewrite Ropp_plus_distr; rewrite Ropp_involutive; rewrite (Rplus_comm x); apply Rmin_r. - unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. + unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel. do 2 rewrite Ropp_involutive; apply Rmin_r. - unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel. + unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel. rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0); [ rewrite <- Rabs_Ropp; apply RRle_abs | apply H6 ]. assumption. @@ -2659,7 +2659,7 @@ Proof. with ((RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) / h0). replace (RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) with (RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))). - unfold Rdiv in |- *; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro. + unfold Rdiv; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro. apply Rle_lt_trans with (RiemannInt (RiemannInt_P16 @@ -2678,8 +2678,8 @@ Proof. apply Rabs_pos. apply RiemannInt_P19; try assumption. intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x). - unfold fct_cte in |- *; case (Req_dec x x1); intro. - rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left; + unfold fct_cte; case (Req_dec x x1); intro. + rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left; assumption. elim H3; intros; left; apply H11. repeat split. @@ -2690,16 +2690,16 @@ Proof. elim H8; intros; assumption. apply Rplus_le_compat_l; apply Rle_trans with del. left; apply Rle_lt_trans with (Rabs h0); [ apply RRle_abs | assumption ]. - unfold del in |- *; apply Rmin_l. + unfold del; apply Rmin_l. apply Rge_minus; apply Rle_ge; left; elim H8; intros; assumption. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. rewrite RiemannInt_P15. rewrite Rmult_assoc; replace ((x + h0 - x) * Rabs (/ h0)) with 1. - rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2; [ prove_sup0 | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; assumption | discrR ] ]. rewrite Rabs_right. @@ -2709,7 +2709,7 @@ Proof. apply Rle_ge; left; apply Rinv_0_lt_compat. elim r; intro. apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption. - elim H5; symmetry in |- *; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r; + elim H5; symmetry ; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r; assumption. apply Rle_lt_trans with (RiemannInt @@ -2733,7 +2733,7 @@ Proof. (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))))); auto with real. - symmetry in |- *; apply RiemannInt_P8. + symmetry ; apply RiemannInt_P8. apply Rle_lt_trans with (RiemannInt (RiemannInt_P14 (x + h0) x (eps / 2)) * Rabs (/ h0)). do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. @@ -2741,8 +2741,8 @@ Proof. apply RiemannInt_P19. auto with real. intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x). - unfold fct_cte in |- *; case (Req_dec x x1); intro. - rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left; + unfold fct_cte; case (Req_dec x x1); intro. + rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left; assumption. elim H3; intros; left; apply H11. repeat split. @@ -2752,22 +2752,22 @@ Proof. [ idtac | ring ]. replace (x1 - x0 + - (x1 - x)) with (x - x0); [ idtac | ring ]. apply Rle_lt_trans with (x + h0). - unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. + unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel. rewrite Ropp_involutive; apply Rle_trans with (Rabs h0). rewrite <- Rabs_Ropp; apply RRle_abs. apply Rle_trans with del; - [ left; assumption | unfold del in |- *; apply Rmin_l ]. + [ left; assumption | unfold del; apply Rmin_l ]. elim H8; intros; assumption. apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; replace (x + (x1 - x)) with x1; [ elim H8; intros; assumption | ring ]. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. rewrite RiemannInt_P15. rewrite Rmult_assoc; replace ((x - (x + h0)) * Rabs (/ h0)) with 1. - rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2; [ prove_sup0 | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; assumption | discrR ] ]. rewrite Rabs_left. @@ -2784,14 +2784,14 @@ Proof. (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) . ring. - unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. + unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring. rewrite RiemannInt_P15; apply Rmult_eq_reg_l with h0; - [ unfold Rdiv in |- *; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc; + [ unfold Rdiv; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | assumption ] | assumption ]. cut (a <= x + h0). cut (x + h0 <= b). - intros; unfold primitive in |- *. + intros; unfold primitive. case (Rle_dec a (x + h0)); case (Rle_dec (x + h0) b); case (Rle_dec a x); case (Rle_dec x b); intros; try (elim n; assumption || left; assumption). rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); ring. @@ -2801,7 +2801,7 @@ Proof. apply RRle_abs. apply Rle_trans with del; [ left; assumption - | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a)); + | unfold del; apply Rle_trans with (Rmin (b - x) (x - a)); [ apply Rmin_r | apply Rmin_l ] ]. apply Ropp_le_cancel; apply Rplus_le_reg_l with x; replace (x + - (x + h0)) with (- h0); [ idtac | ring ]. @@ -2809,7 +2809,7 @@ Proof. [ rewrite <- Rabs_Ropp; apply RRle_abs | apply Rle_trans with del; [ left; assumption - | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a)); + | unfold del; apply Rle_trans with (Rmin (b - x) (x - a)); apply Rmin_r ] ]. Qed. @@ -2826,14 +2826,14 @@ Proof. (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))); rewrite H3. assert (H4 : derivable_pt_lim f_b b (f b)). - unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0). + unfold f_b; pattern (f b) at 2; replace (f b) with (f b + 0). change (derivable_pt_lim ((fct_cte (f b) * (id - fct_cte b))%F + fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b ( - f b + 0)) in |- *. + f b + 0)). apply derivable_pt_lim_plus. - pattern (f b) at 2 in |- *; + pattern (f b) at 2; replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1). apply derivable_pt_lim_mult. apply derivable_pt_lim_const. @@ -2841,26 +2841,26 @@ Proof. apply derivable_pt_lim_minus. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. apply derivable_pt_lim_const. ring. - unfold derivable_pt_lim in |- *; intros; elim (H4 _ H5); intros; + unfold derivable_pt_lim; intros; elim (H4 _ H5); intros; assert (H7 : continuity_pt f b). apply C0; split; [ left; assumption | right; reflexivity ]. assert (H8 : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - elim (H7 _ H8); unfold D_x, no_cond, dist in |- *; simpl in |- *; - unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin x1 (b - a))); + elim (H7 _ H8); unfold D_x, no_cond, dist; simpl; + unfold R_dist; intros; set (del := Rmin x0 (Rmin x1 (b - a))); assert (H10 : 0 < del). - unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - a)); intros. + unfold del; unfold Rmin; case (Rle_dec x1 (b - a)); intros. case (Rle_dec x0 x1); intro; [ apply (cond_pos x0) | elim H9; intros; assumption ]. case (Rle_dec x0 (b - a)); intro; [ apply (cond_pos x0) | apply Rlt_Rminus; assumption ]. split with (mkposreal _ H10); intros; case (Rcase_abs h0); intro. assert (H14 : b + h0 < b). - pattern b at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern b at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. assert (H13 : Riemann_integrable f (b + h0) b). apply continuity_implies_RiemannInt. @@ -2874,11 +2874,11 @@ Proof. apply Rle_trans with (Rabs h0). rewrite <- Rabs_Ropp; apply RRle_abs. left; assumption. - unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. + unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. replace (primitive h (FTC_P1 h C0) (b + h0) - primitive h (FTC_P1 h C0) b) with (- RiemannInt H13). replace (f b) with (- RiemannInt (RiemannInt_P14 (b + h0) b (f b)) / h0). - rewrite <- Rabs_Ropp; unfold Rminus in |- *; unfold Rdiv in |- *; + rewrite <- Rabs_Ropp; unfold Rminus; unfold Rdiv; rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_plus_distr; repeat rewrite Ropp_involutive; replace @@ -2887,7 +2887,7 @@ Proof. ((RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) / h0). replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) with (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))). - unfold Rdiv in |- *; rewrite Rabs_mult; + unfold Rdiv; rewrite Rabs_mult; apply Rle_lt_trans with (RiemannInt (RiemannInt_P16 @@ -2907,8 +2907,8 @@ Proof. apply RiemannInt_P19. left; assumption. intros; replace (f x2 + -1 * fct_cte (f b) x2) with (f x2 - f b). - unfold fct_cte in |- *; case (Req_dec b x2); intro. - rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold fct_cte; case (Req_dec b x2); intro. + rewrite H16; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left; assumption. elim H9; intros; left; apply H18. repeat split. @@ -2919,22 +2919,22 @@ Proof. replace (x2 - x1 + x1) with x2; [ idtac | ring ]. apply Rlt_le_trans with (b + h0). 2: elim H15; intros; left; assumption. - unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel; + unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0). rewrite <- Rabs_Ropp; apply RRle_abs. apply Rlt_le_trans with del; [ assumption - | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); + | unfold del; apply Rle_trans with (Rmin x1 (b - a)); [ apply Rmin_r | apply Rmin_l ] ]. apply Rle_ge; left; apply Rlt_Rminus; elim H15; intros; assumption. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. rewrite RiemannInt_P15. rewrite Rmult_assoc; replace ((b - (b + h0)) * Rabs (/ h0)) with 1. - rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2; [ prove_sup0 | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; assumption | discrR ] ]. rewrite Rabs_left. @@ -2948,16 +2948,16 @@ Proof. (RiemannInt_P13 H13 (RiemannInt_P14 (b + h0) b (f b)) (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))) ; ring. - unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. + unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring. rewrite RiemannInt_P15. rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_eq_reg_l with h0; - [ repeat rewrite (Rmult_comm h0); unfold Rdiv in |- *; + [ repeat rewrite (Rmult_comm h0); unfold Rdiv; repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ ring | assumption ] | assumption ]. cut (a <= b + h0). cut (b + h0 <= b). - intros; unfold primitive in |- *; case (Rle_dec a (b + h0)); + intros; unfold primitive; case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b); case (Rle_dec a b); case (Rle_dec b b); intros; try (elim n; right; reflexivity) || (elim n; left; assumption). rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); ring. @@ -2970,26 +2970,26 @@ Proof. apply Rle_trans with (Rabs h0). rewrite <- Rabs_Ropp; apply RRle_abs. left; assumption. - unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. + unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. cut (primitive h (FTC_P1 h C0) b = f_b b). intro; cut (primitive h (FTC_P1 h C0) (b + h0) = f_b (b + h0)). intro; rewrite H13; rewrite H14; apply H6. assumption. apply Rlt_le_trans with del; - [ assumption | unfold del in |- *; apply Rmin_l ]. + [ assumption | unfold del; apply Rmin_l ]. assert (H14 : b < b + h0). - pattern b at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. + pattern b at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. assert (H14 := Rge_le _ _ r); elim H14; intro. assumption. - elim H11; symmetry in |- *; assumption. - unfold primitive in |- *; case (Rle_dec a (b + h0)); + elim H11; symmetry ; assumption. + unfold primitive; case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b); intros; [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14)) - | unfold f_b in |- *; reflexivity + | unfold f_b; reflexivity | elim n; left; apply Rlt_trans with b; assumption | elim n0; left; apply Rlt_trans with b; assumption ]. - unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; - rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive in |- *; + unfold f_b; unfold Rminus; rewrite Rplus_opp_r; + rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive; case (Rle_dec a b); case (Rle_dec b b); intros; [ apply RiemannInt_P5 | elim n; right; reflexivity @@ -2998,9 +2998,9 @@ Proof. (*****) set (f_a := fun x:R => f a * (x - a)); rewrite <- H2; assert (H3 : derivable_pt_lim f_a a (f a)). - unfold f_a in |- *; + unfold f_a; change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a)) - in |- *; pattern (f a) at 2 in |- *; + ; pattern (f a) at 2; replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1). apply derivable_pt_lim_mult. apply derivable_pt_lim_const. @@ -3008,18 +3008,18 @@ Proof. apply derivable_pt_lim_minus. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte in |- *; ring. - unfold derivable_pt_lim in |- *; intros; elim (H3 _ H4); intros. + unfold fct_cte; ring. + unfold derivable_pt_lim; intros; elim (H3 _ H4); intros. assert (H6 : continuity_pt f a). apply C0; split; [ right; reflexivity | left; assumption ]. assert (H7 : 0 < eps / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - elim (H6 _ H7); unfold D_x, no_cond, dist in |- *; simpl in |- *; - unfold R_dist in |- *; intros. + elim (H6 _ H7); unfold D_x, no_cond, dist; simpl; + unfold R_dist; intros. set (del := Rmin x0 (Rmin x1 (b - a))). assert (H9 : 0 < del). - unfold del in |- *; unfold Rmin in |- *. + unfold del; unfold Rmin. case (Rle_dec x1 (b - a)); intros. case (Rle_dec x0 x1); intro. apply (cond_pos x0). @@ -3030,9 +3030,9 @@ Proof. split with (mkposreal _ H9). intros; case (Rcase_abs h0); intro. assert (H12 : a + h0 < a). - pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern a at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. - unfold primitive in |- *. + unfold primitive. case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b); intros; try (elim n; left; assumption) || (elim n; right; reflexivity). @@ -3042,15 +3042,15 @@ Proof. replace (f a * (a + h0 - a)) with (f_a (a + h0)). apply H5; try assumption. apply Rlt_le_trans with del; - [ assumption | unfold del in |- *; apply Rmin_l ]. - unfold f_a in |- *; ring. - unfold f_a in |- *; ring. + [ assumption | unfold del; apply Rmin_l ]. + unfold f_a; ring. + unfold f_a; ring. elim n; left; apply Rlt_trans with a; assumption. assert (H12 : a < a + h0). - pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. + pattern a at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. assert (H12 := Rge_le _ _ r); elim H12; intro. assumption. - elim H10; symmetry in |- *; assumption. + elim H10; symmetry ; assumption. assert (H13 : Riemann_integrable f a (a + h0)). apply continuity_implies_RiemannInt. left; assumption. @@ -3062,7 +3062,7 @@ Proof. apply Ropp_le_cancel; rewrite Ropp_involutive; rewrite Ropp_minus_distr; apply Rle_trans with del. apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ]. - unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. + unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. replace (primitive h (FTC_P1 h C0) (a + h0) - primitive h (FTC_P1 h C0) a) with (RiemannInt H13). replace (f a) with (RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0). @@ -3071,7 +3071,7 @@ Proof. with ((RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) / h0). replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) with (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))). - unfold Rdiv in |- *; rewrite Rabs_mult; + unfold Rdiv; rewrite Rabs_mult; apply Rle_lt_trans with (RiemannInt (RiemannInt_P16 @@ -3091,8 +3091,8 @@ Proof. apply RiemannInt_P19. left; assumption. intros; replace (f x2 + -1 * fct_cte (f a) x2) with (f x2 - f a). - unfold fct_cte in |- *; case (Req_dec a x2); intro. - rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold fct_cte; case (Req_dec a x2); intro. + rewrite H15; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left; assumption. elim H8; intros; left; apply H17; repeat split. assumption. @@ -3104,42 +3104,42 @@ Proof. apply RRle_abs. apply Rlt_le_trans with del; [ assumption - | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); + | unfold del; apply Rle_trans with (Rmin x1 (b - a)); [ apply Rmin_r | apply Rmin_l ] ]. apply Rle_ge; left; apply Rlt_Rminus; elim H14; intros; assumption. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. rewrite RiemannInt_P15. rewrite Rmult_assoc; replace ((a + h0 - a) * Rabs (/ h0)) with 1. - rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2; [ prove_sup0 | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; - [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; assumption | discrR ] ]. rewrite Rabs_right. - rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym; [ reflexivity | assumption ]. apply Rle_ge; left; apply Rinv_0_lt_compat; assert (H14 := Rge_le _ _ r); elim H14; intro. assumption. - elim H10; symmetry in |- *; assumption. + elim H10; symmetry ; assumption. rewrite (RiemannInt_P13 H13 (RiemannInt_P14 a (a + h0) (f a)) (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))) ; ring. - unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. + unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring. rewrite RiemannInt_P15. - rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; - rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv; rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ ring | assumption ]. cut (a <= a + h0). cut (a + h0 <= b). - intros; unfold primitive in |- *; case (Rle_dec a (a + h0)); + intros; unfold primitive; case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b); intros; try (elim n; right; reflexivity) || (elim n; left; assumption). - rewrite RiemannInt_P9; unfold Rminus in |- *; rewrite Ropp_0; + rewrite RiemannInt_P9; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply RiemannInt_P5. elim n; assumption. elim n; assumption. @@ -3148,15 +3148,15 @@ Proof. [ idtac | ring ]. rewrite Rplus_comm; apply Rle_trans with del; [ apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ] - | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ]. + | unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ]. (*****) assert (H1 : x = a). rewrite <- H0 in H; elim H; intros; apply Rle_antisym; assumption. set (f_a := fun x:R => f a * (x - a)). assert (H2 : derivable_pt_lim f_a a (f a)). - unfold f_a in |- *; + unfold f_a; change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a)) - in |- *; pattern (f a) at 2 in |- *; + ; pattern (f a) at 2; replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1). apply derivable_pt_lim_mult. apply derivable_pt_lim_const. @@ -3164,18 +3164,18 @@ Proof. apply derivable_pt_lim_minus. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. set (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))). assert (H3 : derivable_pt_lim f_b b (f b)). - unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0). + unfold f_b; pattern (f b) at 2; replace (f b) with (f b + 0). change (derivable_pt_lim ((fct_cte (f b) * (id - fct_cte b))%F + fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b ( - f b + 0)) in |- *. + f b + 0)). apply derivable_pt_lim_plus. - pattern (f b) at 2 in |- *; + pattern (f b) at 2; replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1). apply derivable_pt_lim_mult. apply derivable_pt_lim_const. @@ -3183,20 +3183,20 @@ Proof. apply derivable_pt_lim_minus. apply derivable_pt_lim_id. apply derivable_pt_lim_const. - unfold fct_cte in |- *; ring. + unfold fct_cte; ring. apply derivable_pt_lim_const. ring. - unfold derivable_pt_lim in |- *; intros; elim (H2 _ H4); intros; + unfold derivable_pt_lim; intros; elim (H2 _ H4); intros; elim (H3 _ H4); intros; set (del := Rmin x0 x1). assert (H7 : 0 < del). - unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x0 x1); intro. + unfold del; unfold Rmin; case (Rle_dec x0 x1); intro. apply (cond_pos x0). apply (cond_pos x1). split with (mkposreal _ H7); intros; case (Rcase_abs h0); intro. assert (H10 : a + h0 < a). - pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern a at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. - rewrite H1; unfold primitive in |- *; case (Rle_dec a (a + h0)); + rewrite H1; unfold primitive; case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b); intros; try (elim n; right; assumption || reflexivity). elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H10)). @@ -3205,27 +3205,27 @@ Proof. replace (f a * (a + h0 - a)) with (f_a (a + h0)). apply H5; try assumption. apply Rlt_le_trans with del; try assumption. - unfold del in |- *; apply Rmin_l. - unfold f_a in |- *; ring. - unfold f_a in |- *; ring. + unfold del; apply Rmin_l. + unfold f_a; ring. + unfold f_a; ring. elim n; rewrite <- H0; left; assumption. assert (H10 : a < a + h0). - pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. + pattern a at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. assert (H10 := Rge_le _ _ r); elim H10; intro. assumption. - elim H8; symmetry in |- *; assumption. - rewrite H0 in H1; rewrite H1; unfold primitive in |- *; + elim H8; symmetry ; assumption. + rewrite H0 in H1; rewrite H1; unfold primitive; case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b); case (Rle_dec a b); case (Rle_dec b b); intros; try (elim n; right; assumption || reflexivity). rewrite H0 in H10; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)). repeat rewrite RiemannInt_P9. replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b). - fold (f_b (b + h0)) in |- *. + fold (f_b (b + h0)). apply H6; try assumption. apply Rlt_le_trans with del; try assumption. - unfold del in |- *; apply Rmin_r. - unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + unfold del; apply Rmin_r. + unfold f_b; unfold Rminus; rewrite Rplus_opp_r; rewrite Rmult_0_r; rewrite Rplus_0_l; apply RiemannInt_P5. elim n; rewrite <- H0; left; assumption. elim n0; rewrite <- H0; left; assumption. @@ -3236,11 +3236,11 @@ Lemma RiemannInt_P29 : (C0:forall x:R, a <= x <= b -> continuity_pt f x), antiderivative f (primitive h (FTC_P1 h C0)) a b. Proof. - intro f; intros; unfold antiderivative in |- *; split; try assumption; intros; + intro f; intros; unfold antiderivative; split; try assumption; intros; assert (H0 := RiemannInt_P28 h C0 H); assert (H1 : derivable_pt (primitive h (FTC_P1 h C0)) x); - [ unfold derivable_pt in |- *; split with (f x); apply H0 - | split with H1; symmetry in |- *; apply derive_pt_eq_0; apply H0 ]. + [ unfold derivable_pt; split with (f x); apply H0 + | split with H1; symmetry ; apply derive_pt_eq_0; apply H0 ]. Qed. Lemma RiemannInt_P30 : @@ -3259,7 +3259,7 @@ Lemma RiemannInt_P31 : forall (f:C1_fun) (a b:R), a <= b -> antiderivative (derive f (diff0 f)) f a b. Proof. - intro f; intros; unfold antiderivative in |- *; split; try assumption; intros; + intro f; intros; unfold antiderivative; split; try assumption; intros; split with (diff0 f x); reflexivity. Qed. diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v index abcb8d6ff..1e4e83e7f 100644 --- a/theories/Reals/RiemannInt_SF.v +++ b/theories/Reals/RiemannInt_SF.v @@ -33,19 +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; @@ -88,7 +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)) @@ -99,10 +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. @@ -173,7 +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. @@ -181,13 +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. @@ -198,23 +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. @@ -232,7 +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. @@ -242,26 +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. @@ -278,19 +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. @@ -303,10 +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 : @@ -320,12 +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]. @@ -340,32 +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]]; @@ -377,141 +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. @@ -534,7 +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). @@ -544,31 +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 @@ -576,8 +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; @@ -587,18 +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); @@ -606,20 +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); @@ -636,20 +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. @@ -689,10 +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]. @@ -716,34 +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 @@ -752,64 +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 @@ -817,64 +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. @@ -918,12 +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))) ] ]. @@ -931,7 +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))) ] ]. @@ -943,8 +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. @@ -954,38 +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)); @@ -1001,22 +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)). @@ -1024,7 +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)); @@ -1037,16 +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 @@ -1059,7 +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) @@ -1074,8 +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) @@ -1083,23 +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 @@ -1109,10 +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); @@ -1124,13 +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)); @@ -1149,7 +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. @@ -1160,7 +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 @@ -1168,24 +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). @@ -1203,11 +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). @@ -1219,11 +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. @@ -1253,22 +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)). @@ -1276,7 +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)); @@ -1289,16 +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 @@ -1311,7 +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) @@ -1325,8 +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) @@ -1334,23 +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 @@ -1360,10 +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); @@ -1375,13 +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)); @@ -1400,7 +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)). @@ -1409,7 +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 @@ -1417,24 +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). @@ -1452,12 +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). @@ -1469,11 +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; @@ -1509,35 +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. @@ -1556,7 +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. @@ -1565,7 +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. @@ -1574,7 +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))) @@ -1611,10 +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 ]. @@ -1623,8 +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. @@ -1634,8 +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. @@ -1643,7 +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. @@ -1652,7 +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 @@ -1676,18 +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. @@ -1700,7 +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 @@ -1708,9 +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. @@ -1730,7 +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 : @@ -1741,16 +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 ] ]. @@ -1809,27 +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; @@ -1838,30 +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). @@ -1872,17 +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. @@ -1890,21 +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. @@ -1912,7 +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 @@ -1925,16 +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 @@ -1951,34 +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; @@ -1991,28 +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 @@ -2041,13 +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 @@ -2055,21 +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. @@ -2087,22 +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). @@ -2120,7 +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. @@ -2132,7 +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. @@ -2140,13 +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 @@ -2157,14 +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 : @@ -2189,11 +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. @@ -2229,27 +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 @@ -2264,24 +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. @@ -2289,19 +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). @@ -2311,7 +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). @@ -2321,19 +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). @@ -2341,7 +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). @@ -2351,19 +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)). @@ -2371,7 +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). @@ -2384,56 +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 : @@ -2449,7 +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. @@ -2461,11 +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. @@ -2479,22 +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. @@ -2508,42 +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. @@ -2565,7 +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; @@ -2576,11 +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. @@ -2593,32 +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'; diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v index d2fee1fde..bacf42bb1 100644 --- a/theories/Reals/Rlimit.v +++ b/theories/Reals/Rlimit.v @@ -31,7 +31,7 @@ Proof. intro esp. assert (H := double_var esp). unfold Rdiv in H. - symmetry in |- *; exact H. + symmetry ; exact H. Qed. (*********) @@ -39,9 +39,9 @@ Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2. Proof. intro eps. replace (2 + 2) with 4. - pattern eps at 3 in |- *; rewrite double_var. + pattern eps at 3; rewrite double_var. rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)). - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rmult_assoc. rewrite <- Rinv_mult_distr. reflexivity. @@ -54,7 +54,7 @@ Qed. Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps. Proof. intros. - pattern eps at 2 in |- *; rewrite <- Rmult_1_r. + pattern eps at 2; rewrite <- Rmult_1_r. repeat rewrite (Rmult_comm eps). apply Rmult_lt_compat_r. exact H. @@ -70,7 +70,7 @@ Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps. Proof. intros. replace (2 + 2) with 4. - pattern eps at 2 in |- *; rewrite <- Rmult_1_r. + pattern eps at 2; rewrite <- Rmult_1_r. repeat rewrite (Rmult_comm eps). apply Rmult_lt_compat_r. exact H. @@ -113,10 +113,10 @@ Qed. (*********) Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0. Proof. - intros; unfold Rgt in |- *; rewrite <- (Rmult_0_r eps); + intros; unfold Rgt; rewrite <- (Rmult_0_r eps); apply Rmult_lt_compat_l. assumption. - unfold mul_factor in |- *; apply Rinv_0_lt_compat; + unfold mul_factor; apply Rinv_0_lt_compat; cut (1 <= 1 + (Rabs l + Rabs l')). cut (0 < 1). exact (Rlt_le_trans _ _ _). @@ -210,7 +210,7 @@ Qed. (*********) Lemma lim_x : forall (D:R -> Prop) (x0:R), limit1_in (fun x:R => x) D x0 x0. Proof. - unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + unfold limit1_in; unfold limit_in; simpl; intros; split with eps; split; auto; intros; elim H0; intros; auto. Qed. @@ -221,9 +221,9 @@ Lemma limit_plus : limit1_in f D l x0 -> limit1_in g D l' x0 -> limit1_in (fun x:R => f x + g x) D (l + l') x0. Proof. - intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; unfold limit1_in; unfold limit_in; simpl; intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1)); - elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl in |- *; + elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl; clear H H0; intros; elim H; elim H0; clear H H0; intros; split with (Rmin x1 x); split. exact (Rmin_Rgt_r x1 x 0 (conj H H2)). @@ -244,12 +244,12 @@ Lemma limit_Ropp : forall (f:R -> R) (D:R -> Prop) (l x0:R), limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0. Proof. - unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + unfold limit1_in; unfold limit_in; simpl; intros; elim (H eps H0); clear H; intros; elim H; clear H; intros; split with x; split; auto; intros; generalize (H1 x1 H2); - clear H1; intro; unfold R_dist in |- *; unfold Rminus in |- *; + clear H1; intro; unfold R_dist; unfold Rminus; rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l); - fold (l - f x1) in |- *; fold (R_dist l (f x1)) in |- *; + fold (l - f x1); fold (R_dist l (f x1)); rewrite R_dist_sym; assumption. Qed. @@ -259,7 +259,7 @@ Lemma limit_minus : limit1_in f D l x0 -> limit1_in g D l' x0 -> limit1_in (fun x:R => f x - g x) D (l - l') x0. Proof. - intros; unfold Rminus in |- *; generalize (limit_Ropp g D l' x0 H0); intro; + intros; unfold Rminus; generalize (limit_Ropp g D l' x0 H0); intro; exact (limit_plus f (fun x:R => - g x) D l (- l') x0 H H1). Qed. @@ -268,7 +268,7 @@ Lemma limit_free : forall (f:R -> R) (D:R -> Prop) (x x0:R), limit1_in (fun h:R => f x) D (f x) x0. Proof. - unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + unfold limit1_in; unfold limit_in; simpl; intros; split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x)); intros a b; rewrite (b (eq_refl (f x))); unfold Rgt in H; assumption. @@ -280,14 +280,14 @@ Lemma limit_mul : limit1_in f D l x0 -> limit1_in g D l' x0 -> limit1_in (fun x:R => f x * g x) D (l * l') x0. Proof. - intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; unfold limit1_in; unfold limit_in; simpl; intros; elim (H (Rmin 1 (eps * mul_factor l l')) (mul_factor_gt_f eps l l' H1)); elim (H0 (eps * mul_factor l l') (mul_factor_gt eps l l' H1)); - clear H H0; simpl in |- *; intros; elim H; elim H0; + clear H H0; simpl; intros; elim H; elim H0; clear H H0; intros; split with (Rmin x1 x); split. exact (Rmin_Rgt_r x1 x 0 (conj H H2)). - intros; elim H4; clear H4; intros; unfold R_dist in |- *; + intros; elim H4; clear H4; intros; unfold R_dist; replace (f x2 * g x2 - l * l') with (f x2 * (g x2 - l') + l' * (f x2 - l)). cut (Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) < eps). cut @@ -309,7 +309,7 @@ Proof. apply Rmult_ge_0_gt_0_lt_compat. apply Rle_ge. exact (Rabs_pos (g x2 - l')). - rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt in |- *; apply Rle_lt_0_plus_1; + rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt; apply Rle_lt_0_plus_1; exact (Rabs_pos l). unfold R_dist in H9; apply (Rplus_lt_reg_r (- Rabs l) (Rabs (f x2)) (1 + Rabs l)). @@ -323,13 +323,13 @@ Proof. generalize (H3 x2 (conj H4 H6)); trivial. apply Rmult_le_compat_l. exact (Rabs_pos l'). - unfold Rle in |- *; left; assumption. + unfold Rle; left; assumption. rewrite (Rmult_comm (1 + Rabs l) (eps * mul_factor l l')); rewrite (Rmult_comm (Rabs l') (eps * mul_factor l l')); rewrite <- (Rmult_plus_distr_l (eps * mul_factor l l') (1 + Rabs l) (Rabs l')) ; rewrite (Rmult_assoc eps (mul_factor l l') (1 + Rabs l + Rabs l')); - rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor in |- *; + rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor; rewrite (Rinv_l (1 + (Rabs l + Rabs l')) (mul_factor_wd l l')); rewrite (proj1 (Rmult_ne eps)); apply Req_le; trivial. ring. @@ -344,10 +344,10 @@ Lemma single_limit : forall (f:R -> R) (D:R -> Prop) (l l' x0:R), adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'. Proof. - unfold limit1_in in |- *; unfold limit_in in |- *; intros. + unfold limit1_in; unfold limit_in; intros. cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps). - clear H0 H1; unfold dist in |- *; unfold R_met in |- *; unfold R_dist in |- *; - unfold Rabs in |- *; case (Rcase_abs (l - l')); intros. + clear H0 H1; unfold dist; unfold R_met; unfold R_dist; + unfold Rabs; case (Rcase_abs (l - l')); intros. cut (forall eps:R, eps > 0 -> - (l - l') < eps). intro; generalize (prop_eps (- (l - l')) H1); intro; generalize (Ropp_gt_lt_0_contravar (l - l') r); intro; @@ -358,10 +358,10 @@ Proof. rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; - unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3); intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; clear a b; apply (Rlt_trans 0 1 2 H3 H4). - unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); auto. apply (Rinv_0_lt_compat 2); cut (1 < 2). @@ -380,10 +380,10 @@ Proof. rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; - unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3); intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; clear a b; apply (Rlt_trans 0 1 2 H3 H4). - unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); auto. apply (Rinv_0_lt_compat 2); cut (1 < 2). @@ -393,7 +393,7 @@ Proof. (**) intros; unfold adhDa in H; elim (H0 eps H2); intros; elim (H1 eps H2); intros; clear H0 H1; elim H3; elim H4; clear H3 H4; intros; - simpl in |- *; simpl in H1, H4; generalize (Rmin_Rgt x x1 0); + simpl; simpl in H1, H4; generalize (Rmin_Rgt x x1 0); intro; elim H5; intros; clear H5; elim (H (Rmin x x1) (H7 (conj H3 H0))); intros; elim H5; intros; clear H5 H H6 H7; generalize (Rmin_Rgt x x1 (R_dist x2 x0)); intro; @@ -403,10 +403,10 @@ Proof. intros; generalize (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0); - unfold R_dist in |- *; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; + unfold R_dist; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1); elim (Rmult_ne eps); intros a b; rewrite a; clear a b; - generalize (R_dist_tri l l' (f x2)); unfold R_dist in |- *; + generalize (R_dist_tri l l' (f x2)); unfold R_dist; intros; apply (Rle_lt_trans (Rabs (l - l')) (Rabs (l - f x2) + Rabs (f x2 - l')) @@ -419,7 +419,7 @@ Lemma limit_comp : limit1_in f Df l x0 -> limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0. Proof. - unfold limit1_in, limit_in, Dgf in |- *; simpl in |- *. + unfold limit1_in, limit_in, Dgf; simpl. intros f g Df Dg l l' x0 Hf Hg eps eps_pos. elim (Hg eps eps_pos). intros alpg lg. @@ -436,12 +436,12 @@ Lemma limit_inv : forall (f:R -> R) (D:R -> Prop) (l x0:R), limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0. Proof. - unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; - unfold R_dist in |- *; intros; elim (H (Rabs l / 2)). + unfold limit1_in; unfold limit_in; simpl; + unfold R_dist; intros; elim (H (Rabs l / 2)). intros delta1 H2; elim (H (eps * (Rsqr l / 2))). intros delta2 H3; elim H2; elim H3; intros; exists (Rmin delta1 delta2); split. - unfold Rmin in |- *; case (Rle_dec delta1 delta2); intro; assumption. + unfold Rmin; case (Rle_dec delta1 delta2); intro; assumption. intro; generalize (H5 x); clear H5; intro H5; generalize (H7 x); clear H7; intro H7; intro H10; elim H10; intros; cut (D x /\ Rabs (x - x0) < delta1). cut (D x /\ Rabs (x - x0) < delta2). @@ -455,7 +455,7 @@ Proof. (Rplus_lt_compat_l (Rabs (f x) - Rabs l / 2) (Rabs l - Rabs (f x)) (Rabs l / 2) H14); replace (Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))) with (Rabs l / 2). - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro; cut (f x <> 0). intro; replace (/ f x + - / l) with ((l - f x) * / (l * f x)). rewrite Rabs_mult; rewrite Rabs_Rinv. @@ -467,7 +467,7 @@ Proof. (/ Rabs (l * f x)) (2 / Rsqr l) (Rabs_pos (l - f x)) H18 H5 H17); replace (eps * (Rsqr l / 2) * (2 / Rsqr l)) with eps. intro; assumption. - unfold Rdiv in |- *; unfold Rsqr in |- *; rewrite Rinv_mult_distr. + unfold Rdiv; unfold Rsqr; rewrite Rinv_mult_distr. repeat rewrite Rmult_assoc. rewrite (Rmult_comm l). repeat rewrite Rmult_assoc. @@ -487,7 +487,7 @@ Proof. left; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply prod_neq_R0; assumption. rewrite Rmult_comm; rewrite Rabs_mult; rewrite Rinv_mult_distr. - rewrite (Rsqr_abs l); unfold Rsqr in |- *; unfold Rdiv in |- *; + rewrite (Rsqr_abs l); unfold Rsqr; unfold Rdiv; rewrite Rinv_mult_distr. repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r. apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. @@ -496,7 +496,7 @@ Proof. apply Rabs_pos_lt; assumption. apply Rabs_pos_lt; assumption. apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); - [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR; intro H18; assumption | discriminate ]. replace (Rabs (f x) * Rabs l * / 2 * / Rabs (f x)) with (Rabs l / 2). @@ -512,7 +512,7 @@ Proof. discrR. apply Rabs_no_R0. assumption. - unfold Rdiv in |- *. + unfold Rdiv. repeat rewrite Rmult_assoc. rewrite (Rmult_comm (Rabs (f x))). repeat rewrite Rmult_assoc. @@ -526,7 +526,7 @@ Proof. apply Rabs_no_R0; assumption. apply prod_neq_R0; assumption. rewrite (Rinv_mult_distr _ _ H0 H16). - unfold Rminus in |- *; rewrite Rmult_plus_distr_r. + unfold Rminus; rewrite Rmult_plus_distr_r. rewrite <- Rmult_assoc. rewrite <- Rinv_r_sym. rewrite Rmult_1_l. @@ -538,16 +538,16 @@ Proof. reflexivity. assumption. assumption. - red in |- *; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15; + red; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15; cut (0 < Rabs l / 2). intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rabs l / 2) 0 H17 H15)). - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply Rabs_pos_lt; assumption. apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); - [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR; intro; assumption | discriminate ]. - pattern (Rabs l) at 3 in |- *; rewrite double_var. + pattern (Rabs l) at 3; rewrite double_var. ring. split; [ assumption @@ -557,18 +557,18 @@ Proof. [ assumption | apply Rlt_le_trans with (Rmin delta1 delta2); [ assumption | apply Rmin_l ] ]. - change (0 < eps * (Rsqr l / 2)) in |- *; unfold Rdiv in |- *; + change (0 < eps * (Rsqr l / 2)); unfold Rdiv; repeat rewrite Rmult_assoc; apply Rmult_lt_0_compat. assumption. apply Rmult_lt_0_compat. apply Rsqr_pos_lt; assumption. apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); - [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR; intro; assumption | discriminate ]. - change (0 < Rabs l / 2) in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat; + change (0 < Rabs l / 2); unfold Rdiv; apply Rmult_lt_0_compat; [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); - [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR; intro; assumption | discriminate ] ]. Qed. diff --git a/theories/Reals/Rlogic.v b/theories/Reals/Rlogic.v index 2237ea6ef..5cb0a2ffb 100644 --- a/theories/Reals/Rlogic.v +++ b/theories/Reals/Rlogic.v @@ -271,10 +271,10 @@ assert (H2 : ~ is_upper_bound E M'). intro H5. assert (M <= M')%R by (apply H4; exact H5). apply (Rlt_not_le M M'). - unfold M' in |- *. - pattern M at 2 in |- *. + unfold M'. + pattern M at 2. rewrite <- Rplus_0_l. - pattern (0 + M)%R in |- *. + pattern (0 + M)%R. rewrite Rplus_comm. rewrite <- (Rplus_opp_r 1). apply Rplus_lt_compat_l. @@ -284,7 +284,7 @@ assert (H2 : ~ is_upper_bound E M'). apply H2. intros N (n,H7). rewrite H7. -unfold M' in |- *. +unfold M'. assert (H5 : (INR (S n) <= M)%R) by (apply H3; exists (S n); reflexivity). rewrite S_INR in H5. assert (H6 : (INR n + 1 + -1 <= M + -1)%R). diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index 019a2d96e..0e4101aa9 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -26,14 +26,14 @@ Local Open Scope R_scope. Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y). Proof. - intros P x y H1 H2; unfold Rmin in |- *; case (Rle_dec x y); intro; + intros P x y H1 H2; unfold Rmin; case (Rle_dec x y); intro; assumption. Qed. Lemma exp_le_3 : exp 1 <= 3. Proof. assert (exp_1 : exp 1 <> 0). - assert (H0 := exp_pos 1); red in |- *; intro; rewrite H in H0; + assert (H0 := exp_pos 1); red; intro; rewrite H in H0; elim (Rlt_irrefl _ H0). apply Rmult_le_reg_l with (/ exp 1). apply Rinv_0_lt_compat; apply exp_pos. @@ -43,7 +43,7 @@ Proof. rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)). - unfold exp in |- *; case (exist_exp (-1)); intros; simpl in |- *; + unfold exp; case (exist_exp (-1)); intros; simpl; unfold exp_in in e; assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1). cut @@ -73,7 +73,7 @@ Proof. ring. discrR. apply H. - unfold Un_decreasing in |- *; intros; + unfold Un_decreasing; intros; apply Rmult_le_reg_l with (INR (fact n)). apply INR_fact_lt_0. apply Rmult_le_reg_l with (INR (fact (S n))). @@ -84,13 +84,13 @@ Proof. rewrite Rmult_1_r; apply le_INR; apply fact_le; apply le_n_Sn. apply INR_fact_neq_0. apply INR_fact_neq_0. - assert (H0 := cv_speed_pow_fact 1); unfold Un_cv in |- *; unfold Un_cv in H0; + assert (H0 := cv_speed_pow_fact 1); unfold Un_cv; unfold Un_cv in H0; intros; elim (H0 _ H1); intros; exists x0; intros; - unfold R_dist in H2; unfold R_dist in |- *; + unfold R_dist in H2; unfold R_dist; replace (/ INR (fact n)) with (1 ^ n / INR (fact n)). apply (H2 _ H3). - unfold Rdiv in |- *; rewrite pow1; rewrite Rmult_1_l; reflexivity. - unfold infinite_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0); + unfold Rdiv; rewrite pow1; rewrite Rmult_1_l; reflexivity. + unfold infinite_sum in e; unfold Un_cv, tg_alt; intros; elim (e _ H0); intros; exists x0; intros; replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with (sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n). @@ -121,7 +121,7 @@ Proof. intro. replace (derive_pt exp x0 (H0 x0)) with (exp x0). apply exp_pos. - symmetry in |- *; apply derive_pt_eq_0. + symmetry ; apply derive_pt_eq_0. apply (derivable_pt_lim_exp x0). apply H. Qed. @@ -143,11 +143,11 @@ Proof. rewrite Ropp_0; rewrite Rplus_0_r; replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0). rewrite exp_0; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; - pattern x at 1 in |- *; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0)); + pattern x at 1; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0)); apply Rmult_lt_compat_l. apply H. rewrite <- exp_0; apply exp_increasing; elim H3; intros; assumption. - symmetry in |- *; apply derive_pt_eq_0; apply derivable_pt_lim_exp. + symmetry ; apply derive_pt_eq_0; apply derivable_pt_lim_exp. Qed. Lemma ln_exists1 : forall y:R, 1 <= y -> { z:R | y = exp z }. @@ -160,18 +160,18 @@ Proof. cut (f 0 * f y <= 0); [intro H4|]. pose proof (IVT_cor f 0 y H2 (Rlt_le _ _ H0) H4) as (t,(_,H7)); exists t; unfold f in H7; apply Rminus_diag_uniq_sym; exact H7. - pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)); + pattern 0 at 2; rewrite <- (Rmult_0_r (f y)); rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l; assumption. - unfold f in |- *; apply Rplus_le_reg_l with y; left; + unfold f; apply Rplus_le_reg_l with y; left; apply Rlt_trans with (1 + y). rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1. replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H0) | ring ]. - unfold f in |- *; change (continuity (exp - fct_cte y)) in |- *; + unfold f; change (continuity (exp - fct_cte y)); apply continuity_minus; [ apply derivable_continuous; apply derivable_exp | apply derivable_continuous; apply derivable_const ]. - unfold f in |- *; rewrite exp_0; apply Rplus_le_reg_l with y; + unfold f; rewrite exp_0; apply Rplus_le_reg_l with y; rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H | ring ]. Qed. @@ -185,18 +185,18 @@ Proof. apply H. rewrite <- Rinv_r_sym. rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n). - red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). destruct (ln_exists1 _ H0) as (x,p); exists (- x); apply Rmult_eq_reg_l with (exp x / y). - unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. + unfold Rdiv; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc; rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0; - rewrite Rmult_1_r; symmetry in |- *; apply p. - red in |- *; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H). - unfold Rdiv in |- *; apply prod_neq_R0. - assert (H3 := exp_pos x); red in |- *; intro H4; rewrite H4 in H3; + rewrite Rmult_1_r; symmetry ; apply p. + red; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H). + unfold Rdiv; apply prod_neq_R0. + assert (H3 := exp_pos x); red; intro H4; rewrite H4 in H3; elim (Rlt_irrefl _ H3). - apply Rinv_neq_0_compat; red in |- *; intro H3; rewrite H3 in H; + apply Rinv_neq_0_compat; red; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H). Qed. @@ -213,11 +213,11 @@ Definition ln (x:R) : R := Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x. Proof. - intros; unfold ln in |- *; case (Rlt_dec 0 x); intro. - unfold Rln in |- *; + intros; unfold ln; case (Rlt_dec 0 x); intro. + unfold Rln; case (ln_exists (mkposreal x r) (cond_pos (mkposreal x r))); intros. - simpl in e; symmetry in |- *; apply e. + simpl in e; symmetry ; apply e. elim n; apply H. Qed. @@ -231,7 +231,7 @@ Qed. Theorem exp_Ropp : forall x:R, exp (- x) = / exp x. Proof. intros x; assert (H : exp x <> 0). - assert (H := exp_pos x); red in |- *; intro; rewrite H0 in H; + assert (H := exp_pos x); red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). apply Rmult_eq_reg_l with (r := exp x). rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0. @@ -306,11 +306,11 @@ Theorem ln_continue : forall y:R, 0 < y -> continue_in ln (fun x:R => 0 < x) y. Proof. intros y H. - unfold continue_in, limit1_in, limit_in in |- *; intros eps Heps. + unfold continue_in, limit1_in, limit_in; intros eps Heps. cut (1 < exp eps); [ intros H1 | idtac ]. cut (exp (- eps) < 1); [ intros H2 | idtac ]. exists (Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))); split. - red in |- *; apply P_Rmin. + red; apply P_Rmin. apply Rmult_lt_0_compat. assumption. apply Rplus_lt_reg_r with 1. @@ -321,7 +321,7 @@ Proof. apply Rplus_lt_reg_r with (exp (- eps)). rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1; [ apply H2 | ring ]. - unfold dist, R_met, R_dist in |- *; simpl in |- *. + unfold dist, R_met, R_dist; simpl. intros x [[H3 H4] H5]. cut (y * (x * / y) = x). intro Hxyy. @@ -351,7 +351,7 @@ Proof. rewrite Hxyy; rewrite Rmult_1_r; apply Hxy. rewrite Hxy; rewrite Rinv_r. rewrite ln_1; rewrite Rabs_R0; apply Heps. - red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). rewrite Rabs_right. apply exp_lt_inv. rewrite exp_ln. @@ -366,7 +366,7 @@ Proof. left; apply (Rgt_minus _ _ Hxy). apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. rewrite <- ln_1. - apply Rgt_ge; red in |- *; apply ln_increasing. + apply Rgt_ge; red; apply ln_increasing. apply Rlt_0_1. apply Rmult_lt_reg_l with (r := y). apply H. @@ -379,7 +379,7 @@ Proof. apply Rinv_0_lt_compat; assumption. rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. ring. - red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). apply Rmult_lt_reg_l with (exp eps). apply exp_pos. rewrite <- exp_plus; rewrite Rmult_1_r; rewrite Rplus_opp_r; rewrite exp_0; @@ -412,13 +412,13 @@ Local Infix "^R" := Rpower (at level 30, right associativity) : R_scope. Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y. Proof. - intros x y z; unfold Rpower in |- *. + intros x y z; unfold Rpower. rewrite Rmult_plus_distr_r; rewrite exp_plus; auto. Qed. Theorem Rpower_mult : forall x y z:R, (x ^R y) ^R z = x ^R (y * z). Proof. - intros x y z; unfold Rpower in |- *. + intros x y z; unfold Rpower. rewrite ln_exp. replace (z * (y * ln x)) with (y * z * ln x). reflexivity. @@ -427,22 +427,22 @@ Qed. Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1. Proof. - intros x _; unfold Rpower in |- *. + intros x _; unfold Rpower. rewrite Rmult_0_l; apply exp_0. Qed. Theorem Rpower_1 : forall x:R, 0 < x -> x ^R 1 = x. Proof. - intros x H; unfold Rpower in |- *. + intros x H; unfold Rpower. rewrite Rmult_1_l; apply exp_ln; apply H. Qed. Theorem Rpower_pow : forall (n:nat) (x:R), 0 < x -> x ^R INR n = x ^ n. Proof. - intros n; elim n; simpl in |- *; auto; fold INR in |- *. + intros n; elim n; simpl; auto; fold INR. intros x H; apply Rpower_O; auto. intros n1; case n1. - intros H x H0; simpl in |- *; rewrite Rmult_1_r; apply Rpower_1; auto. + intros H x H0; simpl; rewrite Rmult_1_r; apply Rpower_1; auto. intros n0 H x H0; rewrite Rpower_plus; rewrite H; try rewrite Rpower_1; try apply Rmult_comm || assumption. Qed. @@ -451,7 +451,7 @@ Theorem Rpower_lt : forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z. Proof. intros x y z H H0 H1. - unfold Rpower in |- *. + unfold Rpower. apply exp_increasing. apply Rmult_lt_compat_r. rewrite <- ln_1; apply ln_increasing. @@ -464,18 +464,18 @@ Theorem Rpower_sqrt : forall x:R, 0 < x -> x ^R (/ 2) = sqrt x. Proof. intros x H. apply ln_inv. - unfold Rpower in |- *; apply exp_pos. + unfold Rpower; apply exp_pos. apply sqrt_lt_R0; apply H. apply Rmult_eq_reg_l with (INR 2). apply exp_inv. - fold Rpower in |- *. + fold Rpower. cut ((x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2). - unfold Rpower in |- *; auto. + unfold Rpower; auto. rewrite Rpower_mult. rewrite Rinv_l. replace 1 with (INR 1); auto. - repeat rewrite Rpower_pow; simpl in |- *. - pattern x at 1 in |- *; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)). + repeat rewrite Rpower_pow; simpl. + pattern x at 1; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)). ring. apply sqrt_lt_R0; apply H. apply H. @@ -485,7 +485,7 @@ Qed. Theorem Rpower_Ropp : forall x y:R, x ^R (- y) = / x ^R y. Proof. - unfold Rpower in |- *. + unfold Rpower. intros x y; rewrite Ropp_mult_distr_l_reverse. apply exp_Ropp. Qed. @@ -505,11 +505,11 @@ Proof. rewrite Rinv_r. apply exp_lt_inv. apply Rle_lt_trans with (1 := exp_le_3). - change (3 < 2 ^R 2) in |- *. + change (3 < 2 ^R 2). repeat rewrite Rpower_plus; repeat rewrite Rpower_1. repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; repeat rewrite Rmult_1_l. - pattern 3 at 1 in |- *; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1); + pattern 3 at 1; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1); [ apply Rplus_lt_compat_l; apply Rlt_0_1 | ring ]. prove_sup0. discrR. @@ -523,7 +523,7 @@ Theorem limit1_ext : forall (f g:R -> R) (D:R -> Prop) (l x:R), (forall x:R, D x -> f x = g x) -> limit1_in f D l x -> limit1_in g D l x. Proof. - intros f g D l x H; unfold limit1_in, limit_in in |- *. + intros f g D l x H; unfold limit1_in, limit_in. intros H0 eps H1; case (H0 eps); auto. intros x0 [H2 H3]; exists x0; split; auto. intros x1 [H4 H5]; rewrite <- H; auto. @@ -533,7 +533,7 @@ Theorem limit1_imp : forall (f:R -> R) (D D1:R -> Prop) (l x:R), (forall x:R, D1 x -> D x) -> limit1_in f D l x -> limit1_in f D1 l x. Proof. - intros f D D1 l x H; unfold limit1_in, limit_in in |- *. + intros f D D1 l x H; unfold limit1_in, limit_in. intros H0 eps H1; case (H0 eps H1); auto. intros alpha [H2 H3]; exists alpha; split; auto. intros d [H4 H5]; apply H3; split; auto. @@ -541,7 +541,7 @@ Qed. Theorem Rinv_Rdiv : forall x y:R, x <> 0 -> y <> 0 -> / (x / y) = y / x. Proof. - intros x y H1 H2; unfold Rdiv in |- *; rewrite Rinv_mult_distr. + intros x y H1 H2; unfold Rdiv; rewrite Rinv_mult_distr. rewrite Rinv_involutive. apply Rmult_comm. assumption. @@ -551,18 +551,18 @@ Qed. Theorem Dln : forall y:R, 0 < y -> D_in ln Rinv (fun x:R => 0 < x) y. Proof. - intros y Hy; unfold D_in in |- *. + intros y Hy; unfold D_in. apply limit1_ext with (f := fun x:R => / ((exp (ln x) - exp (ln y)) / (ln x - ln y))). intros x [HD1 HD2]; repeat rewrite exp_ln. - unfold Rdiv in |- *; rewrite Rinv_mult_distr. + unfold Rdiv; rewrite Rinv_mult_distr. rewrite Rinv_involutive. apply Rmult_comm. apply Rminus_eq_contra. - red in |- *; intros H2; case HD2. - symmetry in |- *; apply (ln_inv _ _ HD1 Hy H2). + red; intros H2; case HD2. + symmetry ; apply (ln_inv _ _ HD1 Hy H2). apply Rminus_eq_contra; apply (not_eq_sym HD2). - apply Rinv_neq_0_compat; apply Rminus_eq_contra; red in |- *; intros H2; + apply Rinv_neq_0_compat; apply Rminus_eq_contra; red; intros H2; case HD2; apply ln_inv; auto. assumption. assumption. @@ -574,17 +574,17 @@ Proof. intros x [H1 H2]; split. split; auto. split; auto. - red in |- *; intros H3; case H2; apply ln_inv; auto. + red; intros H3; case H2; apply ln_inv; auto. apply limit_comp with (l := ln y) (g := fun x:R => (exp x - exp (ln y)) / (x - ln y)) (f := ln). apply ln_continue; auto. assert (H0 := derivable_pt_lim_exp (ln y)); unfold derivable_pt_lim in H0; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros; elim (H0 _ H); + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros; elim (H0 _ H); intros; exists (pos x); split. apply (cond_pos x). - intros; pattern y at 3 in |- *; rewrite <- exp_ln. - pattern x0 at 1 in |- *; replace x0 with (ln y + (x0 - ln y)); + intros; pattern y at 3; rewrite <- exp_ln. + pattern x0 at 1; replace x0 with (ln y + (x0 - ln y)); [ idtac | ring ]. apply H1. elim H2; intros H3 _; unfold D_x in H3; elim H3; clear H3; intros _ H3; @@ -592,36 +592,36 @@ Proof. apply H3. elim H2; clear H2; intros _ H2; apply H2. assumption. - red in |- *; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy). + red; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy). Qed. Lemma derivable_pt_lim_ln : forall x:R, 0 < x -> derivable_pt_lim ln x (/ x). Proof. intros; assert (H0 := Dln x H); unfold D_in in H0; unfold limit1_in in H0; unfold limit_in in H0; simpl in H0; unfold R_dist in H0; - unfold derivable_pt_lim in |- *; intros; elim (H0 _ H1); + unfold derivable_pt_lim; intros; elim (H0 _ H1); intros; elim H2; clear H2; intros; set (alp := Rmin x0 (x / 2)); assert (H4 : 0 < alp). - unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec x0 (x / 2)); intro. + unfold alp; unfold Rmin; case (Rle_dec x0 (x / 2)); intro. apply H2. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. - exists (mkposreal _ H4); intros; pattern h at 2 in |- *; + exists (mkposreal _ H4); intros; pattern h at 2; replace h with (x + h - x); [ idtac | ring ]. apply H3; split. - unfold D_x in |- *; split. + unfold D_x; split. case (Rcase_abs h); intro. assert (H7 : Rabs h < x / 2). apply Rlt_le_trans with alp. apply H6. - unfold alp in |- *; apply Rmin_r. + unfold alp; apply Rmin_r. apply Rlt_trans with (x / 2). - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. rewrite Rabs_left in H7. apply Rplus_lt_reg_r with (- h - x / 2). replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ]. - pattern x at 2 in |- *; rewrite double_var. + pattern x at 2; rewrite double_var. replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ]. apply r. apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply r ]. @@ -629,7 +629,7 @@ Proof. [ apply H5 | ring ]. replace (x + h - x) with h; [ apply Rlt_le_trans with alp; - [ apply H6 | unfold alp in |- *; apply Rmin_l ] + [ apply H6 | unfold alp; apply Rmin_l ] | ring ]. Qed. @@ -637,7 +637,7 @@ Theorem D_in_imp : forall (f g:R -> R) (D D1:R -> Prop) (x:R), (forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x. Proof. - intros f g D D1 x H; unfold D_in in |- *. + intros f g D D1 x H; unfold D_in. intros H0; apply limit1_imp with (D := D_x D x); auto. intros x1 [H1 H2]; split; auto. Qed. @@ -646,7 +646,7 @@ Theorem D_in_ext : forall (f g h:R -> R) (D:R -> Prop) (x:R), f x = g x -> D_in h f D x -> D_in h g D x. Proof. - intros f g h D x H; unfold D_in in |- *. + intros f g h D x H; unfold D_in. rewrite H; auto. Qed. @@ -661,7 +661,7 @@ Proof. intros x H0; repeat split. assumption. apply D_in_ext with (f := fun x:R => / x * (z * exp (z * ln x))). - unfold Rminus in |- *; rewrite Rpower_plus; rewrite Rpower_Ropp; + unfold Rminus; rewrite Rpower_plus; rewrite Rpower_Ropp; rewrite (Rpower_1 _ H); unfold Rpower; ring. apply Dcomp with (f := ln) @@ -674,7 +674,7 @@ Proof. intros x H1; repeat split; auto. apply (Dcomp (fun _:R => True) (fun _:R => True) (fun x => z) exp - (fun x:R => z * x) exp); simpl in |- *. + (fun x:R => z * x) exp); simpl. apply D_in_ext with (f := fun x:R => z * 1). apply Rmult_1_r. apply (Dmult_const (fun x => True) (fun x => x) (fun x => 1)); apply Dx. @@ -687,16 +687,16 @@ Theorem derivable_pt_lim_power : 0 < x -> derivable_pt_lim (fun x => x ^R y) x (y * x ^R (y - 1)). Proof. intros x y H. - unfold Rminus in |- *; rewrite Rpower_plus. + unfold Rminus; rewrite Rpower_plus. rewrite Rpower_Ropp. rewrite Rpower_1; auto. rewrite <- Rmult_assoc. - unfold Rpower in |- *. + unfold Rpower. apply derivable_pt_lim_comp with (f1 := ln) (f2 := fun x => exp (y * x)). apply derivable_pt_lim_ln; assumption. rewrite (Rmult_comm y). apply derivable_pt_lim_comp with (f1 := fun x => y * x) (f2 := exp). - pattern y at 2 in |- *; replace y with (0 * ln x + y * 1). + pattern y at 2; replace y with (0 * ln x + y * 1). apply derivable_pt_lim_mult with (f1 := fun x:R => y) (f2 := fun x:R => x). apply derivable_pt_lim_const with (a := y). apply derivable_pt_lim_id. diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v index 6bcf4bd4c..e97894af3 100644 --- a/theories/Reals/Rprod.v +++ b/theories/Reals/Rprod.v @@ -36,7 +36,7 @@ Proof. replace (S n - k - 1)%nat with O; [rewrite H1; simpl|omega]. replace (n+1+0)%nat with (S n); ring. replace (S n - k-1)%nat with (S (n - k-1));[idtac|omega]. - simpl in |- *; replace (k + S (n - k))%nat with (S n). + simpl; replace (k + S (n - k))%nat with (S n). replace (k + 1 + S (n - k - 1))%nat with (S n). rewrite Hrecn; [ ring | assumption ]. omega. @@ -49,8 +49,8 @@ Lemma prod_SO_pos : (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N. Proof. intros; induction N as [| N HrecN]. - simpl in |- *; apply H; trivial. - simpl in |- *; apply Rmult_le_pos. + simpl; apply H; trivial. + simpl; apply Rmult_le_pos. apply HrecN; intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ]. apply H; apply le_n. @@ -64,7 +64,7 @@ Lemma prod_SO_Rle : Proof. intros; induction N as [| N HrecN]. elim H with O; trivial. - simpl in |- *; apply Rle_trans with (prod_f_R0 An N * Bn (S N)). + simpl; apply Rle_trans with (prod_f_R0 An N * Bn (S N)). apply Rmult_le_compat_l. apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros; assumption. @@ -114,7 +114,7 @@ Proof. (if eq_nat_dec n 0 then 1 else INR n) = INR n). intros n; case (eq_nat_dec n 0); auto with real. intros; absurd (0 < n)%nat; omega. - intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO. + intros; unfold Rsqr; repeat rewrite fact_prodSO. cut ((k=N)%nat \/ (k < N)%nat \/ (N < k)%nat). intro H2; elim H2; intro H3. rewrite H3; replace (2*N-N)%nat with N;[right; ring|omega]. @@ -164,14 +164,14 @@ Qed. (**********) Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n). Proof. - intro; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; - elim (fact_neq_0 n); symmetry in |- *; assumption. + intro; apply lt_INR_0; apply neq_O_lt; red; intro; + elim (fact_neq_0 n); symmetry ; assumption. Qed. (** We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *) Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N. Proof. - intros; unfold C in |- *; unfold Rdiv in |- *; apply Rmult_le_compat_l. + intros; unfold C; unfold Rdiv; apply Rmult_le_compat_l. apply pos_INR. replace (2 * N - N)%nat with N. apply Rmult_le_reg_l with (INR (fact N) * INR (fact N)). diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v index e67f118f6..2f80ac13d 100644 --- a/theories/Reals/Rseries.v +++ b/theories/Reals/Rseries.v @@ -54,20 +54,20 @@ Section sequence. (*********) Lemma EUn_noempty : exists r : R, EUn r. Proof. - unfold EUn in |- *; split with (Un 0); split with 0%nat; trivial. + unfold EUn; split with (Un 0); split with 0%nat; trivial. Qed. (*********) Lemma Un_in_EUn : forall n:nat, EUn (Un n). Proof. - intro; unfold EUn in |- *; split with n; trivial. + intro; unfold EUn; split with n; trivial. Qed. (*********) Lemma Un_bound_imp : forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x. Proof. - intros; unfold is_upper_bound in |- *; intros; unfold EUn in H0; elim H0; + intros; unfold is_upper_bound; intros; unfold EUn in H0; elim H0; clear H0; intros; generalize (H x1); intro; rewrite <- H0 in H1; trivial. Qed. @@ -77,7 +77,7 @@ Section sequence. forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m. Proof. double induction n m; intros. - unfold Rge in |- *; right; trivial. + unfold Rge; right; trivial. exfalso; unfold ge in H1; generalize (le_Sn_O n0); intro; auto. cut (n0 >= 0)%nat. generalize H0; intros; unfold Un_growing in H0; @@ -89,7 +89,7 @@ Section sequence. elim y; clear y; intro y. unfold ge in H2; generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2)); intro; exfalso; auto. - rewrite y; unfold Rge in |- *; right; trivial. + rewrite y; unfold Rge; right; trivial. unfold ge in H0; generalize (H0 (S n0) H1 (lt_le_S n0 n1 y)); intro; unfold Un_growing in H1; apply @@ -285,7 +285,7 @@ Section sequence. (*********) Lemma cauchy_bound : Cauchy_crit -> bound EUn. Proof. - unfold Cauchy_crit, bound in |- *; intros; unfold is_upper_bound in |- *; + unfold Cauchy_crit, bound; intros; unfold is_upper_bound; unfold Rgt in H; elim (H 1 Rlt_0_1); clear H; intros; generalize (H x); intro; generalize (le_dec x); intro; elim (finite_greater x); intros; split with (Rmax x0 (Un x + 1)); @@ -324,12 +324,12 @@ End Isequence. Lemma GP_infinite : forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)). Proof. - intros; unfold Pser in |- *; unfold infinite_sum in |- *; intros; + intros; unfold Pser; unfold infinite_sum; intros; elim (Req_dec x 0). intros; exists 0%nat; intros; rewrite H1; rewrite Rminus_0_r; rewrite Rinv_1; cut (sum_f_R0 (fun n0:nat => 1 * 0 ^ n0) n = 1). intros; rewrite H3; rewrite R_dist_eq; auto. - elim n; simpl in |- *. + elim n; simpl. ring. intros; rewrite H3; ring. intro; cut (0 < eps * (Rabs (1 - x) * Rabs (/ x))). @@ -344,11 +344,11 @@ Proof. apply Rabs_pos_lt. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. - unfold R_dist in |- *; rewrite <- Rabs_mult. + unfold R_dist; rewrite <- Rabs_mult. rewrite Rmult_minus_distr_l. cut ((1 - x) * sum_f_R0 (fun n0:nat => x ^ n0) n = @@ -359,7 +359,7 @@ Proof. cut (- (x ^ (n + 1) - 1) - 1 = - x ^ (n + 1)). intro; rewrite H7. rewrite Rabs_Ropp; cut ((n + 1)%nat = S n); auto. - intro H8; rewrite H8; simpl in |- *; rewrite Rabs_mult; + intro H8; rewrite H8; simpl; rewrite Rabs_mult; apply (Rlt_le_trans (Rabs x * Rabs (x ^ n)) (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x)))) ( @@ -373,7 +373,7 @@ Proof. Rabs x * Rabs (/ x) * (eps * Rabs (1 - x))). clear H8; intros; rewrite H8; rewrite <- Rabs_mult; rewrite Rinv_r. rewrite Rabs_R1; cut (1 * (eps * Rabs (1 - x)) = Rabs (1 - x) * eps). - intros; rewrite H9; unfold Rle in |- *; right; reflexivity. + intros; rewrite H9; unfold Rle; right; reflexivity. ring. assumption. ring. @@ -381,12 +381,12 @@ Proof. ring. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. ring; ring. - elim n; simpl in |- *. + elim n; simpl. ring. intros; rewrite H5. ring. @@ -396,7 +396,7 @@ Proof. apply Rabs_pos_lt. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. diff --git a/theories/Reals/Rsigma.v b/theories/Reals/Rsigma.v index d26dcde21..9810c6117 100644 --- a/theories/Reals/Rsigma.v +++ b/theories/Reals/Rsigma.v @@ -28,8 +28,8 @@ Section Sigma. Proof. intros; induction k as [| k Hreck]. cut (low = 0%nat). - intro; rewrite H1; unfold sigma in |- *; rewrite <- minus_n_n; - rewrite <- minus_n_O; simpl in |- *; replace (high - 1)%nat with (pred high). + intro; rewrite H1; unfold sigma; rewrite <- minus_n_n; + rewrite <- minus_n_O; simpl; replace (high - 1)%nat with (pred high). apply (decomp_sum (fun k:nat => f k)). assumption. apply pred_of_minus. @@ -42,8 +42,8 @@ Section Sigma. apply Hreck. assumption. apply lt_trans with (S k); [ apply lt_n_Sn | assumption ]. - unfold sigma in |- *; replace (high - S (S k))%nat with (pred (high - S k)). - pattern (S k) at 3 in |- *; replace (S k) with (S k + 0)%nat; + unfold sigma; replace (high - S (S k))%nat with (pred (high - S k)). + pattern (S k) at 3; replace (S k) with (S k + 0)%nat; [ idtac | ring ]. replace (sum_f_R0 (fun k0:nat => f (S (S k) + k0)) (pred (high - S k))) with (sum_f_R0 (fun k0:nat => f (S k + S k0)) (pred (high - S k))). @@ -55,12 +55,12 @@ Section Sigma. replace (high - S (S k))%nat with (high - S k - 1)%nat. apply pred_of_minus. omega. - unfold sigma in |- *; replace (S k - low)%nat with (S (k - low)). - pattern (S k) at 1 in |- *; replace (S k) with (low + S (k - low))%nat. - symmetry in |- *; apply (tech5 (fun i:nat => f (low + i))). + unfold sigma; replace (S k - low)%nat with (S (k - low)). + pattern (S k) at 1; replace (S k) with (low + S (k - low))%nat. + symmetry ; apply (tech5 (fun i:nat => f (low + i))). omega. omega. - rewrite <- H2; unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *; + rewrite <- H2; unfold sigma; rewrite <- minus_n_n; simpl; replace (high - S low)%nat with (pred (high - low)). replace (sum_f_R0 (fun k0:nat => f (S (low + k0))) (pred (high - low))) with (sum_f_R0 (fun k0:nat => f (low + S k0)) (pred (high - low))). @@ -79,7 +79,7 @@ Section Sigma. (low <= k)%nat -> (k < high)%nat -> sigma low high - sigma low k = sigma (S k) high. Proof. - intros low high k H1 H2; symmetry in |- *; rewrite (sigma_split H1 H2); ring. + intros low high k H1 H2; symmetry ; rewrite (sigma_split H1 H2); ring. Qed. Theorem sigma_diff_neg : @@ -100,8 +100,8 @@ Section Sigma. apply sigma_split. apply le_n. assumption. - unfold sigma in |- *; rewrite <- minus_n_n. - simpl in |- *. + unfold sigma; rewrite <- minus_n_n. + simpl. replace (low + 0)%nat with low; [ reflexivity | ring ]. Qed. @@ -113,20 +113,20 @@ Section Sigma. generalize (lt_le_weak low high H1); intro H3; replace (f high) with (sigma high high). rewrite Rplus_comm; cut (high = S (pred high)). - intro; pattern high at 3 in |- *; rewrite H. + intro; pattern high at 3; rewrite H. apply sigma_split. apply le_S_n; rewrite <- H; apply lt_le_S; assumption. apply lt_pred_n_n; apply le_lt_trans with low; [ apply le_O_n | assumption ]. apply S_pred with 0%nat; apply le_lt_trans with low; [ apply le_O_n | assumption ]. - unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *; + unfold sigma; rewrite <- minus_n_n; simpl; replace (high + 0)%nat with high; [ reflexivity | ring ]. Qed. Theorem sigma_eq_arg : forall low:nat, sigma low low = f low. Proof. - intro; unfold sigma in |- *; rewrite <- minus_n_n. - simpl in |- *; replace (low + 0)%nat with low; [ reflexivity | ring ]. + intro; unfold sigma; rewrite <- minus_n_n. + simpl; replace (low + 0)%nat with low; [ reflexivity | ring ]. Qed. End Sigma. diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v index ecde4f8bb..d5e5f0905 100644 --- a/theories/Reals/Rsqrt_def.v +++ b/theories/Reals/Rsqrt_def.v @@ -41,18 +41,18 @@ Lemma dicho_comp : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *; assumption. - simpl in |- *. + simpl; assumption. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. - pattern 2 at 1 in |- *; rewrite Rmult_comm. + pattern 2 at 1; rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. rewrite Rmult_1_r. rewrite double. apply Rplus_le_compat_l. assumption. - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. @@ -67,14 +67,14 @@ Lemma dicho_lb_growing : forall (x y:R) (P:R -> bool), x <= y -> Un_growing (dicho_lb x y P). Proof. intros. - unfold Un_growing in |- *. + unfold Un_growing. intro. - simpl in |- *. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). right; reflexivity. - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. - pattern 2 at 1 in |- *; rewrite Rmult_comm. + pattern 2 at 1; rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. rewrite Rmult_1_r. rewrite double. @@ -87,11 +87,11 @@ Lemma dicho_up_decreasing : forall (x y:R) (P:R -> bool), x <= y -> Un_decreasing (dicho_up x y P). Proof. intros. - unfold Un_decreasing in |- *. + unfold Un_decreasing. intro. - simpl in |- *. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. @@ -112,17 +112,17 @@ Lemma dicho_lb_maj_y : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *; assumption. - simpl in |- *. + simpl; assumption. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). assumption. - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ]. rewrite double; apply Rplus_le_compat. assumption. - pattern y at 2 in |- *; replace y with (Dichotomy_ub x y P 0); + pattern y at 2; replace y with (Dichotomy_ub x y P 0); [ idtac | reflexivity ]. apply decreasing_prop. assert (H0 := dicho_up_decreasing x y P H). @@ -136,10 +136,10 @@ Proof. intros. cut (forall n:nat, dicho_lb x y P n <= y). intro. - unfold has_ub in |- *. - unfold bound in |- *. + unfold has_ub. + unfold bound. exists y. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. elim H1; intros. rewrite H2; apply H0. @@ -151,15 +151,15 @@ Lemma dicho_up_min_x : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *; assumption. - simpl in |- *. + simpl; assumption. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). - unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. + unfold Rdiv; apply Rmult_le_reg_l with 2. prove_sup0. - pattern 2 at 1 in |- *; rewrite Rmult_comm. + pattern 2 at 1; rewrite Rmult_comm. rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ]. rewrite double; apply Rplus_le_compat. - pattern x at 1 in |- *; replace x with (Dichotomy_lb x y P 0); + pattern x at 1; replace x with (Dichotomy_lb x y P 0); [ idtac | reflexivity ]. apply tech9. assert (H0 := dicho_lb_growing x y P H). @@ -175,14 +175,14 @@ Proof. intros. cut (forall n:nat, x <= dicho_up x y P n). intro. - unfold has_lb in |- *. - unfold bound in |- *. + unfold has_lb. + unfold bound. exists (- x). - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. elim H1; intros. rewrite H2. - unfold opp_seq in |- *. + unfold opp_seq. apply Ropp_le_contravar. apply H0. apply dicho_up_min_x; assumption. @@ -214,35 +214,35 @@ Lemma dicho_lb_dicho_up : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *. - unfold Rdiv in |- *; rewrite Rinv_1; ring. - simpl in |- *. + simpl. + unfold Rdiv; rewrite Rinv_1; ring. + simpl. case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). - unfold Rdiv in |- *. + unfold Rdiv. replace ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 - Dichotomy_lb x y P n) with ((dicho_up x y P n - dicho_lb x y P n) / 2). - unfold Rdiv in |- *; rewrite Hrecn. - unfold Rdiv in |- *. + unfold Rdiv; rewrite Hrecn. + unfold Rdiv. rewrite Rinv_mult_distr. ring. discrR. apply pow_nonzero; discrR. - pattern (Dichotomy_lb x y P n) at 2 in |- *; + pattern (Dichotomy_lb x y P n) at 2; rewrite (double_var (Dichotomy_lb x y P n)); - unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring. + unfold dicho_up, dicho_lb, Rminus, Rdiv; ring. replace (Dichotomy_ub x y P n - (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) with ((dicho_up x y P n - dicho_lb x y P n) / 2). - unfold Rdiv in |- *; rewrite Hrecn. - unfold Rdiv in |- *. + unfold Rdiv; rewrite Hrecn. + unfold Rdiv. rewrite Rinv_mult_distr. ring. discrR. apply pow_nonzero; discrR. - pattern (Dichotomy_ub x y P n) at 1 in |- *; + pattern (Dichotomy_ub x y P n) at 1; rewrite (double_var (Dichotomy_ub x y P n)); - unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring. + unfold dicho_up, dicho_lb, Rminus, Rdiv; ring. Qed. Definition pow_2_n (n:nat) := 2 ^ n. @@ -250,23 +250,23 @@ Definition pow_2_n (n:nat) := 2 ^ n. Lemma pow_2_n_neq_R0 : forall n:nat, pow_2_n n <> 0. Proof. intro. - unfold pow_2_n in |- *. + unfold pow_2_n. apply pow_nonzero. discrR. Qed. Lemma pow_2_n_growing : Un_growing pow_2_n. Proof. - unfold Un_growing in |- *. + unfold Un_growing. intro. replace (S n) with (n + 1)%nat; - [ unfold pow_2_n in |- *; rewrite pow_add | ring ]. - pattern (2 ^ n) at 1 in |- *; rewrite <- Rmult_1_r. + [ unfold pow_2_n; rewrite pow_add | ring ]. + pattern (2 ^ n) at 1; rewrite <- Rmult_1_r. apply Rmult_le_compat_l. left; apply pow_lt; prove_sup0. - simpl in |- *. + simpl. rewrite Rmult_1_r. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply Rlt_0_1. Qed. @@ -274,7 +274,7 @@ Lemma pow_2_n_infty : cv_infty pow_2_n. Proof. cut (forall N:nat, INR N <= 2 ^ N). intros. - unfold cv_infty in |- *. + unfold cv_infty. intro. case (total_order_T 0 M); intro. elim s; intro. @@ -287,41 +287,41 @@ Proof. apply Rlt_le_trans with (INR N0). rewrite INR_IZR_INZ. rewrite <- H1. - unfold N in |- *. + unfold N. assert (H3 := archimed M). elim H3; intros; assumption. apply Rle_trans with (pow_2_n N0). - unfold pow_2_n in |- *; apply H. + unfold pow_2_n; apply H. apply Rge_le. apply growing_prop. apply pow_2_n_growing. assumption. apply le_IZR. - unfold N in |- *. - simpl in |- *. + unfold N. + simpl. assert (H0 := archimed M); elim H0; intros. left; apply Rlt_trans with M; assumption. exists 0%nat; intros. rewrite <- b. - unfold pow_2_n in |- *; apply pow_lt; prove_sup0. + unfold pow_2_n; apply pow_lt; prove_sup0. exists 0%nat; intros. apply Rlt_trans with 0. assumption. - unfold pow_2_n in |- *; apply pow_lt; prove_sup0. + unfold pow_2_n; apply pow_lt; prove_sup0. simple induction N. - simpl in |- *. + simpl. left; apply Rlt_0_1. intros. - pattern (S n) at 2 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + pattern (S n) at 2; replace (S n) with (n + 1)%nat; [ idtac | ring ]. rewrite S_INR; rewrite pow_add. - simpl in |- *. + simpl. rewrite Rmult_1_r. apply Rle_trans with (2 ^ n). rewrite <- (Rplus_comm 1). rewrite <- (Rmult_1_r (INR n)). apply (poly n 1). apply Rlt_0_1. - pattern (2 ^ n) at 1 in |- *; rewrite <- Rplus_0_r. + pattern (2 ^ n) at 1; rewrite <- Rplus_0_r. rewrite <- (Rmult_comm 2). rewrite double. apply Rplus_le_compat_l. @@ -338,8 +338,8 @@ Proof. cut (Un_cv (fun i:nat => dicho_lb x y P i - dicho_up x y P i) 0). intro. assert (H4 := UL_sequence _ _ _ H2 H3). - symmetry in |- *; apply Rminus_diag_uniq_sym; assumption. - unfold Un_cv in |- *; unfold R_dist in |- *. + symmetry ; apply Rminus_diag_uniq_sym; assumption. + unfold Un_cv; unfold R_dist. intros. assert (H4 := cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty). case (total_order_T x y); intro. @@ -356,7 +356,7 @@ Proof. rewrite <- Rabs_Ropp. rewrite Ropp_minus_distr'. rewrite dicho_lb_dicho_up. - unfold Rdiv in |- *; rewrite Rabs_mult. + unfold Rdiv; rewrite Rabs_mult. rewrite (Rabs_right (y - x)). apply Rmult_lt_reg_l with (/ (y - x)). apply Rinv_0_lt_compat; assumption. @@ -366,12 +366,12 @@ Proof. [ unfold pow_2_n, Rdiv in H6; rewrite <- (Rmult_comm eps); apply H6; assumption | ring ]. - red in |- *; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp). + red; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp). apply Rle_ge. apply Rplus_le_reg_l with x; rewrite Rplus_0_r. replace (x + (y - x)) with y; [ assumption | ring ]. assumption. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; assumption ]. apply Rplus_lt_reg_r with x; rewrite Rplus_0_r. replace (x + (y - x)) with y; [ assumption | ring ]. @@ -382,7 +382,7 @@ Proof. rewrite Ropp_minus_distr'. rewrite dicho_lb_dicho_up. rewrite b. - unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; rewrite Rmult_0_l; + unfold Rminus, Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rabs_R0; assumption. assumption. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). @@ -399,23 +399,23 @@ Lemma continuity_seq : forall (f:R -> R) (Un:nat -> R) (l:R), continuity_pt f l -> Un_cv Un l -> Un_cv (fun i:nat => f (Un i)) (f l). Proof. - unfold continuity_pt, Un_cv in |- *; unfold continue_in in |- *. - unfold limit1_in in |- *. - unfold limit_in in |- *. - unfold dist in |- *. - simpl in |- *. - unfold R_dist in |- *. + unfold continuity_pt, Un_cv; unfold continue_in. + unfold limit1_in. + unfold limit_in. + unfold dist. + simpl. + unfold R_dist. intros. elim (H eps H1); intros alp H2. elim H2; intros. elim (H0 alp H3); intros N H5. exists N; intros. case (Req_dec (Un n) l); intro. - rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. apply H4. split. - unfold D_x, no_cond in |- *. + unfold D_x, no_cond. split. trivial. apply (not_eq_sym (A:=R)); assumption. @@ -428,9 +428,9 @@ Lemma dicho_lb_car : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *. + simpl. assumption. - simpl in |- *. + simpl. assert (X := sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))). @@ -447,9 +447,9 @@ Lemma dicho_up_car : Proof. intros. induction n as [| n Hrecn]. - simpl in |- *. + simpl. assumption. - simpl in |- *. + simpl. assert (X := sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))). @@ -480,7 +480,7 @@ Proof. split. split. apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0). - simpl in |- *. + simpl. right; reflexivity. apply growing_ineq. apply dicho_lb_growing; assumption. @@ -503,7 +503,7 @@ Proof. assert (H10 := H5 H7). apply Rle_antisym; assumption. intro. - unfold Wn in |- *. + unfold Wn. cut (forall z:R, cond_positivity z = true <-> 0 <= z). intro. assert (H8 := dicho_up_car x y (fun z:R => cond_positivity (f z)) n). @@ -514,7 +514,7 @@ Proof. apply H12. left; assumption. intro. - unfold cond_positivity in |- *. + unfold cond_positivity. case (Rle_dec 0 z); intro. split. intro; assumption. @@ -523,7 +523,7 @@ Proof. intro feqt;discriminate feqt. intro. elim n0; assumption. - unfold Vn in |- *. + unfold Vn. cut (forall z:R, cond_positivity z = false <-> z < 0). intros. assert (H8 := dicho_lb_car x y (fun z:R => cond_positivity (f z)) n). @@ -535,7 +535,7 @@ Proof. apply H12. assumption. intro. - unfold cond_positivity in |- *. + unfold cond_positivity. case (Rle_dec 0 z); intro. split. intro feqt; discriminate feqt. @@ -554,7 +554,7 @@ Proof. cut (0 < - f x0). intro. elim (H7 (- f x0) H8); intros. - cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ]. + cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ]. assert (H11 := H9 x2 H10). rewrite Rabs_right in H11. pattern (- f x0) at 1 in H11; rewrite <- Rplus_0_r in H11. @@ -562,11 +562,11 @@ Proof. assert (H12 := Rplus_lt_reg_r _ _ _ H11). assert (H13 := H6 x2). elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H12)). - apply Rle_ge; left; unfold Rminus in |- *; apply Rplus_le_lt_0_compat. + apply Rle_ge; left; unfold Rminus; apply Rplus_le_lt_0_compat. apply H6. exact H8. apply Ropp_0_gt_lt_contravar; assumption. - unfold Wn in |- *; assumption. + unfold Wn; assumption. cut (Un_cv Vn x0). intros. assert (H7 := continuity_seq f Vn x0 (H x0) H5). @@ -574,7 +574,7 @@ Proof. elim s; intro. unfold Un_cv in H7; unfold R_dist in H7. elim (H7 (f x0) a); intros. - cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ]. + cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ]. assert (H10 := H8 x2 H9). rewrite Rabs_left in H10. pattern (f x0) at 2 in H10; rewrite <- Rplus_0_r in H10. @@ -589,12 +589,12 @@ Proof. apply Ropp_0_gt_lt_contravar; assumption. apply Rplus_lt_reg_r with (f x0 - f (Vn x2)). rewrite Rplus_0_r; replace (f x0 - f (Vn x2) + (f (Vn x2) - f x0)) with 0; - [ unfold Rminus in |- *; apply Rplus_lt_le_0_compat | ring ]. + [ unfold Rminus; apply Rplus_lt_le_0_compat | ring ]. assumption. apply Ropp_0_ge_le_contravar; apply Rle_ge; apply H6. right; rewrite <- b; reflexivity. left; assumption. - unfold Vn in |- *; assumption. + unfold Vn; assumption. Qed. Lemma IVT_cor : @@ -613,11 +613,11 @@ Proof. exists y. split. split; [ assumption | right; reflexivity ]. - symmetry in |- *; exact b. + symmetry ; exact b. exists x. split. split; [ right; reflexivity | assumption ]. - symmetry in |- *; exact b. + symmetry ; exact b. elim s; intro. cut (x < y). intro. @@ -633,8 +633,8 @@ Proof. unfold opp_fct in H7. rewrite <- (Ropp_involutive (f x0)). apply Ropp_eq_0_compat; assumption. - unfold opp_fct in |- *; apply Ropp_0_gt_lt_contravar; assumption. - unfold opp_fct in |- *. + unfold opp_fct; apply Ropp_0_gt_lt_contravar; assumption. + unfold opp_fct. apply Rplus_lt_reg_r with (f x); rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption. inversion H0. @@ -644,7 +644,7 @@ Proof. exists x. split. split; [ right; reflexivity | assumption ]. - symmetry in |- *; assumption. + symmetry ; assumption. case (total_order_T 0 (f y)); intro. elim s; intro. cut (x < y). @@ -657,7 +657,7 @@ Proof. exists y. split. split; [ assumption | right; reflexivity ]. - symmetry in |- *; assumption. + symmetry ; assumption. cut (0 < f x * f y). intro. elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H2 H1)). @@ -690,18 +690,18 @@ Proof. elim H5; intros; assumption. unfold f in H6. apply Rminus_diag_uniq_sym; exact H6. - rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f 1)). + rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f 1)). apply Rmult_le_compat_l; assumption. - unfold f in |- *. + unfold f. rewrite Rsqr_1. apply Rplus_le_reg_l with y. - rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; left; assumption. exists 1. split. left; apply Rlt_0_1. - rewrite b; symmetry in |- *; apply Rsqr_1. + rewrite b; symmetry ; apply Rsqr_1. cut (0 <= f y). intro. cut (f 0 * f y <= 0). @@ -714,14 +714,14 @@ Proof. elim H5; intros; assumption. unfold f in H6. apply Rminus_diag_uniq_sym; exact H6. - rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)). + rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f y)). apply Rmult_le_compat_l; assumption. - unfold f in |- *. + unfold f. apply Rplus_le_reg_l with y. - rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. - pattern y at 1 in |- *; rewrite <- Rmult_1_r. - unfold Rsqr in |- *; apply Rmult_le_compat_l. + pattern y at 1; rewrite <- Rmult_1_r. + unfold Rsqr; apply Rmult_le_compat_l. assumption. left; exact r. replace f with (Rsqr - fct_cte y)%F. @@ -729,8 +729,8 @@ Proof. apply derivable_continuous; apply derivable_Rsqr. apply derivable_continuous; apply derivable_const. reflexivity. - unfold f in |- *; rewrite Rsqr_0. - unfold Rminus in |- *; rewrite Rplus_0_l. + unfold f; rewrite Rsqr_0. + unfold Rminus; rewrite Rplus_0_l. apply Rge_le. apply Ropp_0_le_ge_contravar; assumption. Qed. @@ -749,7 +749,7 @@ Proof. intros. elim p; intros. rewrite H in H0; assumption. - unfold Rsqrt in |- *. + unfold Rsqrt. case (Rsqrt_exists x (cond_nonneg x)). intros. elim p; elim a; intros. @@ -770,7 +770,7 @@ Proof. rewrite <- H. elim p; intros. rewrite H1; reflexivity. - unfold Rsqrt in |- *. + unfold Rsqrt. case (Rsqrt_exists x (cond_nonneg x)). intros. elim p; elim a; intros. diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v index 77a4d5fbb..f1ca105da 100644 --- a/theories/Reals/Rtopology.v +++ b/theories/Reals/Rtopology.v @@ -30,16 +30,16 @@ Definition interior (D:R -> Prop) (x:R) : Prop := neighbourhood D x. Lemma interior_P1 : forall D:R -> Prop, included (interior D) D. Proof. - intros; unfold included in |- *; unfold interior in |- *; intros; + intros; unfold included; unfold interior; intros; unfold neighbourhood in H; elim H; intros; unfold included in H0; - apply H0; unfold disc in |- *; unfold Rminus in |- *; + apply H0; unfold disc; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos x0). Qed. Lemma interior_P2 : forall D:R -> Prop, open_set D -> included D (interior D). Proof. - intros; unfold open_set in H; unfold included in |- *; intros; - assert (H1 := H _ H0); unfold interior in |- *; apply H1. + intros; unfold open_set in H; unfold included; intros; + assert (H1 := H _ H0); unfold interior; apply H1. Qed. Definition point_adherent (D:R -> Prop) (x:R) : Prop := @@ -49,11 +49,11 @@ Definition adherence (D:R -> Prop) (x:R) : Prop := point_adherent D x. Lemma adherence_P1 : forall D:R -> Prop, included D (adherence D). Proof. - intro; unfold included in |- *; intros; unfold adherence in |- *; - unfold point_adherent in |- *; intros; exists x; - unfold intersection_domain in |- *; split. + intro; unfold included; intros; unfold adherence; + unfold point_adherent; intros; exists x; + unfold intersection_domain; split. unfold neighbourhood in H0; elim H0; intros; unfold included in H1; apply H1; - unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + unfold disc; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos x0). apply H. Qed. @@ -62,29 +62,29 @@ Lemma included_trans : forall D1 D2 D3:R -> Prop, included D1 D2 -> included D2 D3 -> included D1 D3. Proof. - unfold included in |- *; intros; apply H0; apply H; apply H1. + unfold included; intros; apply H0; apply H; apply H1. Qed. Lemma interior_P3 : forall D:R -> Prop, open_set (interior D). Proof. - intro; unfold open_set, interior in |- *; unfold neighbourhood in |- *; + intro; unfold open_set, interior; unfold neighbourhood; intros; elim H; intros. - exists x0; unfold included in |- *; intros. + exists x0; unfold included; intros. set (del := x0 - Rabs (x - x1)). cut (0 < del). intro; exists (mkposreal del H2); intros. cut (included (disc x1 (mkposreal del H2)) (disc x x0)). intro; assert (H5 := included_trans _ _ _ H4 H0). apply H5; apply H3. - unfold included in |- *; unfold disc in |- *; intros. + unfold included; unfold disc; intros. apply Rle_lt_trans with (Rabs (x3 - x1) + Rabs (x1 - x)). replace (x3 - x) with (x3 - x1 + (x1 - x)); [ apply Rabs_triang | ring ]. replace (pos x0) with (del + Rabs (x1 - x)). do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l; apply H4. - unfold del in |- *; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr; + unfold del; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr; ring. - unfold del in |- *; apply Rplus_lt_reg_r with (Rabs (x - x1)); + unfold del; apply Rplus_lt_reg_r with (Rabs (x - x1)); rewrite Rplus_0_r; replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0); [ idtac | ring ]. @@ -95,7 +95,7 @@ Lemma complementary_P1 : forall D:R -> Prop, ~ (exists y : R, intersection_domain D (complementary D) y). Proof. - intro; red in |- *; intro; elim H; intros; + intro; red; intro; elim H; intros; unfold intersection_domain, complementary in H0; elim H0; intros; elim H2; assumption. Qed. @@ -103,8 +103,8 @@ Qed. Lemma adherence_P2 : forall D:R -> Prop, closed_set D -> included (adherence D) D. Proof. - unfold closed_set in |- *; unfold open_set, complementary in |- *; intros; - unfold included, adherence in |- *; intros; assert (H1 := classic (D x)); + unfold closed_set; unfold open_set, complementary; intros; + unfold included, adherence; intros; assert (H1 := classic (D x)); elim H1; intro. assumption. assert (H3 := H _ H2); assert (H4 := H0 _ H3); elim H4; intros; @@ -114,8 +114,8 @@ Qed. Lemma adherence_P3 : forall D:R -> Prop, closed_set (adherence D). Proof. - intro; unfold closed_set, adherence in |- *; - unfold open_set, complementary, point_adherent in |- *; + intro; unfold closed_set, adherence; + unfold open_set, complementary, point_adherent; intros; set (P := @@ -123,21 +123,21 @@ Proof. neighbourhood V x -> exists y : R, intersection_domain V D y); assert (H0 := not_all_ex_not _ P H); elim H0; intros V0 H1; unfold P in H1; assert (H2 := imply_to_and _ _ H1); - unfold neighbourhood in |- *; elim H2; intros; unfold neighbourhood in H3; - elim H3; intros; exists x0; unfold included in |- *; - intros; red in |- *; intro. + unfold neighbourhood; elim H2; intros; unfold neighbourhood in H3; + elim H3; intros; exists x0; unfold included; + intros; red; intro. assert (H8 := H7 V0); cut (exists delta : posreal, (forall x:R, disc x1 delta x -> V0 x)). intro; assert (H10 := H8 H9); elim H4; assumption. cut (0 < x0 - Rabs (x - x1)). intro; set (del := mkposreal _ H9); exists del; intros; - unfold included in H5; apply H5; unfold disc in |- *; + unfold included in H5; apply H5; unfold disc; apply Rle_lt_trans with (Rabs (x2 - x1) + Rabs (x1 - x)). replace (x2 - x) with (x2 - x1 + (x1 - x)); [ apply Rabs_triang | ring ]. replace (pos x0) with (del + Rabs (x1 - x)). do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l; apply H10. - unfold del in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x1)); + unfold del; simpl; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr; ring. apply Rplus_lt_reg_r with (Rabs (x - x1)); rewrite Rplus_0_r; replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0); @@ -152,10 +152,10 @@ Infix "=_D" := eq_Dom (at level 70, no associativity). Lemma open_set_P1 : forall D:R -> Prop, open_set D <-> D =_D interior D. Proof. intro; split. - intro; unfold eq_Dom in |- *; split. + intro; unfold eq_Dom; split. apply interior_P2; assumption. apply interior_P1. - intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set in |- *; + intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set; intros; unfold included, interior in H; unfold included in H0; apply (H _ H1). Qed. @@ -163,20 +163,20 @@ Qed. Lemma closed_set_P1 : forall D:R -> Prop, closed_set D <-> D =_D adherence D. Proof. intro; split. - intro; unfold eq_Dom in |- *; split. + intro; unfold eq_Dom; split. apply adherence_P1. apply adherence_P2; assumption. - unfold eq_Dom in |- *; unfold included in |- *; intros; + unfold eq_Dom; unfold included; intros; assert (H0 := adherence_P3 D); unfold closed_set in H0; - unfold closed_set in |- *; unfold open_set in |- *; + unfold closed_set; unfold open_set; unfold open_set in H0; intros; assert (H2 : complementary (adherence D) x). - unfold complementary in |- *; unfold complementary in H1; red in |- *; intro; + unfold complementary; unfold complementary in H1; red; intro; elim H; clear H; intros _ H; elim H1; apply (H _ H2). - assert (H3 := H0 _ H2); unfold neighbourhood in |- *; + assert (H3 := H0 _ H2); unfold neighbourhood; unfold neighbourhood in H3; elim H3; intros; exists x0; - unfold included in |- *; unfold included in H4; intros; + unfold included; unfold included in H4; intros; assert (H6 := H4 _ H5); unfold complementary in H6; - unfold complementary in |- *; red in |- *; intro; + unfold complementary; red; intro; elim H; clear H; intros H _; elim H6; apply (H _ H7). Qed. @@ -184,8 +184,8 @@ Lemma neighbourhood_P1 : forall (D1 D2:R -> Prop) (x:R), included D1 D2 -> neighbourhood D1 x -> neighbourhood D2 x. Proof. - unfold included, neighbourhood in |- *; intros; elim H0; intros; exists x0; - intros; unfold included in |- *; unfold included in H1; + unfold included, neighbourhood; intros; elim H0; intros; exists x0; + intros; unfold included; unfold included in H1; intros; apply (H _ (H1 _ H2)). Qed. @@ -193,12 +193,12 @@ Lemma open_set_P2 : forall D1 D2:R -> Prop, open_set D1 -> open_set D2 -> open_set (union_domain D1 D2). Proof. - unfold open_set in |- *; intros; unfold union_domain in H1; elim H1; intro. + unfold open_set; intros; unfold union_domain in H1; elim H1; intro. apply neighbourhood_P1 with D1. - unfold included, union_domain in |- *; tauto. + unfold included, union_domain; tauto. apply H; assumption. apply neighbourhood_P1 with D2. - unfold included, union_domain in |- *; tauto. + unfold included, union_domain; tauto. apply H0; assumption. Qed. @@ -206,53 +206,53 @@ Lemma open_set_P3 : forall D1 D2:R -> Prop, open_set D1 -> open_set D2 -> open_set (intersection_domain D1 D2). Proof. - unfold open_set in |- *; intros; unfold intersection_domain in H1; elim H1; + unfold open_set; intros; unfold intersection_domain in H1; elim H1; intros. assert (H4 := H _ H2); assert (H5 := H0 _ H3); - unfold intersection_domain in |- *; unfold neighbourhood in H4, H5; + unfold intersection_domain; unfold neighbourhood in H4, H5; elim H4; clear H; intros del1 H; elim H5; clear H0; intros del2 H0; cut (0 < Rmin del1 del2). intro; set (del := mkposreal _ H6). - exists del; unfold included in |- *; intros; unfold included in H, H0; + exists del; unfold included; intros; unfold included in H, H0; unfold disc in H, H0, H7. split. apply H; apply Rlt_le_trans with (pos del). apply H7. - unfold del in |- *; simpl in |- *; apply Rmin_l. + unfold del; simpl; apply Rmin_l. apply H0; apply Rlt_le_trans with (pos del). apply H7. - unfold del in |- *; simpl in |- *; apply Rmin_r. - unfold Rmin in |- *; case (Rle_dec del1 del2); intro. + unfold del; simpl; apply Rmin_r. + unfold Rmin; case (Rle_dec del1 del2); intro. apply (cond_pos del1). apply (cond_pos del2). Qed. Lemma open_set_P4 : open_set (fun x:R => False). Proof. - unfold open_set in |- *; intros; elim H. + unfold open_set; intros; elim H. Qed. Lemma open_set_P5 : open_set (fun x:R => True). Proof. - unfold open_set in |- *; intros; unfold neighbourhood in |- *. - exists (mkposreal 1 Rlt_0_1); unfold included in |- *; intros; trivial. + unfold open_set; intros; unfold neighbourhood. + exists (mkposreal 1 Rlt_0_1); unfold included; intros; trivial. Qed. Lemma disc_P1 : forall (x:R) (del:posreal), open_set (disc x del). Proof. intros; assert (H := open_set_P1 (disc x del)). elim H; intros; apply H1. - unfold eq_Dom in |- *; split. - unfold included, interior, disc in |- *; intros; + unfold eq_Dom; split. + unfold included, interior, disc; intros; cut (0 < del - Rabs (x - x0)). intro; set (del2 := mkposreal _ H3). - exists del2; unfold included in |- *; intros. + exists del2; unfold included; intros. apply Rle_lt_trans with (Rabs (x1 - x0) + Rabs (x0 - x)). replace (x1 - x) with (x1 - x0 + (x0 - x)); [ apply Rabs_triang | ring ]. replace (pos del) with (del2 + Rabs (x0 - x)). do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l. apply H4. - unfold del2 in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x0)); + unfold del2; simpl; rewrite <- (Rabs_Ropp (x - x0)); rewrite Ropp_minus_distr; ring. apply Rplus_lt_reg_r with (Rabs (x - x0)); rewrite Rplus_0_r; replace (Rabs (x - x0) + (del - Rabs (x - x0))) with (pos del); @@ -278,19 +278,19 @@ Proof. elim H3; intros. exists (disc x (mkposreal del2 H4)). intros; unfold included in H1; split. - unfold neighbourhood, disc in |- *. + unfold neighbourhood, disc. exists (mkposreal del2 H4). - unfold included in |- *; intros; assumption. - intros; apply H1; unfold disc in |- *; case (Req_dec y x); intro. - rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold included; intros; assumption. + intros; apply H1; unfold disc; case (Req_dec y x); intro. + rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos del1). apply H5; split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. apply (not_eq_sym (A:=R)); apply H7. unfold disc in H6; apply H6. - intros; unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; + intros; unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; intros. assert (H1 := H (disc (f x) (mkposreal eps H0))). cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)). @@ -299,10 +299,10 @@ Proof. intros del1 H7. exists (pos del1); split. apply (cond_pos del1). - intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl in |- *; - unfold R_dist in |- *; apply (H6 _ (H7 _ H10)). - unfold neighbourhood, disc in |- *; exists (mkposreal eps H0); - unfold included in |- *; intros; assumption. + intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl; + unfold R_dist; apply (H6 _ (H7 _ H10)). + unfold neighbourhood, disc; exists (mkposreal eps H0); + unfold included; intros; assumption. Qed. Definition image_rec (f:R -> R) (D:R -> Prop) (x:R) : Prop := D (f x). @@ -312,13 +312,13 @@ Lemma continuity_P2 : forall (f:R -> R) (D:R -> Prop), continuity f -> open_set D -> open_set (image_rec f D). Proof. - intros; unfold open_set in H0; unfold open_set in |- *; intros; + intros; unfold open_set in H0; unfold open_set; intros; assert (H2 := continuity_P1 f x); elim H2; intros H3 _; - assert (H4 := H3 (H x)); unfold neighbourhood, image_rec in |- *; + assert (H4 := H3 (H x)); unfold neighbourhood, image_rec; unfold image_rec in H1; assert (H5 := H4 D (H0 (f x) H1)); elim H5; intros V0 H6; elim H6; intros; unfold neighbourhood in H7; elim H7; intros del H9; exists del; unfold included in H9; - unfold included in |- *; intros; apply (H8 _ (H9 _ H10)). + unfold included; intros; apply (H8 _ (H9 _ H10)). Qed. (**********) @@ -329,9 +329,9 @@ Lemma continuity_P3 : Proof. intros; split. intros; apply continuity_P2; assumption. - intros; unfold continuity in |- *; unfold continuity_pt in |- *; - unfold continue_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros; unfold continuity; unfold continuity_pt; + unfold continue_in; unfold limit1_in; + unfold limit_in; simpl; unfold R_dist; intros; cut (open_set (disc (f x) (mkposreal _ H0))). intro; assert (H2 := H _ H1). unfold open_set, image_rec in H2; cut (disc (f x) (mkposreal _ H0) (f x)). @@ -340,7 +340,7 @@ Proof. exists (pos del); split. apply (cond_pos del). intros; unfold included in H5; apply H5; elim H6; intros; apply H8. - unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + unfold disc; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0. apply disc_P1. Qed. @@ -358,23 +358,23 @@ Proof. cut (0 < D / 2). intro; exists (disc x (mkposreal _ H)). exists (disc y (mkposreal _ H)); split. - unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *; + unfold neighbourhood; exists (mkposreal _ H); unfold included; tauto. split. - unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *; + unfold neighbourhood; exists (mkposreal _ H); unfold included; tauto. - red in |- *; intro; elim H0; intros; unfold intersection_domain in H1; + red; intro; elim H0; intros; unfold intersection_domain in H1; elim H1; intros. cut (D < D). intro; elim (Rlt_irrefl _ H4). - change (Rabs (x - y) < D) in |- *; + change (Rabs (x - y) < D); apply Rle_lt_trans with (Rabs (x - x0) + Rabs (x0 - y)). replace (x - y) with (x - x0 + (x0 - y)); [ apply Rabs_triang | ring ]. rewrite (double_var D); apply Rplus_lt_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2. apply H3. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. - unfold D in |- *; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep). + unfold Rdiv; apply Rmult_lt_0_compat. + unfold D; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep). apply Rinv_0_lt_compat; prove_sup0. Qed. @@ -404,7 +404,7 @@ Lemma restriction_family : (exists y : R, (fun z1 z2:R => f z1 z2 /\ D z1) x y) -> intersection_domain (ind f) D x. Proof. - intros; elim H; intros; unfold intersection_domain in |- *; elim H0; intros; + intros; elim H; intros; unfold intersection_domain; elim H0; intros; split. apply (cond_fam f0); exists x0; assumption. assumption. @@ -424,19 +424,19 @@ Lemma family_P1 : forall (f:family) (D:R -> Prop), family_open_set f -> family_open_set (subfamily f D). Proof. - unfold family_open_set in |- *; intros; unfold subfamily in |- *; - simpl in |- *; assert (H0 := classic (D x)). + unfold family_open_set; intros; unfold subfamily; + simpl; assert (H0 := classic (D x)). elim H0; intro. cut (open_set (f0 x) -> open_set (fun y:R => f0 x y /\ D x)). intro; apply H2; apply H. - unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3; + unfold open_set; unfold neighbourhood; intros; elim H3; intros; assert (H6 := H2 _ H4); elim H6; intros; exists x1; - unfold included in |- *; intros; split. + unfold included; intros; split. apply (H7 _ H8). assumption. cut (open_set (fun y:R => False) -> open_set (fun y:R => f0 x y /\ D x)). intro; apply H2; apply open_set_P4. - unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3; + unfold open_set; unfold neighbourhood; intros; elim H3; intros; elim H1; assumption. Qed. @@ -446,7 +446,7 @@ Definition bounded (D:R -> Prop) : Prop := Lemma open_set_P6 : forall D1 D2:R -> Prop, open_set D1 -> D1 =_D D2 -> open_set D2. Proof. - unfold open_set in |- *; unfold neighbourhood in |- *; intros. + unfold open_set; unfold neighbourhood; intros. unfold eq_Dom in H0; elim H0; intros. assert (H4 := H _ (H3 _ H1)). elim H4; intros. @@ -465,7 +465,7 @@ Proof. intro; assert (H3 := H1 H2); elim H3; intros D' H4; unfold covering_finite in H4; elim H4; intros; unfold family_finite in H6; unfold domain_finite in H6; elim H6; intros l H7; - unfold bounded in |- *; set (r := MaxRlist l). + unfold bounded; set (r := MaxRlist l). exists (- r); exists r; intros. unfold covering in H5; assert (H9 := H5 _ H8); elim H9; intros; unfold subfamily in H10; simpl in H10; elim H10; intros; @@ -484,25 +484,25 @@ Proof. left; apply H11. assumption. apply (MaxRlist_P1 l x0 H16). - unfold intersection_domain, D in |- *; tauto. - unfold covering_open_set in |- *; split. - unfold covering in |- *; intros; simpl in |- *; exists (Rabs x + 1); - unfold g in |- *; pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; + unfold intersection_domain, D; tauto. + unfold covering_open_set; split. + unfold covering; intros; simpl; exists (Rabs x + 1); + unfold g; pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. - unfold family_open_set in |- *; intro; case (Rtotal_order 0 x); intro. + unfold family_open_set; intro; case (Rtotal_order 0 x); intro. apply open_set_P6 with (disc 0 (mkposreal _ H2)). apply disc_P1. - unfold eq_Dom in |- *; unfold f0 in |- *; simpl in |- *; - unfold g, disc in |- *; split. - unfold included in |- *; intros; unfold Rminus in H3; rewrite Ropp_0 in H3; + unfold eq_Dom; unfold f0; simpl; + unfold g, disc; split. + unfold included; intros; unfold Rminus in H3; rewrite Ropp_0 in H3; rewrite Rplus_0_r in H3; apply H3. - unfold included in |- *; intros; unfold Rminus in |- *; rewrite Ropp_0; + unfold included; intros; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply H3. apply open_set_P6 with (fun x:R => False). apply open_set_P4. - unfold eq_Dom in |- *; split. - unfold included in |- *; intros; elim H3. - unfold included, f0 in |- *; simpl in |- *; unfold g in |- *; intros; elim H2; + unfold eq_Dom; split. + unfold included; intros; elim H3. + unfold included, f0; simpl; unfold g; intros; elim H2; intro; [ rewrite <- H4 in H3; assert (H5 := Rabs_pos x0); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)) @@ -515,10 +515,10 @@ Lemma compact_P2 : forall X:R -> Prop, compact X -> closed_set X. Proof. intros; assert (H0 := closed_set_P1 X); elim H0; clear H0; intros _ H0; apply H0; clear H0. - unfold eq_Dom in |- *; split. + unfold eq_Dom; split. apply adherence_P1. - unfold included in |- *; unfold adherence in |- *; - unfold point_adherent in |- *; intros; unfold compact in H; + unfold included; unfold adherence; + unfold point_adherent; intros; unfold compact in H; assert (H1 := classic (X x)); elim H1; clear H1; intro. assumption. cut (forall y:R, X y -> 0 < Rabs (y - x) / 2). @@ -548,44 +548,44 @@ Proof. replace (y0 - x) with (y0 - y + (y - x)); [ apply Rabs_triang | ring ]. rewrite (double_var (Rabs (y0 - x))); apply Rplus_lt_compat; assumption. apply (MinRlist_P1 (AbsList l x) (Rabs (y0 - x) / 2)); apply AbsList_P1; - elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain in |- *; + elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain; split; assumption. assert (H11 := disc_P1 x (mkposreal alp H9)); unfold open_set in H11; apply H11. - unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + unfold disc; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply H9. - unfold alp in |- *; apply MinRlist_P2; intros; + unfold alp; apply MinRlist_P2; intros; assert (H10 := AbsList_P2 _ _ _ H9); elim H10; clear H10; intros z H10; elim H10; clear H10; intros; rewrite H11; apply H2; elim (H8 z); clear H8; intros; assert (H13 := H12 H10); unfold intersection_domain, D in H13; elim H13; clear H13; intros; assumption. - unfold covering_open_set in |- *; split. - unfold covering in |- *; intros; exists x0; simpl in |- *; unfold g in |- *; + unfold covering_open_set; split. + unfold covering; intros; exists x0; simpl; unfold g; split. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; unfold Rminus in H2; apply (H2 _ H5). apply H5. - unfold family_open_set in |- *; intro; simpl in |- *; unfold g in |- *; + unfold family_open_set; intro; simpl; unfold g; elim (classic (D x0)); intro. apply open_set_P6 with (disc x0 (mkposreal _ (H2 _ H5))). apply disc_P1. - unfold eq_Dom in |- *; split. - unfold included, disc in |- *; simpl in |- *; intros; split. + unfold eq_Dom; split. + unfold included, disc; simpl; intros; split. rewrite <- (Rabs_Ropp (x0 - x1)); rewrite Ropp_minus_distr; apply H6. apply H5. - unfold included, disc in |- *; simpl in |- *; intros; elim H6; intros; + unfold included, disc; simpl; intros; elim H6; intros; rewrite <- (Rabs_Ropp (x1 - x0)); rewrite Ropp_minus_distr; apply H7. apply open_set_P6 with (fun z:R => False). apply open_set_P4. - unfold eq_Dom in |- *; split. - unfold included in |- *; intros; elim H6. - unfold included in |- *; intros; elim H6; intros; elim H5; assumption. + unfold eq_Dom; split. + unfold included; intros; elim H6. + unfold included; intros; elim H6; intros; elim H5; assumption. intros; elim H3; intros; unfold g in H4; elim H4; clear H4; intros _ H4; apply H4. - intros; unfold Rdiv in |- *; apply Rmult_lt_0_compat. - apply Rabs_pos_lt; apply Rminus_eq_contra; red in |- *; intro; + intros; unfold Rdiv; apply Rmult_lt_0_compat. + apply Rabs_pos_lt; apply Rminus_eq_contra; red; intro; rewrite H3 in H2; elim H1; apply H2. apply Rinv_0_lt_compat; prove_sup0. Qed. @@ -593,29 +593,29 @@ Qed. (**********) Lemma compact_EMP : compact (fun _:R => False). Proof. - unfold compact in |- *; intros; exists (fun x:R => False); - unfold covering_finite in |- *; split. - unfold covering in |- *; intros; elim H0. - unfold family_finite in |- *; unfold domain_finite in |- *; exists nil; intro. + unfold compact; intros; exists (fun x:R => False); + unfold covering_finite; split. + unfold covering; intros; elim H0. + unfold family_finite; unfold domain_finite; exists nil; intro. split. - simpl in |- *; unfold intersection_domain in |- *; intros; elim H0. + simpl; unfold intersection_domain; intros; elim H0. elim H0; clear H0; intros _ H0; elim H0. - simpl in |- *; intro; elim H0. + simpl; intro; elim H0. Qed. Lemma compact_eqDom : forall X1 X2:R -> Prop, compact X1 -> X1 =_D X2 -> compact X2. Proof. - unfold compact in |- *; intros; unfold eq_Dom in H0; elim H0; clear H0; - unfold included in |- *; intros; assert (H3 : covering_open_set X1 f0). - unfold covering_open_set in |- *; unfold covering_open_set in H1; elim H1; + unfold compact; intros; unfold eq_Dom in H0; elim H0; clear H0; + unfold included; intros; assert (H3 : covering_open_set X1 f0). + unfold covering_open_set; unfold covering_open_set in H1; elim H1; clear H1; intros; split. - unfold covering in H1; unfold covering in |- *; intros; + unfold covering in H1; unfold covering; intros; apply (H1 _ (H0 _ H4)). apply H3. - elim (H _ H3); intros D H4; exists D; unfold covering_finite in |- *; + elim (H _ H3); intros D H4; exists D; unfold covering_finite; unfold covering_finite in H4; elim H4; intros; split. - unfold covering in H5; unfold covering in |- *; intros; + unfold covering in H5; unfold covering; intros; apply (H5 _ (H2 _ H7)). apply H6. Qed. @@ -624,7 +624,7 @@ Qed. Lemma compact_P3 : forall a b:R, compact (fun c:R => a <= c <= b). Proof. intros; case (Rle_dec a b); intro. - unfold compact in |- *; intros; + unfold compact; intros; set (A := fun x:R => @@ -647,92 +647,92 @@ Proof. rewrite H11 in H10; rewrite H11 in H8; unfold A in H9; elim H9; clear H9; intros; elim H12; clear H12; intros Dx H12; set (Db := fun x:R => Dx x \/ x = y0); exists Db; - unfold covering_finite in |- *; split. - unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12; + unfold covering_finite; split. + unfold covering; unfold covering_finite in H12; elim H12; clear H12; intros; unfold covering in H12; case (Rle_dec x0 x); intro. cut (a <= x0 <= x). intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1; - simpl in H16; simpl in |- *; unfold Db in |- *; elim H16; + simpl in H16; simpl; unfold Db; elim H16; clear H16; intros; split; [ apply H16 | left; apply H17 ]. split. elim H14; intros; assumption. assumption. - exists y0; simpl in |- *; split. - apply H8; unfold disc in |- *; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; + exists y0; simpl; split. + apply H8; unfold disc; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right. apply Rlt_trans with (b - x). - unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; + unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; auto with real. elim H10; intros H15 _; apply Rplus_lt_reg_r with (x - eps); replace (x - eps + (b - x)) with (b - eps); [ replace (x - eps + eps) with x; [ apply H15 | ring ] | ring ]. apply Rge_minus; apply Rle_ge; elim H14; intros _ H15; apply H15. - unfold Db in |- *; right; reflexivity. - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold Db; right; reflexivity. + unfold family_finite; unfold domain_finite; unfold covering_finite in H12; elim H12; clear H12; intros; unfold family_finite in H13; unfold domain_finite in H13; elim H13; clear H13; intros l H13; exists (cons y0 l); intro; split. intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0); clear H13; intros; case (Req_dec x0 y0); intro. - simpl in |- *; left; apply H16. - simpl in |- *; right; apply H13. - simpl in |- *; unfold intersection_domain in |- *; unfold Db in H14; + simpl; left; apply H16. + simpl; right; apply H13. + simpl; unfold intersection_domain; unfold Db in H14; decompose [and or] H14. split; assumption. elim H16; assumption. - intro; simpl in H14; elim H14; intro; simpl in |- *; - unfold intersection_domain in |- *. + intro; simpl in H14; elim H14; intro; simpl; + unfold intersection_domain. split. apply (cond_fam f0); rewrite H15; exists m; apply H6. - unfold Db in |- *; right; assumption. - simpl in |- *; unfold intersection_domain in |- *; elim (H13 x0). + unfold Db; right; assumption. + simpl; unfold intersection_domain; elim (H13 x0). intros _ H16; assert (H17 := H16 H15); simpl in H17; unfold intersection_domain in H17; split. elim H17; intros; assumption. - unfold Db in |- *; left; elim H17; intros; assumption. + unfold Db; left; elim H17; intros; assumption. set (m' := Rmin (m + eps / 2) b); cut (A m'). intro; elim H3; intros; unfold is_upper_bound in H13; assert (H15 := H13 m' H12); cut (m < m'). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H15 H16)). - unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro. - pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold m'; unfold Rmin; case (Rle_dec (m + eps / 2) b); intro. + pattern m at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. elim H4; intros. elim H17; intro. assumption. elim H11; assumption. - unfold A in |- *; split. + unfold A; split. split. apply Rle_trans with m. elim H4; intros; assumption. - unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro. - pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold m'; unfold Rmin; case (Rle_dec (m + eps / 2) b); intro. + pattern m at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. elim H4; intros. elim H13; intro. assumption. elim H11; assumption. - unfold m' in |- *; apply Rmin_r. + unfold m'; apply Rmin_r. unfold A in H9; elim H9; clear H9; intros; elim H12; clear H12; intros Dx H12; set (Db := fun x:R => Dx x \/ x = y0); exists Db; - unfold covering_finite in |- *; split. - unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12; + unfold covering_finite; split. + unfold covering; unfold covering_finite in H12; elim H12; clear H12; intros; unfold covering in H12; case (Rle_dec x0 x); intro. cut (a <= x0 <= x). intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1; - simpl in H16; simpl in |- *; unfold Db in |- *. + simpl in H16; simpl; unfold Db. elim H16; clear H16; intros; split; [ apply H16 | left; apply H17 ]. elim H14; intros; split; assumption. - exists y0; simpl in |- *; split. - apply H8; unfold disc in |- *; unfold Rabs in |- *; case (Rcase_abs (x0 - m)); + exists y0; simpl; split. + apply H8; unfold disc; unfold Rabs; case (Rcase_abs (x0 - m)); intro. rewrite Ropp_minus_distr; apply Rlt_trans with (m - x). - unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; + unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; auto with real. apply Rplus_lt_reg_r with (x - eps); replace (x - eps + (m - x)) with (m - eps). @@ -741,56 +741,56 @@ Proof. ring. ring. apply Rle_lt_trans with (m' - m). - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- m)); + unfold Rminus; do 2 rewrite <- (Rplus_comm (- m)); apply Rplus_le_compat_l; elim H14; intros; assumption. apply Rplus_lt_reg_r with m; replace (m + (m' - m)) with m'. apply Rle_lt_trans with (m + eps / 2). - unfold m' in |- *; apply Rmin_l. + unfold m'; apply Rmin_l. apply Rplus_lt_compat_l; 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; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite Rmult_1_l; pattern (pos eps) at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps). discrR. ring. - unfold Db in |- *; right; reflexivity. - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold Db; right; reflexivity. + unfold family_finite; unfold domain_finite; unfold covering_finite in H12; elim H12; clear H12; intros; unfold family_finite in H13; unfold domain_finite in H13; elim H13; clear H13; intros l H13; exists (cons y0 l); intro; split. intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0); clear H13; intros; case (Req_dec x0 y0); intro. - simpl in |- *; left; apply H16. - simpl in |- *; right; apply H13; simpl in |- *; - unfold intersection_domain in |- *; unfold Db in H14; + simpl; left; apply H16. + simpl; right; apply H13; simpl; + unfold intersection_domain; unfold Db in H14; decompose [and or] H14. split; assumption. elim H16; assumption. - intro; simpl in H14; elim H14; intro; simpl in |- *; - unfold intersection_domain in |- *. + intro; simpl in H14; elim H14; intro; simpl; + unfold intersection_domain. split. apply (cond_fam f0); rewrite H15; exists m; apply H6. - unfold Db in |- *; right; assumption. + unfold Db; right; assumption. elim (H13 x0); intros _ H16. assert (H17 := H16 H15). simpl in H17. unfold intersection_domain in H17. split. elim H17; intros; assumption. - unfold Db in |- *; left; elim H17; intros; assumption. + unfold Db; left; elim H17; intros; assumption. elim (classic (exists x : R, A x /\ m - eps < x <= m)); intro. assumption. elim H3; intros; cut (is_upper_bound A (m - eps)). intro; assert (H13 := H11 _ H12); cut (m - eps < m). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H14)). - pattern m at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *; + pattern m at 2; rewrite <- Rplus_0_r; unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_involutive; rewrite Ropp_0; apply (cond_pos eps). set (P := fun n:R => A n /\ m - eps < n <= m); assert (H12 := not_ex_all_not _ P H9); unfold P in H12; - unfold is_upper_bound in |- *; intros; + unfold is_upper_bound; intros; assert (H14 := not_and_or _ _ (H12 x)); elim H14; intro. elim H15; apply H13. @@ -803,44 +803,44 @@ Proof. unfold is_upper_bound in H3. split. apply (H3 _ H0). - apply (H4 b); unfold is_upper_bound in |- *; intros; unfold A in H5; elim H5; + apply (H4 b); unfold is_upper_bound; intros; unfold A in H5; elim H5; clear H5; intros H5 _; elim H5; clear H5; intros _ H5; apply H5. exists a; apply H0. - unfold bound in |- *; exists b; unfold is_upper_bound in |- *; intros; + unfold bound; exists b; unfold is_upper_bound; intros; unfold A in H1; elim H1; clear H1; intros H1 _; elim H1; clear H1; intros _ H1; apply H1. - unfold A in |- *; split. + unfold A; split. split; [ right; reflexivity | apply r ]. unfold covering_open_set in H; elim H; clear H; intros; unfold covering in H; cut (a <= a <= b). intro; elim (H _ H1); intros y0 H2; set (D' := fun x:R => x = y0); exists D'; - unfold covering_finite in |- *; split. - unfold covering in |- *; simpl in |- *; intros; cut (x = a). + unfold covering_finite; split. + unfold covering; simpl; intros; cut (x = a). intro; exists y0; split. rewrite H4; apply H2. - unfold D' in |- *; reflexivity. + unfold D'; reflexivity. elim H3; intros; apply Rle_antisym; assumption. - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold family_finite; unfold domain_finite; exists (cons y0 nil); intro; split. - simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; clear H3; + simpl; unfold intersection_domain; intro; elim H3; clear H3; intros; unfold D' in H4; left; apply H4. - simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; intro. + simpl; unfold intersection_domain; intro; elim H3; intro. split; [ rewrite H4; apply (cond_fam f0); exists a; apply H2 | apply H4 ]. elim H4. split; [ right; reflexivity | apply r ]. apply compact_eqDom with (fun c:R => False). apply compact_EMP. - unfold eq_Dom in |- *; split. - unfold included in |- *; intros; elim H. - unfold included in |- *; intros; elim H; clear H; intros; + unfold eq_Dom; split. + unfold included; intros; elim H. + unfold included; intros; elim H; clear H; intros; assert (H1 := Rle_trans _ _ _ H H0); elim n; apply H1. Qed. Lemma compact_P4 : forall X F:R -> Prop, compact X -> closed_set F -> included F X -> compact F. Proof. - unfold compact in |- *; intros; elim (classic (exists z : R, F z)); + unfold compact; intros; elim (classic (exists z : R, F z)); intro Hyp_F_NE. set (D := ind f0); set (g := f f0); unfold closed_set in H0. set (g' := fun x y:R => f0 x y \/ complementary F y /\ D x). @@ -848,61 +848,61 @@ Proof. cut (forall x:R, (exists y : R, g' x y) -> D' x). intro; set (f' := mkfamily D' g' H3); cut (covering_open_set X f'). intro; elim (H _ H4); intros DX H5; exists DX. - unfold covering_finite in |- *; unfold covering_finite in H5; elim H5; + unfold covering_finite; unfold covering_finite in H5; elim H5; clear H5; intros. split. - unfold covering in |- *; unfold covering in H5; intros. - elim (H5 _ (H1 _ H7)); intros y0 H8; exists y0; simpl in H8; simpl in |- *; + unfold covering; unfold covering in H5; intros. + elim (H5 _ (H1 _ H7)); intros y0 H8; exists y0; simpl in H8; simpl; elim H8; clear H8; intros. split. unfold g' in H8; elim H8; intro. apply H10. elim H10; intros H11 _; unfold complementary in H11; elim H11; apply H7. apply H9. - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold family_finite; unfold domain_finite; unfold family_finite in H6; unfold domain_finite in H6; elim H6; clear H6; intros l H6; exists l; intro; assert (H7 := H6 x); elim H7; clear H7; intros. split. - intro; apply H7; simpl in |- *; unfold intersection_domain in |- *; - simpl in H9; unfold intersection_domain in H9; unfold D' in |- *; + intro; apply H7; simpl; unfold intersection_domain; + simpl in H9; unfold intersection_domain in H9; unfold D'; apply H9. intro; assert (H10 := H8 H9); simpl in H10; unfold intersection_domain in H10; - simpl in |- *; unfold intersection_domain in |- *; + simpl; unfold intersection_domain; unfold D' in H10; apply H10. - unfold covering_open_set in |- *; unfold covering_open_set in H2; elim H2; + unfold covering_open_set; unfold covering_open_set in H2; elim H2; clear H2; intros. split. - unfold covering in |- *; unfold covering in H2; intros. + unfold covering; unfold covering in H2; intros. elim (classic (F x)); intro. - elim (H2 _ H6); intros y0 H7; exists y0; simpl in |- *; unfold g' in |- *; + elim (H2 _ H6); intros y0 H7; exists y0; simpl; unfold g'; left; assumption. cut (exists z : R, D z). - intro; elim H7; clear H7; intros x0 H7; exists x0; simpl in |- *; - unfold g' in |- *; right. + intro; elim H7; clear H7; intros x0 H7; exists x0; simpl; + unfold g'; right. split. - unfold complementary in |- *; apply H6. + unfold complementary; apply H6. apply H7. elim Hyp_F_NE; intros z0 H7. assert (H8 := H2 _ H7). elim H8; clear H8; intros t H8; exists t; apply (cond_fam f0); exists z0; apply H8. - unfold family_open_set in |- *; intro; simpl in |- *; unfold g' in |- *; + unfold family_open_set; intro; simpl; unfold g'; elim (classic (D x)); intro. apply open_set_P6 with (union_domain (f0 x) (complementary F)). apply open_set_P2. unfold family_open_set in H4; apply H4. apply H0. - unfold eq_Dom in |- *; split. - unfold included, union_domain, complementary in |- *; intros. + unfold eq_Dom; split. + unfold included, union_domain, complementary; intros. elim H6; intro; [ left; apply H7 | right; split; assumption ]. - unfold included, union_domain, complementary in |- *; intros. + unfold included, union_domain, complementary; intros. elim H6; intro; [ left; apply H7 | right; elim H7; intros; apply H8 ]. apply open_set_P6 with (f0 x). unfold family_open_set in H4; apply H4. - unfold eq_Dom in |- *; split. - unfold included, complementary in |- *; intros; left; apply H6. - unfold included, complementary in |- *; intros. + unfold eq_Dom; split. + unfold included, complementary; intros; left; apply H6. + unfold included, complementary; intros. elim H6; intro. apply H7. elim H7; intros _ H8; elim H5; apply H8. @@ -914,9 +914,9 @@ Proof. intro; apply (H3 f0 H2). apply compact_eqDom with (fun _:R => False). apply compact_EMP. - unfold eq_Dom in |- *; split. - unfold included in |- *; intros; elim H3. - assert (H3 := not_ex_all_not _ _ Hyp_F_NE); unfold included in |- *; intros; + unfold eq_Dom; split. + unfold included; intros; elim H3. + assert (H3 := not_ex_all_not _ _ Hyp_F_NE); unfold included; intros; elim (H3 x); apply H4. Qed. @@ -947,7 +947,7 @@ Lemma continuity_compact : forall (f:R -> R) (X:R -> Prop), (forall x:R, continuity_pt f x) -> compact X -> compact (image_dir f X). Proof. - unfold compact in |- *; intros; unfold covering_open_set in H1. + unfold compact; intros; unfold covering_open_set in H1. elim H1; clear H1; intros. set (D := ind f1). set (g := fun x y:R => image_rec f0 (f1 x) y). @@ -956,24 +956,24 @@ Proof. cut (covering_open_set X f'). intro; elim (H0 f' H4); intros D' H5; exists D'. unfold covering_finite in H5; elim H5; clear H5; intros; - unfold covering_finite in |- *; split. - unfold covering, image_dir in |- *; simpl in |- *; unfold covering in H5; + unfold covering_finite; split. + unfold covering, image_dir; simpl; unfold covering in H5; intros; elim H7; intros y H8; elim H8; intros; assert (H11 := H5 _ H10); simpl in H11; elim H11; intros z H12; exists z; unfold g in H12; unfold image_rec in H12; rewrite H9; apply H12. unfold family_finite in H6; unfold domain_finite in H6; - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold family_finite; unfold domain_finite; elim H6; intros l H7; exists l; intro; elim (H7 x); intros; split; intro. - apply H8; simpl in H10; simpl in |- *; apply H10. + apply H8; simpl in H10; simpl; apply H10. apply (H9 H10). - unfold covering_open_set in |- *; split. - unfold covering in |- *; intros; simpl in |- *; unfold covering in H1; - unfold image_dir in H1; unfold g in |- *; unfold image_rec in |- *; + unfold covering_open_set; split. + unfold covering; intros; simpl; unfold covering in H1; + unfold image_dir in H1; unfold g; unfold image_rec; apply H1. exists x; split; [ reflexivity | apply H4 ]. - unfold family_open_set in |- *; unfold family_open_set in H2; intro; - simpl in |- *; unfold g in |- *; + unfold family_open_set; unfold family_open_set in H2; intro; + simpl; unfold g; cut ((fun y:R => image_rec f0 (f1 x) y) = image_rec f0 (f1 x)). intro; rewrite H4. apply (continuity_P2 f0 (f1 x) H (H2 x)). @@ -1010,16 +1010,16 @@ Proof. assert (H2 : 0 < b - a). apply Rlt_Rminus; assumption. exists h; split. - unfold continuity in |- *; intro; case (Rtotal_order x a); intro. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros; exists (a - x); + unfold continuity; intro; case (Rtotal_order x a); intro. + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros; exists (a - x); split. - change (0 < a - x) in |- *; apply Rlt_Rminus; assumption. - intros; elim H5; clear H5; intros _ H5; unfold h in |- *. + change (0 < a - x); apply Rlt_Rminus; assumption. + intros; elim H5; clear H5; intros _ H5; unfold h. case (Rle_dec x a); intro. case (Rle_dec x0 a); intro. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. elim n; left; apply Rplus_lt_reg_r with (- x); do 2 rewrite (Rplus_comm (- x)); apply Rle_lt_trans with (Rabs (x0 - x)). apply RRle_abs. @@ -1030,23 +1030,23 @@ Proof. split; [ right; reflexivity | left; assumption ]. assert (H6 := H0 _ H5); unfold continuity_pt in H6; unfold continue_in in H6; unfold limit1_in in H6; unfold limit_in in H6; simpl in H6; - unfold R_dist in H6; unfold continuity_pt in |- *; - unfold continue_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + unfold R_dist in H6; unfold continuity_pt; + unfold continue_in; unfold limit1_in; + unfold limit_in; simpl; unfold R_dist; intros; elim (H6 _ H7); intros; exists (Rmin x0 (b - a)); split. - unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro. + unfold Rmin; case (Rle_dec x0 (b - a)); intro. elim H8; intros; assumption. - change (0 < b - a) in |- *; apply Rlt_Rminus; assumption. + change (0 < b - a); apply Rlt_Rminus; assumption. intros; elim H9; clear H9; intros _ H9; cut (x1 < b). - intro; unfold h in |- *; case (Rle_dec x a); intro. + intro; unfold h; case (Rle_dec x a); intro. case (Rle_dec x1 a); intro. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. case (Rle_dec x1 b); intro. elim H8; intros; apply H12; split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. - red in |- *; intro; elim n; right; symmetry in |- *; assumption. + red; intro; elim n; right; symmetry ; assumption. apply Rlt_le_trans with (Rmin x0 (b - a)). rewrite H4 in H9; apply H9. apply Rmin_l. @@ -1063,9 +1063,9 @@ Proof. split; left; assumption. assert (H7 := H0 _ H6); unfold continuity_pt in H7; unfold continue_in in H7; unfold limit1_in in H7; unfold limit_in in H7; simpl in H7; - unfold R_dist in H7; unfold continuity_pt in |- *; - unfold continue_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + unfold R_dist in H7; unfold continuity_pt; + unfold continue_in; unfold limit1_in; + unfold limit_in; simpl; unfold R_dist; intros; elim (H7 _ H8); intros; elim H9; clear H9; intros. assert (H11 : 0 < x - a). @@ -1073,7 +1073,7 @@ Proof. assert (H12 : 0 < b - x). apply Rlt_Rminus; assumption. exists (Rmin x0 (Rmin (x - a) (b - x))); split. - unfold Rmin in |- *; case (Rle_dec (x - a) (b - x)); intro. + unfold Rmin; case (Rle_dec (x - a) (b - x)); intro. case (Rle_dec x0 (x - a)); intro. assumption. assumption. @@ -1081,7 +1081,7 @@ Proof. assumption. assumption. intros; elim H13; clear H13; intros; cut (a < x1 < b). - intro; elim H15; clear H15; intros; unfold h in |- *; case (Rle_dec x a); + intro; elim H15; clear H15; intros; unfold h; case (Rle_dec x a); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). case (Rle_dec x b); intro. @@ -1115,16 +1115,16 @@ Proof. split; [ left; assumption | right; reflexivity ]. assert (H8 := H0 _ H7); unfold continuity_pt in H8; unfold continue_in in H8; unfold limit1_in in H8; unfold limit_in in H8; simpl in H8; - unfold R_dist in H8; unfold continuity_pt in |- *; - unfold continue_in in |- *; unfold limit1_in in |- *; - unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + unfold R_dist in H8; unfold continuity_pt; + unfold continue_in; unfold limit1_in; + unfold limit_in; simpl; unfold R_dist; intros; elim (H8 _ H9); intros; exists (Rmin x0 (b - a)); split. - unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro. + unfold Rmin; case (Rle_dec x0 (b - a)); intro. elim H10; intros; assumption. - change (0 < b - a) in |- *; apply Rlt_Rminus; assumption. + change (0 < b - a); apply Rlt_Rminus; assumption. intros; elim H11; clear H11; intros _ H11; cut (a < x1). - intro; unfold h in |- *; case (Rle_dec x a); intro. + intro; unfold h; case (Rle_dec x a); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). case (Rle_dec x1 a); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H12)). @@ -1132,15 +1132,15 @@ Proof. case (Rle_dec x1 b); intro. rewrite H6; elim H10; intros; elim r0; intro. apply H14; split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. - red in |- *; intro; rewrite <- H16 in H15; elim (Rlt_irrefl _ H15). + red; intro; rewrite <- H16 in H15; elim (Rlt_irrefl _ H15). rewrite H6 in H11; apply Rlt_le_trans with (Rmin x0 (b - a)). apply H11. apply Rmin_l. - rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H15; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. - rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. elim n1; right; assumption. rewrite H6 in H11; apply Ropp_lt_cancel; apply Rplus_lt_reg_r with b; @@ -1149,18 +1149,18 @@ Proof. apply Rlt_le_trans with (Rmin x0 (b - a)). assumption. apply Rmin_r. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros; exists (x - b); + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros; exists (x - b); split. - change (0 < x - b) in |- *; apply Rlt_Rminus; assumption. + change (0 < x - b); apply Rlt_Rminus; assumption. intros; elim H8; clear H8; intros. assert (H10 : b < x0). apply Ropp_lt_cancel; apply Rplus_lt_reg_r with x; apply Rle_lt_trans with (Rabs (x0 - x)). rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. assumption. - unfold h in |- *; case (Rle_dec x a); intro. + unfold h; case (Rle_dec x a); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). case (Rle_dec x b); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H6)). @@ -1168,8 +1168,8 @@ Proof. elim (Rlt_irrefl _ (Rlt_trans _ _ _ H1 (Rlt_le_trans _ _ _ H10 r))). case (Rle_dec x0 b); intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)). - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. - intros; elim H3; intros; unfold h in |- *; case (Rle_dec c a); intro. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + intros; elim H3; intros; unfold h; case (Rle_dec c a); intro. elim r; intro. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 H6)). rewrite H6; reflexivity. @@ -1210,7 +1210,7 @@ Proof. intros; rewrite <- (Heq c H10); rewrite <- (Heq Mxx H9); intros; rewrite <- H8; unfold is_lub in H7; elim H7; clear H7; intros H7 _; unfold is_upper_bound in H7; apply H7; - unfold image_dir in |- *; exists c; split; [ reflexivity | apply H10 ]. + unfold image_dir; exists c; split; [ reflexivity | apply H10 ]. apply H9. elim (classic (image_dir g (fun c:R => a <= c <= b) M)); intro. assumption. @@ -1225,13 +1225,13 @@ Proof. cut (is_upper_bound (image_dir g (fun c:R => a <= c <= b)) (M - eps)). intro; assert (H12 := H10 _ H11); cut (M - eps < M). intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H12 H13)). - pattern M at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *; + pattern M at 2; rewrite <- Rplus_0_r; unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_0; rewrite Ropp_involutive; apply (cond_pos eps). - unfold is_upper_bound, image_dir in |- *; intros; cut (x <= M). + unfold is_upper_bound, image_dir; intros; cut (x <= M). intro; case (Rle_dec x (M - eps)); intro. apply r. - elim (H9 x); unfold intersection_domain, disc, image_dir in |- *; split. + elim (H9 x); unfold intersection_domain, disc, image_dir; split. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right. apply Rplus_lt_reg_r with (x - eps); replace (x - eps + (M - x)) with (M - eps). @@ -1249,8 +1249,8 @@ Proof. ~ intersection_domain V (image_dir g (fun c:R => a <= c <= b)) y)). intro; elim H9; intros V H10; elim H10; clear H10; intros. unfold neighbourhood in H10; elim H10; intros del H12; exists del; intros; - red in |- *; intro; elim (H11 y). - unfold intersection_domain in |- *; unfold intersection_domain in H13; + red; intro; elim (H11 y). + unfold intersection_domain; unfold intersection_domain in H13; elim H13; clear H13; intros; split. apply (H12 _ H13). apply H14. @@ -1268,18 +1268,18 @@ Proof. split. apply H12. apply (not_ex_all_not _ _ H13). - red in |- *; intro; cut (adherence (image_dir g (fun c:R => a <= c <= b)) M). + red; intro; cut (adherence (image_dir g (fun c:R => a <= c <= b)) M). intro; elim (closed_set_P1 (image_dir g (fun c:R => a <= c <= b))); intros H11 _; assert (H12 := H11 H3). elim H8. unfold eq_Dom in H12; elim H12; clear H12; intros. apply (H13 _ H10). apply H9. - exists (g a); unfold image_dir in |- *; exists a; split. + exists (g a); unfold image_dir; exists a; split. reflexivity. split; [ right; reflexivity | apply H ]. - unfold bound in |- *; unfold bounded in H4; elim H4; clear H4; intros m H4; - elim H4; clear H4; intros M H4; exists M; unfold is_upper_bound in |- *; + unfold bound; unfold bounded in H4; elim H4; clear H4; intros m H4; + elim H4; clear H4; intros M H4; exists M; unfold is_upper_bound; intros; elim (H4 _ H5); intros _ H6; apply H6. apply prolongement_C0; assumption. Qed. @@ -1327,8 +1327,8 @@ Proof. intros; elim H; intros; unfold f in H0; unfold adherence in H0; unfold point_adherent in H0; assert (H1 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0). - unfold neighbourhood, disc in |- *; exists (mkposreal _ Rlt_0_1); - unfold included in |- *; trivial. + unfold neighbourhood, disc; exists (mkposreal _ Rlt_0_1); + unfold included; trivial. elim (H0 _ H1); intros; unfold intersection_domain in H2; elim H2; intros; elim H4; intros; apply H6. Qed. @@ -1345,17 +1345,17 @@ Lemma ValAdh_un_prop : forall (un:nat -> R) (x:R), ValAdh un x <-> ValAdh_un un x. Proof. intros; split; intro. - unfold ValAdh in H; unfold ValAdh_un in |- *; - unfold intersection_family in |- *; simpl in |- *; - intros; elim H0; intros N H1; unfold adherence in |- *; - unfold point_adherent in |- *; intros; elim (H V N H2); - intros; exists (un x0); unfold intersection_domain in |- *; + unfold ValAdh in H; unfold ValAdh_un; + unfold intersection_family; simpl; + intros; elim H0; intros N H1; unfold adherence; + unfold point_adherent; intros; elim (H V N H2); + intros; exists (un x0); unfold intersection_domain; elim H3; clear H3; intros; split. assumption. split. exists x0; split; [ reflexivity | rewrite H1; apply (le_INR _ _ H3) ]. exists N; assumption. - unfold ValAdh in |- *; intros; unfold ValAdh_un in H; + unfold ValAdh; intros; unfold ValAdh_un in H; unfold intersection_family in H; simpl in H; assert (H1 : @@ -1376,8 +1376,8 @@ Qed. Lemma adherence_P4 : forall F G:R -> Prop, included F G -> included (adherence F) (adherence G). Proof. - unfold adherence, included in |- *; unfold point_adherent in |- *; intros; - elim (H0 _ H1); unfold intersection_domain in |- *; + unfold adherence, included; unfold point_adherent; intros; + elim (H0 _ H1); unfold intersection_domain; intros; elim H2; clear H2; intros; exists x0; split; [ assumption | apply (H _ H3) ]. Qed. @@ -1410,36 +1410,36 @@ Proof. intros; elim H2; intros; unfold f' in H3; elim H3; intros; assumption. set (f0 := mkfamily D' f' H2). unfold compact in H; assert (H3 : covering_open_set X f0). - unfold covering_open_set in |- *; split. - unfold covering in |- *; intros; unfold intersection_vide_in in H1; + unfold covering_open_set; split. + unfold covering; intros; unfold intersection_vide_in in H1; elim (H1 x); intros; unfold intersection_family in H5; assert (H6 := not_ex_all_not _ (fun y:R => forall y0:R, ind g y0 -> g y0 y) H5 x); assert (H7 := not_all_ex_not _ (fun y0:R => ind g y0 -> g y0 x) H6); elim H7; intros; exists x0; elim (imply_to_and _ _ H8); - intros; unfold f0 in |- *; simpl in |- *; unfold f' in |- *; + intros; unfold f0; simpl; unfold f'; split; [ apply H10 | apply H9 ]. - unfold family_open_set in |- *; intro; elim (classic (D' x)); intro. + unfold family_open_set; intro; elim (classic (D' x)); intro. apply open_set_P6 with (complementary (g x)). unfold family_closed_set in H0; unfold closed_set in H0; apply H0. - unfold f0 in |- *; simpl in |- *; unfold f' in |- *; unfold eq_Dom in |- *; + unfold f0; simpl; unfold f'; unfold eq_Dom; split. - unfold included in |- *; intros; split; [ apply H4 | apply H3 ]. - unfold included in |- *; intros; elim H4; intros; assumption. + unfold included; intros; split; [ apply H4 | apply H3 ]. + unfold included; intros; elim H4; intros; assumption. apply open_set_P6 with (fun _:R => False). apply open_set_P4. - unfold eq_Dom in |- *; unfold included in |- *; split; intros; + unfold eq_Dom; unfold included; split; intros; [ elim H4 | simpl in H4; unfold f' in H4; elim H4; intros; elim H3; assumption ]. elim (H _ H3); intros SF H4; exists SF; - unfold intersection_vide_finite_in in |- *; split. - unfold intersection_vide_in in |- *; simpl in |- *; intros; split. - intros; unfold included in |- *; intros; unfold intersection_vide_in in H1; + unfold intersection_vide_finite_in; split. + unfold intersection_vide_in; simpl; intros; split. + intros; unfold included; intros; unfold intersection_vide_in in H1; elim (H1 x); intros; elim H6; intros; apply H7. unfold intersection_domain in H5; elim H5; intros; assumption. assumption. elim (classic (exists y : R, intersection_domain (ind g) SF y)); intro Hyp'. - red in |- *; intro; elim H5; intros; unfold intersection_family in H6; + red; intro; elim H5; intros; unfold intersection_family in H6; simpl in H6. cut (X x0). intro; unfold covering_finite in H4; elim H4; clear H4; intros H4 _; @@ -1462,16 +1462,16 @@ Proof. cut (exists z : R, X z). intro; elim H5; clear H5; intros; unfold covering in H4; elim (H4 x0 H5); intros; simpl in H6; elim Hyp'; exists x1; elim H6; - intros; unfold intersection_domain in |- *; split. + intros; unfold intersection_domain; split. apply (cond_fam f0); exists x0; apply H7. apply H8. apply Hyp. unfold covering_finite in H4; elim H4; clear H4; intros; unfold family_finite in H5; unfold domain_finite in H5; - unfold family_finite in |- *; unfold domain_finite in |- *; + unfold family_finite; unfold domain_finite; elim H5; clear H5; intros l H5; exists l; intro; elim (H5 x); intros; split; intro; - [ apply H6; simpl in |- *; simpl in H8; apply H8 | apply (H7 H8) ]. + [ apply H6; simpl; simpl in H8; apply H8 | apply (H7 H8) ]. Qed. Theorem Bolzano_Weierstrass : @@ -1492,8 +1492,8 @@ Proof. intros; elim H2; intros; unfold g in H3; unfold adherence in H3; unfold point_adherent in H3. assert (H4 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0). - unfold neighbourhood in |- *; exists (mkposreal _ Rlt_0_1); - unfold included in |- *; trivial. + unfold neighbourhood; exists (mkposreal _ Rlt_0_1); + unfold included; trivial. elim (H3 _ H4); intros; unfold intersection_domain in H5; decompose [and] H5; assumption. set (f0 := mkfamily D g H2). @@ -1509,19 +1509,19 @@ Proof. unfold domain_finite in H9; elim H9; clear H9; intros l H9; set (r := MaxRlist l); cut (D r). intro; unfold D in H11; elim H11; intros; exists (un x); - unfold intersection_family in |- *; simpl in |- *; - unfold intersection_domain in |- *; intros; split. - unfold g in |- *; apply adherence_P1; split. + unfold intersection_family; simpl; + unfold intersection_domain; intros; split. + unfold g; apply adherence_P1; split. exists x; split; [ reflexivity - | rewrite <- H12; unfold r in |- *; apply MaxRlist_P1; elim (H9 y); intros; - apply H14; simpl in |- *; apply H13 ]. + | rewrite <- H12; unfold r; apply MaxRlist_P1; elim (H9 y); intros; + apply H14; simpl; apply H13 ]. elim H13; intros; assumption. elim H13; intros; assumption. elim (H9 r); intros. simpl in H12; unfold intersection_domain in H12; cut (In r l). intro; elim (H12 H13); intros; assumption. - unfold r in |- *; apply MaxRlist_P2; + unfold r; apply MaxRlist_P2; cut (exists z : R, intersection_domain (ind f0) SF z). intro; elim H13; intros; elim (H9 x); intros; simpl in H15; assert (H17 := H15 H14); exists x; apply H17. @@ -1541,16 +1541,16 @@ Proof. not_all_ex_not _ (fun y:R => intersection_domain D SF y -> g y x /\ SF y) H18); elim H19; intros; assert (H21 := imply_to_and _ _ H20); elim (H17 x0); elim H21; intros; assumption. - unfold intersection_vide_in in |- *; intros; split. - intro; simpl in H6; unfold f0 in |- *; simpl in |- *; unfold g in |- *; + unfold intersection_vide_in; intros; split. + intro; simpl in H6; unfold f0; simpl; unfold g; apply included_trans with (adherence X). apply adherence_P4. - unfold included in |- *; intros; elim H7; intros; elim H8; intros; elim H10; + unfold included; intros; elim H7; intros; elim H8; intros; elim H10; intros; rewrite H11; apply H0. apply adherence_P2; apply compact_P2; assumption. apply H4. - unfold family_closed_set in |- *; unfold f0 in |- *; simpl in |- *; - unfold g in |- *; intro; apply adherence_P3. + unfold family_closed_set; unfold f0; simpl; + unfold g; intro; apply adherence_P3. Qed. (********************************************************) @@ -1566,7 +1566,7 @@ Definition uniform_continuity (f:R -> R) (X:R -> Prop) : Prop := Lemma is_lub_u : forall (E:R -> Prop) (x y:R), is_lub E x -> is_lub E y -> x = y. Proof. - unfold is_lub in |- *; intros; elim H; elim H0; intros; apply Rle_antisym; + unfold is_lub; intros; elim H; elim H0; intros; apply Rle_antisym; [ apply (H4 _ H1) | apply (H2 _ H3) ]. Qed. @@ -1597,14 +1597,14 @@ Theorem Heine : Proof. intros f0 X H0 H; elim (domain_P1 X); intro Hyp. (* X is empty *) - unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1); + unfold uniform_continuity; intros; exists (mkposreal _ Rlt_0_1); intros; elim Hyp; exists x; assumption. elim Hyp; clear Hyp; intro Hyp. (* X has only one element *) - unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1); + unfold uniform_continuity; intros; exists (mkposreal _ Rlt_0_1); intros; elim Hyp; clear Hyp; intros; elim H4; clear H4; intros; assert (H6 := H5 _ H1); assert (H7 := H5 _ H2); - rewrite H6; rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite H6; rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos eps). (* X has at least two distinct elements *) assert @@ -1624,9 +1624,9 @@ Proof. elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H13 H14) r)). elim X_enc; clear X_enc; intros m X_enc; elim X_enc; clear X_enc; intros M X_enc; elim X_enc; clear X_enc Hyp; intros X_enc Hyp; - unfold uniform_continuity in |- *; intro; + unfold uniform_continuity; intro; assert (H1 : forall t:posreal, 0 < t / 2). - intro; unfold Rdiv in |- *; apply Rmult_lt_0_compat; + intro; unfold Rdiv; apply Rmult_lt_0_compat; [ apply (cond_pos t) | apply Rinv_0_lt_compat; prove_sup0 ]. set (g := @@ -1644,8 +1644,8 @@ Proof. apply H3. set (f' := mkfamily X g H2); unfold compact in H0; assert (H3 : covering_open_set X f'). - unfold covering_open_set in |- *; split. - unfold covering in |- *; intros; exists x; simpl in |- *; unfold g in |- *; + unfold covering_open_set; split. + unfold covering; intros; exists x; simpl; unfold g; split. assumption. assert (H4 := H _ H3); unfold continuity_pt in H4; unfold continue_in in H4; @@ -1658,22 +1658,22 @@ Proof. 0 < zeta <= M - m /\ (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2)); assert (H6 : bound E). - unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *; - unfold E in |- *; intros; elim H6; clear H6; intros H6 _; + unfold bound; exists (M - m); unfold is_upper_bound; + unfold E; intros; elim H6; clear H6; intros H6 _; elim H6; clear H6; intros _ H6; apply H6. assert (H7 : exists x : R, E x). - elim H5; clear H5; intros; exists (Rmin x0 (M - m)); unfold E in |- *; intros; + elim H5; clear H5; intros; exists (Rmin x0 (M - m)); unfold E; intros; split. split. - unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro. + unfold Rmin; case (Rle_dec x0 (M - m)); intro. apply H5. apply Rlt_Rminus; apply Hyp. apply Rmin_r. intros; case (Req_dec x z); intro. - rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (H1 eps). apply H7; split. - unfold D_x, no_cond in |- *; split; [ trivial | assumption ]. + unfold D_x, no_cond; split; [ trivial | assumption ]. apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H8 | apply Rmin_l ]. assert (H8 := completeness _ H6 H7); elim H8; clear H8; intros; cut (0 < x1 <= M - m). @@ -1690,15 +1690,15 @@ Proof. unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))). intro; assert (H16 := H14 _ H15); elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H10 H16)). - unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H13; + unfold is_upper_bound; intros; unfold is_upper_bound in H13; assert (H16 := H13 _ H15); case (Rle_dec x2 (Rabs (z - x))); intro. assumption. elim (H12 x2); split; [ split; [ auto with real | assumption ] | assumption ]. split. apply p. - unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; - rewrite Rabs_R0; simpl in |- *; unfold Rdiv in |- *; + unfold disc; unfold Rminus; rewrite Rplus_opp_r; + rewrite Rabs_R0; simpl; unfold Rdiv; apply Rmult_lt_0_compat; [ apply H8 | apply Rinv_0_lt_compat; prove_sup0 ]. elim H7; intros; unfold E in H8; elim H8; intros H9 _; elim H9; intros H10 _; unfold is_lub in p; elim p; intros; unfold is_upper_bound in H12; @@ -1706,13 +1706,13 @@ Proof. apply Rlt_le_trans with x2; [ assumption | apply (H11 _ H8) ]. apply H12; intros; unfold E in H13; elim H13; intros; elim H14; intros; assumption. - unfold family_open_set in |- *; intro; simpl in |- *; elim (classic (X x)); + unfold family_open_set; intro; simpl; elim (classic (X x)); intro. - unfold g in |- *; unfold open_set in |- *; intros; elim H4; clear H4; + unfold g; unfold open_set; intros; elim H4; clear H4; intros _ H4; elim H4; clear H4; intros; elim H4; clear H4; - intros; unfold neighbourhood in |- *; case (Req_dec x x0); + intros; unfold neighbourhood; case (Req_dec x x0); intro. - exists (mkposreal _ (H1 x1)); rewrite <- H6; unfold included in |- *; intros; + exists (mkposreal _ (H1 x1)); rewrite <- H6; unfold included; intros; split. assumption. exists x1; split. @@ -1721,24 +1721,24 @@ Proof. elim H5; intros; apply H8. apply H7. set (d := x1 / 2 - Rabs (x0 - x)); assert (H7 : 0 < d). - unfold d in |- *; apply Rlt_Rminus; elim H5; clear H5; intros; + unfold d; apply Rlt_Rminus; elim H5; clear H5; intros; unfold disc in H7; apply H7. - exists (mkposreal _ H7); unfold included in |- *; intros; split. + exists (mkposreal _ H7); unfold included; intros; split. assumption. exists x1; split. apply H4. elim H5; intros; split. assumption. - unfold disc in H8; simpl in H8; unfold disc in |- *; simpl in |- *; + unfold disc in H8; simpl in H8; unfold disc; simpl; unfold disc in H10; simpl in H10; apply Rle_lt_trans with (Rabs (x2 - x0) + Rabs (x0 - x)). replace (x2 - x) with (x2 - x0 + (x0 - x)); [ apply Rabs_triang | ring ]. - replace (x1 / 2) with (d + Rabs (x0 - x)); [ idtac | unfold d in |- *; ring ]. + replace (x1 / 2) with (d + Rabs (x0 - x)); [ idtac | unfold d; ring ]. do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l; apply H8. apply open_set_P6 with (fun _:R => False). apply open_set_P4. - unfold eq_Dom in |- *; unfold included in |- *; intros; split. + unfold eq_Dom; unfold included; intros; split. intros; elim H4. intros; unfold g in H4; elim H4; clear H4; intros H4 _; elim H3; apply H4. elim (H0 _ H3); intros DF H4; unfold covering_finite in H4; elim H4; clear H4; @@ -1776,10 +1776,10 @@ Proof. apply Rlt_trans with (pos_Rl l' i / 2). apply H21. elim H13; clear H13; intros; assumption. - unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. + unfold Rdiv; apply Rmult_lt_reg_l with 2. prove_sup0. rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. - rewrite Rmult_1_r; pattern (pos_Rl l' i) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite Rmult_1_r; pattern (pos_Rl l' i) at 1; rewrite <- Rplus_0_r; rewrite double; apply Rplus_lt_compat_l; apply H19. discrR. assert (H19 := H8 i H17); elim H19; clear H19; intros; rewrite <- H18 in H20; @@ -1791,15 +1791,15 @@ Proof. rewrite (double_var (pos_Rl l' i)); apply Rplus_lt_compat. apply Rlt_le_trans with (D / 2). rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H12. - unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2)); + unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_le_compat_l. left; apply Rinv_0_lt_compat; prove_sup0. - unfold D in |- *; apply MinRlist_P1; elim (pos_Rl_P2 l' (pos_Rl l' i)); + unfold D; apply MinRlist_P1; elim (pos_Rl_P2 l' (pos_Rl l' i)); intros; apply H26; exists i; split; [ rewrite <- H7; assumption | reflexivity ]. assumption. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; - [ unfold D in |- *; apply MinRlist_P2; intros; elim (pos_Rl_P2 l' y); intros; + unfold Rdiv; apply Rmult_lt_0_compat; + [ unfold D; apply MinRlist_P2; intros; elim (pos_Rl_P2 l' y); intros; elim (H10 H9); intros; elim H12; intros; rewrite H14; rewrite <- H7 in H13; elim (H8 x H13); intros; apply H15 @@ -1811,25 +1811,25 @@ Proof. 0 < zeta <= M - m /\ (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2)); assert (H11 : bound E). - unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *; - unfold E in |- *; intros; elim H11; clear H11; intros H11 _; + unfold bound; exists (M - m); unfold is_upper_bound; + unfold E; intros; elim H11; clear H11; intros H11 _; elim H11; clear H11; intros _ H11; apply H11. assert (H12 : exists x : R, E x). assert (H13 := H _ H9); unfold continuity_pt in H13; unfold continue_in in H13; unfold limit1_in in H13; unfold limit_in in H13; simpl in H13; unfold R_dist in H13; elim (H13 _ (H1 eps)); intros; elim H12; clear H12; - intros; exists (Rmin x0 (M - m)); unfold E in |- *; + intros; exists (Rmin x0 (M - m)); unfold E; intros; split. split; - [ unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro; + [ unfold Rmin; case (Rle_dec x0 (M - m)); intro; [ apply H12 | apply Rlt_Rminus; apply Hyp ] | apply Rmin_r ]. intros; case (Req_dec x z); intro. - rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + rewrite H16; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; apply (H1 eps). apply H14; split; - [ unfold D_x, no_cond in |- *; split; [ trivial | assumption ] + [ unfold D_x, no_cond; split; [ trivial | assumption ] | apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H15 | apply Rmin_l ] ]. assert (H13 := completeness _ H11 H12); elim H13; clear H13; intros; cut (0 < x0 <= M - m). @@ -1847,14 +1847,14 @@ Proof. unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))). intro; assert (H21 := H19 _ H20); elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H15 H21)). - unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H18; + unfold is_upper_bound; intros; unfold is_upper_bound in H18; assert (H21 := H18 _ H20); case (Rle_dec x1 (Rabs (z - x))); intro. assumption. elim (H17 x1); split. split; [ auto with real | assumption ]. assumption. - unfold included, g in |- *; intros; elim H15; intros; elim H17; intros; + unfold included, g; intros; elim H15; intros; elim H17; intros; decompose [and] H18; cut (x0 = x2). intro; rewrite H20; apply H22. unfold E in p; eapply is_lub_u. diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v index 3bb07fe0c..2d79a929f 100644 --- a/theories/Reals/Rtrigo_alt.v +++ b/theories/Reals/Rtrigo_alt.v @@ -46,12 +46,12 @@ Theorem pre_sin_bound : a <= 4 -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)). Proof. intros; case (Req_dec a 0); intro Hyp_a. - rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx in |- *; - apply sum_eq_R0 || (symmetry in |- *; apply sum_eq_R0); - intros; unfold sin_term in |- *; rewrite pow_add; - simpl in |- *; unfold Rdiv in |- *; rewrite Rmult_0_l; + rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx; + apply sum_eq_R0 || (symmetry ; apply sum_eq_R0); + intros; unfold sin_term; rewrite pow_add; + simpl; unfold Rdiv; rewrite Rmult_0_l; ring. - unfold sin_approx in |- *; cut (0 < a). + unfold sin_approx; cut (0 < a). intro Hyp_a_pos. rewrite (decomp_sum (sin_term a) (2 * n + 1)). rewrite (decomp_sum (sin_term a) (2 * (n + 1))). @@ -76,14 +76,14 @@ Proof. - sum_f_R0 (tg_alt Un) (S (2 * n))). intro; apply H2. apply alternated_series_ineq. - unfold Un_decreasing, Un in |- *; intro; + unfold Un_decreasing, Un; intro; cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))). intro; rewrite H3. replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)). - unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. + unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l. left; apply pow_lt; assumption. apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))). - rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H5 := eq_sym H4); elim (fact_neq_0 _ H5). rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1)))); rewrite Rmult_assoc; rewrite <- Rinv_l_sym. @@ -91,7 +91,7 @@ Proof. repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. rewrite Rmult_1_r. do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; - simpl in |- *; + simpl; replace (((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with @@ -106,7 +106,7 @@ Proof. left; prove_sup0. rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4); [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 20); pattern 20 at 1 in |- *; rewrite <- Rplus_0_r; + rewrite <- (Rplus_comm 20); pattern 20 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. apply Rplus_le_le_0_compat. repeat apply Rmult_le_pos. @@ -119,14 +119,14 @@ Proof. replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply INR_fact_neq_0. apply INR_fact_neq_0. - simpl in |- *; ring. + simpl; ring. ring. - assert (H3 := cv_speed_pow_fact a); unfold Un in |- *; unfold Un_cv in H3; - unfold R_dist in H3; unfold Un_cv in |- *; unfold R_dist in |- *; + assert (H3 := cv_speed_pow_fact a); unfold Un; unfold Un_cv in H3; + unfold R_dist in H3; unfold Un_cv; unfold R_dist; intros; elim (H3 eps H4); intros N H5. exists N; intros; apply H5. replace (2 * S n0 + 1)%nat with (S (2 * S n0)). - unfold ge in |- *; apply le_trans with (2 * S n0)%nat. + unfold ge; apply le_trans with (2 * S n0)%nat. apply le_trans with (2 * S N)%nat. apply le_trans with (2 * N)%nat. apply le_n_2n. @@ -137,49 +137,49 @@ Proof. assert (X := exist_sin (Rsqr a)); elim X; intros. cut (x = sin a / a). intro; rewrite H3 in p; unfold sin_in in p; unfold infinite_sum in p; - unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; + unfold R_dist in p; unfold Un_cv; unfold R_dist; intros. cut (0 < eps / Rabs a). intro; elim (p _ H5); intros N H6. exists N; intros. replace (sum_f_R0 (tg_alt Un) n0) with (a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))). - unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + unfold Rminus; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; rewrite Ropp_plus_distr; rewrite Ropp_involutive; repeat rewrite Rplus_assoc; rewrite (Rplus_comm a); rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a). apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. - pattern (/ Rabs a) at 1 in |- *; rewrite <- (Rabs_Rinv a Hyp_a). + pattern (/ Rabs a) at 1; rewrite <- (Rabs_Rinv a Hyp_a). rewrite <- Rabs_mult; rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_l | assumption ]; rewrite (Rmult_comm (/ a)); rewrite (Rmult_comm (/ Rabs a)); rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive; - unfold Rminus, Rdiv in H6; apply H6; unfold ge in |- *; + unfold Rminus, Rdiv in H6; apply H6; unfold ge; apply le_trans with n0; [ exact H7 | apply le_n_Sn ]. rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)). replace (sin_n 0) with 1. - simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *; + simpl; rewrite Rmult_1_r; unfold Rminus; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse; rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum; apply sum_eq. - intros; unfold sin_n, Un, tg_alt in |- *; + intros; unfold sin_n, Un, tg_alt; replace ((-1) ^ S i) with (- (-1) ^ i). replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a). - unfold Rdiv in |- *; ring. - rewrite pow_add; rewrite pow_Rsqr; simpl in |- *; ring. - simpl in |- *; ring. - unfold sin_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1; + unfold Rdiv; ring. + rewrite pow_add; rewrite pow_Rsqr; simpl; ring. + simpl; ring. + unfold sin_n; unfold Rdiv; simpl; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. apply lt_O_Sn. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. - unfold sin in |- *; case (exist_sin (Rsqr a)). + unfold sin; case (exist_sin (Rsqr a)). intros; cut (x = x0). - intro; rewrite H3; unfold Rdiv in |- *. - symmetry in |- *; apply Rinv_r_simpl_m; assumption. + intro; rewrite H3; unfold Rdiv. + symmetry ; apply Rinv_r_simpl_m; assumption. unfold sin_in in p; unfold sin_in in s; eapply uniqueness_sum. apply p. apply s. @@ -188,16 +188,16 @@ Proof. split; apply Ropp_le_contravar; assumption. replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ]. - apply sum_eq; intros; unfold sin_term, Un, tg_alt in |- *; + apply sum_eq; intros; unfold sin_term, Un, tg_alt; replace ((-1) ^ S i) with (-1 * (-1) ^ i). - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ]. apply sum_eq; intros. - unfold sin_term, Un, tg_alt in |- *; + unfold sin_term, Un, tg_alt; replace ((-1) ^ S i) with (-1 * (-1) ^ i). - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. reflexivity. replace (2 * (n + 1))%nat with (S (S (2 * n))). reflexivity. @@ -213,7 +213,7 @@ Proof. apply Rplus_le_reg_l with (- a). rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite (Rplus_comm (- a)); apply H3. - unfold sin_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1; + unfold sin_term; simpl; unfold Rdiv; rewrite Rinv_1; ring. replace (2 * (n + 1))%nat with (S (S (2 * n))). apply lt_O_Sn. @@ -221,7 +221,7 @@ Proof. replace (2 * n + 1)%nat with (S (2 * n)). apply lt_O_Sn. ring. - inversion H; [ assumption | elim Hyp_a; symmetry in |- *; assumption ]. + inversion H; [ assumption | elim Hyp_a; symmetry ; assumption ]. Qed. (**********) @@ -240,7 +240,7 @@ Proof. a <= 2 -> cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))). intros H a n; apply H. - intros; unfold cos_approx in |- *. + intros; unfold cos_approx. rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)). rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))). replace (cos_term a0 0) with 1. @@ -266,21 +266,21 @@ Proof. - sum_f_R0 (tg_alt Un) (S (2 * n0))). intro; apply H3. apply alternated_series_ineq. - unfold Un_decreasing in |- *; intro; unfold Un in |- *. + unfold Un_decreasing; intro; unfold Un. cut ((2 * S (S n1))%nat = S (S (2 * S n1))). intro; rewrite H4; replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)). - unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. + unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l. apply pow_le; assumption. apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))). - rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H6 := eq_sym H5); elim (fact_neq_0 _ H6). rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1))))); rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR; repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR; - simpl in |- *; + simpl; replace (((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1)) with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ]. @@ -293,9 +293,9 @@ Proof. discrR. assumption. left; prove_sup0. - pattern 4 at 1 in |- *; rewrite <- Rplus_0_r; replace 12 with (4 + 8); + pattern 4 at 1; rewrite <- Rplus_0_r; replace 12 with (4 + 8); [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 12); pattern 12 at 1 in |- *; rewrite <- Rplus_0_r; + rewrite <- (Rplus_comm 12); pattern 12 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. apply Rplus_le_le_0_compat. repeat apply Rmult_le_pos. @@ -308,12 +308,12 @@ Proof. replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply INR_fact_neq_0. apply INR_fact_neq_0. - simpl in |- *; ring. + simpl; ring. ring. - assert (H4 := cv_speed_pow_fact a0); unfold Un in |- *; unfold Un_cv in H4; - unfold R_dist in H4; unfold Un_cv in |- *; unfold R_dist in |- *; + assert (H4 := cv_speed_pow_fact a0); unfold Un; unfold Un_cv in H4; + unfold R_dist in H4; unfold Un_cv; unfold R_dist; intros; elim (H4 eps H5); intros N H6; exists N; intros. - apply H6; unfold ge in |- *; apply le_trans with (2 * S N)%nat. + apply H6; unfold ge; apply le_trans with (2 * S N)%nat. apply le_trans with (2 * N)%nat. apply le_n_2n. apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn. @@ -321,40 +321,40 @@ Proof. assert (X := exist_cos (Rsqr a0)); elim X; intros. cut (x = cos a0). intro; rewrite H4 in p; unfold cos_in in p; unfold infinite_sum in p; - unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; + unfold R_dist in p; unfold Un_cv; unfold R_dist; intros. elim (p _ H5); intros N H6. exists N; intros. replace (sum_f_R0 (tg_alt Un) n1) with (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)). - unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite Ropp_involutive; + unfold Rminus; rewrite Ropp_plus_distr; rewrite Ropp_involutive; repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1); rewrite (Rplus_comm (-1)); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive; unfold Rminus in H6; apply H6. - unfold ge in |- *; apply le_trans with n1. + unfold ge; apply le_trans with n1. exact H7. apply le_n_Sn. rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)). replace (cos_n 0) with 1. - simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *; + simpl; rewrite Rmult_1_r; unfold Rminus; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l; replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1) with (-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1); [ idtac | ring ]; rewrite scal_sum; apply sum_eq; - intros; unfold cos_n, Un, tg_alt in |- *. + intros; unfold cos_n, Un, tg_alt. replace ((-1) ^ S i) with (- (-1) ^ i). replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i). - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. rewrite pow_Rsqr; reflexivity. - simpl in |- *; ring. - unfold cos_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1; + simpl; ring. + unfold cos_n; unfold Rdiv; simpl; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. apply lt_O_Sn. - unfold cos in |- *; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p; + unfold cos; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p; unfold cos_in in c; eapply uniqueness_sum. apply p. apply c. @@ -363,15 +363,15 @@ Proof. split; apply Ropp_le_contravar; assumption. replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ]. - apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *; + apply sum_eq; intros; unfold cos_term, Un, tg_alt; replace ((-1) ^ S i) with (-1 * (-1) ^ i). - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ]; - apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *; + apply sum_eq; intros; unfold cos_term, Un, tg_alt; replace ((-1) ^ S i) with (-1 * (-1) ^ i). - unfold Rdiv in |- *; ring. + unfold Rdiv; ring. reflexivity. replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). reflexivity. @@ -386,7 +386,7 @@ Proof. apply Rplus_le_reg_l with (-1). rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite (Rplus_comm (-1)); apply H4. - unfold cos_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1; + unfold cos_term; simpl; unfold Rdiv; rewrite Rinv_1; ring. replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). apply lt_O_Sn. @@ -403,8 +403,8 @@ Proof. intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H. left; assumption. rewrite <- (Ropp_involutive 2); apply Ropp_le_contravar; exact H0. - intros; unfold cos_approx in |- *; apply sum_eq; intros; - unfold cos_term in |- *; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg; - unfold Rdiv in |- *; reflexivity. + intros; unfold cos_approx; apply sum_eq; intros; + unfold cos_term; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg; + unfold Rdiv; reflexivity. apply Ropp_0_gt_lt_contravar; assumption. Qed. diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v index e77ea6e2d..2691364b6 100644 --- a/theories/Reals/Rtrigo_calc.v +++ b/theories/Reals/Rtrigo_calc.v @@ -15,7 +15,7 @@ Local Open Scope R_scope. Lemma tan_PI : tan PI = 0. Proof. - unfold tan in |- *; rewrite sin_PI; rewrite cos_PI; unfold Rdiv in |- *; + unfold tan; rewrite sin_PI; rewrite cos_PI; unfold Rdiv; apply Rmult_0_l. Qed. @@ -23,12 +23,12 @@ Lemma sin_3PI2 : sin (3 * (PI / 2)) = -1. Proof. replace (3 * (PI / 2)) with (PI + PI / 2). rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; rewrite sin_PI2; ring. - pattern PI at 1 in |- *; rewrite (double_var PI); ring. + pattern PI at 1; rewrite (double_var PI); ring. Qed. Lemma tan_2PI : tan (2 * PI) = 0. Proof. - unfold tan in |- *; rewrite sin_2PI; unfold Rdiv in |- *; apply Rmult_0_l. + unfold tan; rewrite sin_2PI; unfold Rdiv; apply Rmult_0_l. Qed. Lemma sin_cos_PI4 : sin (PI / 4) = cos (PI / 4). @@ -37,9 +37,9 @@ Proof with trivial. replace (PI / 2 + PI / 4) with (- (PI / 4) + PI)... rewrite neg_sin; rewrite sin_neg; ring... cut (PI = PI / 2 + PI / 2); [ intro | apply double_var ]... - pattern PI at 2 3 in |- *; rewrite H; pattern PI at 2 3 in |- *; rewrite H... + pattern PI at 2 3; rewrite H; pattern PI at 2 3; rewrite H... assert (H0 : 2 <> 0); - [ discrR | unfold Rdiv in |- *; rewrite Rinv_mult_distr; try ring ]... + [ discrR | unfold Rdiv; rewrite Rinv_mult_distr; try ring ]... Qed. Lemma sin_PI3_cos_PI6 : sin (PI / 3) = cos (PI / 6). @@ -51,7 +51,7 @@ Proof with trivial. assert (H2 : 2 <> 0); [ discrR | idtac ]... apply Rmult_eq_reg_l with 6... rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... - unfold Rdiv in |- *; repeat rewrite Rmult_assoc... + unfold Rdiv; repeat rewrite Rmult_assoc... rewrite <- Rinv_l_sym... rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; @@ -68,7 +68,7 @@ Proof with trivial. assert (H2 : 2 <> 0); [ discrR | idtac ]... apply Rmult_eq_reg_l with 6... rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... - unfold Rdiv in |- *; repeat rewrite Rmult_assoc... + unfold Rdiv; repeat rewrite Rmult_assoc... rewrite <- Rinv_l_sym... rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; @@ -78,13 +78,13 @@ Qed. Lemma PI6_RGT_0 : 0 < PI / 6. Proof. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. Lemma PI6_RLT_PI2 : PI / 6 < PI / 2. Proof. - unfold Rdiv in |- *; apply Rmult_lt_compat_l. + unfold Rdiv; apply Rmult_lt_compat_l. apply PI_RGT_0. apply Rinv_lt_contravar; prove_sup. Qed. @@ -97,11 +97,11 @@ Proof with trivial. (2 * sin (PI / 6) * cos (PI / 6))... rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3)... rewrite sin_PI3_cos_PI6... - unfold Rdiv in |- *; rewrite Rmult_1_l; rewrite Rmult_assoc; - pattern 2 at 2 in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc; + unfold Rdiv; rewrite Rmult_1_l; rewrite Rmult_assoc; + pattern 2 at 2; rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_r... - unfold Rdiv in |- *; rewrite Rinv_mult_distr... + unfold Rdiv; rewrite Rinv_mult_distr... rewrite (Rmult_comm (/ 2)); rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_r... @@ -119,7 +119,7 @@ Lemma sqrt2_neq_0 : sqrt 2 <> 0. Proof. assert (Hyp : 0 < 2); [ prove_sup0 - | generalize (Rlt_le 0 2 Hyp); intro H1; red in |- *; intro H2; + | generalize (Rlt_le 0 2 Hyp); intro H1; red; intro H2; generalize (sqrt_eq_0 2 H1 H2); intro H; absurd (2 = 0); [ discrR | assumption ] ]. Qed. @@ -137,7 +137,7 @@ Proof. [ discrR | assert (Hyp : 0 < 3); [ prove_sup0 - | generalize (Rlt_le 0 3 Hyp); intro H1; red in |- *; intro H2; + | generalize (Rlt_le 0 3 Hyp); intro H1; red; intro H2; generalize (sqrt_eq_0 3 H1 H2); intro H; absurd (3 = 0); [ discrR | assumption ] ] ]. Qed. @@ -162,7 +162,7 @@ Proof. [ prove_sup0 | generalize (Rlt_le 0 3 Hyp2); intro H2; generalize (lt_INR_0 1 (neq_O_lt 1 H0)); - unfold INR in |- *; intro H3; + unfold INR; intro H3; generalize (Rplus_lt_compat_l 2 0 1 H3); rewrite Rplus_comm; rewrite Rplus_0_l; replace (2 + 1) with 3; [ intro H4; generalize (sqrt_lt_1 2 3 H1 H2 H4); clear H3; intro H3; @@ -173,7 +173,7 @@ Qed. Lemma PI4_RGT_0 : 0 < PI / 4. Proof. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. @@ -189,17 +189,17 @@ Proof with trivial. rewrite Rsqr_div... rewrite Rsqr_1; rewrite Rsqr_sqrt... assert (H : 2 <> 0); [ discrR | idtac ]... - unfold Rsqr in |- *; pattern (cos (PI / 4)) at 1 in |- *; + unfold Rsqr; pattern (cos (PI / 4)) at 1; rewrite <- sin_cos_PI4; replace (sin (PI / 4) * cos (PI / 4)) with (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4)))... rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2)... rewrite sin_PI2... apply Rmult_1_r... - unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr... + unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr... repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_r... - unfold Rdiv in |- *; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc... + unfold Rdiv; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc... rewrite <- Rinv_l_sym... rewrite Rmult_1_l... left; prove_sup... @@ -213,18 +213,18 @@ Qed. Lemma tan_PI4 : tan (PI / 4) = 1. Proof. - unfold tan in |- *; rewrite sin_cos_PI4. - unfold Rdiv in |- *; apply Rinv_r. - change (cos (PI / 4) <> 0) in |- *; rewrite cos_PI4; apply R1_sqrt2_neq_0. + unfold tan; rewrite sin_cos_PI4. + unfold Rdiv; apply Rinv_r. + change (cos (PI / 4) <> 0); rewrite cos_PI4; apply R1_sqrt2_neq_0. Qed. Lemma cos3PI4 : cos (3 * (PI / 4)) = -1 / sqrt 2. Proof with trivial. replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4... - unfold Rdiv in |- *; rewrite Ropp_mult_distr_l_reverse... - unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; - rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + unfold Rdiv; rewrite Ropp_mult_distr_l_reverse... + unfold Rminus; rewrite Ropp_involutive; pattern PI at 1; + rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; [ ring | discrR | discrR ]... Qed. @@ -233,8 +233,8 @@ Lemma sin3PI4 : sin (3 * (PI / 4)) = 1 / sqrt 2. Proof with trivial. replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... rewrite sin_shift; rewrite cos_neg; rewrite cos_PI4... - unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; - rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + unfold Rminus; rewrite Ropp_involutive; pattern PI at 1; + rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; [ ring | discrR | discrR ]... Qed. @@ -251,8 +251,8 @@ Proof with trivial. assert (H : 2 <> 0); [ discrR | idtac ]... assert (H1 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... rewrite Rsqr_div... - rewrite cos2; unfold Rsqr in |- *; rewrite sin_PI6; rewrite sqrt_def... - unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... + rewrite cos2; unfold Rsqr; rewrite sin_PI6; rewrite sqrt_def... + unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... rewrite Rmult_minus_distr_l; rewrite (Rmult_comm 3); repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... rewrite Rmult_1_l; rewrite Rmult_1_r... @@ -265,14 +265,14 @@ Qed. Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3. Proof. - unfold tan in |- *; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv in |- *; + unfold tan; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv; repeat rewrite Rmult_1_l; rewrite Rinv_mult_distr. rewrite Rinv_involutive. rewrite (Rmult_comm (/ 2)); rewrite Rmult_assoc; rewrite <- Rinv_r_sym. apply Rmult_1_r. discrR. discrR. - red in |- *; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1; + red; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1; elim (Rlt_irrefl 0 H1). apply Rinv_neq_0_compat; discrR. Qed. @@ -289,7 +289,7 @@ Qed. Lemma tan_PI3 : tan (PI / 3) = sqrt 3. Proof. - unfold tan in |- *; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv in |- *; + unfold tan; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv; rewrite Rmult_1_l; rewrite Rinv_involutive. rewrite Rmult_assoc; rewrite <- Rinv_l_sym. apply Rmult_1_r. @@ -300,7 +300,7 @@ Qed. Lemma sin_2PI3 : sin (2 * (PI / 3)) = sqrt 3 / 2. Proof. rewrite double; rewrite sin_plus; rewrite sin_PI3; rewrite cos_PI3; - unfold Rdiv in |- *; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2)); + unfold Rdiv; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2)); repeat rewrite <- Rmult_assoc; rewrite double_var; reflexivity. Qed. @@ -310,12 +310,12 @@ Proof with trivial. assert (H : 2 <> 0); [ discrR | idtac ]... assert (H0 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... rewrite double; rewrite cos_plus; rewrite sin_PI3; rewrite cos_PI3; - unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... + unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... rewrite Rmult_minus_distr_l; repeat rewrite Rmult_assoc; rewrite (Rmult_comm 2)... repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_r; rewrite <- Rinv_r_sym... - pattern 2 at 4 in |- *; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; + pattern 2 at 4; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_r; rewrite Ropp_mult_distr_r_reverse; rewrite Rmult_1_r... rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... @@ -329,7 +329,7 @@ Qed. Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3. Proof with trivial. assert (H : 2 <> 0); [ discrR | idtac ]... - unfold tan in |- *; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv in |- *; + unfold tan; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv; rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; rewrite <- Ropp_inv_permute... rewrite Rinv_involutive... @@ -341,21 +341,21 @@ Qed. Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2. Proof with trivial. replace (5 * (PI / 4)) with (PI / 4 + PI)... - rewrite neg_cos; rewrite cos_PI4; unfold Rdiv in |- *; + rewrite neg_cos; rewrite cos_PI4; unfold Rdiv; rewrite Ropp_mult_distr_l_reverse... - pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + pattern PI at 2; rewrite double_var; pattern PI at 2 3; rewrite double_var; assert (H : 2 <> 0); - [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... + [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]... Qed. Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2. Proof with trivial. replace (5 * (PI / 4)) with (PI / 4 + PI)... - rewrite neg_sin; rewrite sin_PI4; unfold Rdiv in |- *; + rewrite neg_sin; rewrite sin_PI4; unfold Rdiv; rewrite Ropp_mult_distr_l_reverse... - pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + pattern PI at 2; rewrite double_var; pattern PI at 2 3; rewrite double_var; assert (H : 2 <> 0); - [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... + [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]... Qed. Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)). @@ -367,7 +367,7 @@ Lemma Rgt_3PI2_0 : 0 < 3 * (PI / 2). Proof. apply Rmult_lt_0_compat; [ prove_sup0 - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ] ]. Qed. @@ -382,7 +382,7 @@ Proof. generalize (Rplus_lt_compat_l PI 0 (PI / 2) H1); replace (PI + PI / 2) with (3 * (PI / 2)). rewrite Rplus_0_r; intro H2; assumption. - pattern PI at 2 in |- *; rewrite double_var; ring. + pattern PI at 2; rewrite double_var; ring. Qed. Lemma Rlt_3PI2_2PI : 3 * (PI / 2) < 2 * PI. @@ -391,7 +391,7 @@ Proof. generalize (Rplus_lt_compat_l (3 * (PI / 2)) 0 (PI / 2) H1); replace (3 * (PI / 2) + PI / 2) with (2 * PI). rewrite Rplus_0_r; intro H2; assumption. - rewrite double; pattern PI at 1 2 in |- *; rewrite double_var; ring. + rewrite double; pattern PI at 1 2; rewrite double_var; ring. Qed. (***************************************************************) @@ -404,13 +404,13 @@ Definition toDeg (x:R) : R := x * plat * / PI. Lemma rad_deg : forall x:R, toRad (toDeg x) = x. Proof. - intro; unfold toRad, toDeg in |- *; + intro; unfold toRad, toDeg; replace (x * plat * / PI * PI * / plat) with (x * (plat * / plat) * (PI * / PI)); [ idtac | ring ]. repeat rewrite <- Rinv_r_sym. ring. apply PI_neq0. - unfold plat in |- *; discrR. + unfold plat; discrR. Qed. Lemma toRad_inj : forall x y:R, toRad x = toRad y -> x = y. @@ -420,7 +420,7 @@ Proof. apply Rmult_eq_reg_l with (/ plat). rewrite <- (Rmult_comm (x * PI)); rewrite <- (Rmult_comm (y * PI)); assumption. - apply Rinv_neq_0_compat; unfold plat in |- *; discrR. + apply Rinv_neq_0_compat; unfold plat; discrR. apply PI_neq0. Qed. @@ -435,7 +435,7 @@ Definition tand (x:R) : R := tan (toRad x). Lemma Rsqr_sin_cos_d_one : forall x:R, Rsqr (sind x) + Rsqr (cosd x) = 1. Proof. - intro x; unfold sind in |- *; unfold cosd in |- *; apply sin2_cos2. + intro x; unfold sind; unfold cosd; apply sin2_cos2. Qed. (***************************************************) @@ -447,10 +447,10 @@ Proof. intros; case (Rtotal_order 0 a); intro. left; apply sin_lb_gt_0; assumption. elim H1; intro. - rewrite <- H2; unfold sin_lb in |- *; unfold sin_approx in |- *; - unfold sum_f_R0 in |- *; unfold sin_term in |- *; + rewrite <- H2; unfold sin_lb; unfold sin_approx; + unfold sum_f_R0; unfold sin_term; repeat rewrite pow_ne_zero. - unfold Rdiv in |- *; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r; + unfold Rdiv; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r; repeat rewrite Rplus_0_r; right; reflexivity. discriminate. discriminate. diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v index 2486e140f..6cd9ce67b 100644 --- a/theories/Reals/Rtrigo_def.v +++ b/theories/Reals/Rtrigo_def.v @@ -27,7 +27,7 @@ Proof. intro; generalize (Alembert_C3 (fun n:nat => / INR (fact n)) x exp_cof_no_R0 Alembert_exp). - unfold Pser, exp_in in |- *. + unfold Pser, exp_in. trivial. Defined. @@ -36,24 +36,24 @@ Definition exp (x:R) : R := proj1_sig (exist_exp x). Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0. Proof. intros; apply pow_ne_zero. - red in |- *; intro; rewrite H0 in H; elim (lt_irrefl _ H). + red; intro; rewrite H0 in H; elim (lt_irrefl _ H). Qed. Lemma exist_exp0 : { l:R | exp_in 0 l }. Proof. exists 1. - unfold exp_in in |- *; unfold infinite_sum in |- *; intros. + unfold exp_in; unfold infinite_sum; intros. exists 0%nat. intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1. - unfold R_dist in |- *; replace (1 - 1) with 0; + unfold R_dist; replace (1 - 1) with 0; [ rewrite Rabs_R0; assumption | ring ]. induction n as [| n Hrecn]. - simpl in |- *; rewrite Rinv_1; ring. + simpl; rewrite Rinv_1; ring. rewrite tech5. rewrite <- Hrecn. - simpl in |- *. + simpl. ring. - unfold ge in |- *; apply le_O_n. + unfold ge; apply le_O_n. Defined. (* Value of [exp 0] *) @@ -61,7 +61,7 @@ Lemma exp_0 : exp 0 = 1. Proof. cut (exp_in 0 (exp 0)). cut (exp_in 0 1). - unfold exp_in in |- *; intros; eapply uniqueness_sum. + unfold exp_in; intros; eapply uniqueness_sum. apply H0. apply H. exact (proj2_sig exist_exp0). @@ -77,14 +77,14 @@ Definition tanh (x:R) : R := sinh x / cosh x. Lemma cosh_0 : cosh 0 = 1. Proof. - unfold cosh in |- *; rewrite Ropp_0; rewrite exp_0. - unfold Rdiv in |- *; rewrite <- Rinv_r_sym; [ reflexivity | discrR ]. + unfold cosh; rewrite Ropp_0; rewrite exp_0. + unfold Rdiv; rewrite <- Rinv_r_sym; [ reflexivity | discrR ]. Qed. Lemma sinh_0 : sinh 0 = 0. Proof. - unfold sinh in |- *; rewrite Ropp_0; rewrite exp_0. - unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; apply Rmult_0_l. + unfold sinh; rewrite Ropp_0; rewrite exp_0. + unfold Rminus, Rdiv; rewrite Rplus_opp_r; apply Rmult_0_l. Qed. Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)). @@ -92,8 +92,8 @@ Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)). Lemma simpl_cos_n : forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)). Proof. - intro; unfold cos_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. - rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr. + intro; unfold cos_n; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr. rewrite Rinv_involutive. replace ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1))) * @@ -101,7 +101,7 @@ Proof. ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1))) * INR (fact (2 * n)) * (-1) ^ 1); [ idtac | ring ]. rewrite <- Rinv_r_sym. - rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r. + rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r. replace (2 * (n + 1))%nat with (S (S (2 * n))); [ idtac | ring ]. do 2 rewrite fact_simpl; do 2 rewrite mult_INR; repeat rewrite Rinv_mult_distr; try (apply not_O_INR; discriminate). @@ -135,7 +135,7 @@ Proof. apply Rmult_le_reg_l with (IZR (Z.of_nat x)). assumption. rewrite <- Rinv_r_sym; - [ idtac | red in |- *; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ]. + [ idtac | red; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ]. apply Rmult_le_reg_l with (IZR (Z.of_nat (max x 1))). apply Rlt_le_trans with (IZR (Z.of_nat x)). assumption. @@ -145,14 +145,14 @@ Proof. rewrite Rmult_1_r; repeat rewrite <- INR_IZR_INZ; apply le_INR; apply le_max_l. rewrite <- INR_IZR_INZ; apply not_O_INR. - red in |- *; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat; + red; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat; [ intro | apply lt_O_Sn ]; assert (H8 := lt_le_trans _ _ _ H7 H6); rewrite H5 in H8; elim (lt_irrefl _ H8). - pattern eps at 1 in |- *; rewrite <- Rinv_involutive. + pattern eps at 1; rewrite <- Rinv_involutive. apply Rinv_lt_contravar. apply Rmult_lt_0_compat; [ apply Rinv_0_lt_compat; assumption | assumption ]. rewrite H3 in H0; assumption. - red in |- *; intro; rewrite H5 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H5 in H; elim (Rlt_irrefl _ H). apply Rlt_trans with (/ eps). apply Rinv_0_lt_compat; assumption. rewrite H3 in H0; assumption. @@ -166,10 +166,10 @@ Qed. Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0. Proof. - unfold Un_cv in |- *; intros. + unfold Un_cv; intros. assert (H0 := archimed_cor1 eps H). elim H0; intros; exists x. - intros; rewrite simpl_cos_n; unfold R_dist in |- *; unfold Rminus in |- *; + intros; rewrite simpl_cos_n; unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; rewrite Rabs_Ropp; rewrite Rabs_right. rewrite mult_INR; rewrite Rinv_mult_distr. @@ -177,7 +177,7 @@ Proof. intro; cut (/ INR (2 * n + 1) < eps). intro; rewrite <- (Rmult_1_l eps). apply Rmult_gt_0_lt_compat; try assumption. - change (0 < / INR (2 * n + 1)) in |- *; apply Rinv_0_lt_compat; + change (0 < / INR (2 * n + 1)); apply Rinv_0_lt_compat; apply lt_INR_0. replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. apply Rlt_0_1. @@ -221,7 +221,7 @@ Proof. Qed. Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0. - intro; unfold cos_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0. + intro; unfold cos_n; unfold Rdiv; apply prod_neq_R0. apply pow_nonzero; discrR. apply Rinv_neq_0_compat. apply INR_fact_neq_0. @@ -234,7 +234,7 @@ Definition cos_in (x l:R) : Prop := (**********) Lemma exist_cos : forall x:R, { l:R | cos_in x l }. intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). - unfold Pser, cos_in in |- *; trivial. + unfold Pser, cos_in; trivial. Qed. @@ -246,8 +246,8 @@ Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)). Lemma simpl_sin_n : forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)). Proof. - intro; unfold sin_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. - rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr. + intro; unfold sin_n; replace (S n) with (n + 1)%nat; [ idtac | ring ]. + rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr. rewrite Rinv_involutive. replace ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1) + 1)) * @@ -255,7 +255,7 @@ Proof. ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1) + 1)) * INR (fact (2 * n + 1)) * (-1) ^ 1); [ idtac | ring ]. rewrite <- Rinv_r_sym. - rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r; + rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r; replace (2 * (n + 1) + 1)%nat with (S (S (2 * n + 1))). do 2 rewrite fact_simpl; do 2 rewrite mult_INR; repeat rewrite Rinv_mult_distr. @@ -291,9 +291,9 @@ Qed. Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0. Proof. - unfold Un_cv in |- *; intros; assert (H0 := archimed_cor1 eps H). + unfold Un_cv; intros; assert (H0 := archimed_cor1 eps H). elim H0; intros; exists x. - intros; rewrite simpl_sin_n; unfold R_dist in |- *; unfold Rminus in |- *; + intros; rewrite simpl_sin_n; unfold R_dist; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; rewrite Rabs_Ropp; rewrite Rabs_right. rewrite mult_INR; rewrite Rinv_mult_distr. @@ -301,7 +301,7 @@ Proof. intro; cut (/ INR (2 * S n + 1) < eps). intro; rewrite <- (Rmult_1_l eps); rewrite (Rmult_comm (/ INR (2 * S n + 1))); apply Rmult_gt_0_lt_compat; try assumption. - change (0 < / INR (2 * S n + 1)) in |- *; apply Rinv_0_lt_compat; + change (0 < / INR (2 * S n + 1)); apply Rinv_0_lt_compat; apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n)); [ apply lt_O_Sn | ring ]. apply Rlt_0_1. @@ -329,7 +329,7 @@ Proof. apply not_O_INR; discriminate. apply not_O_INR; discriminate. apply not_O_INR; discriminate. - left; change (0 < / INR ((2 * S n + 1) * (2 * S n))) in |- *; + left; change (0 < / INR ((2 * S n + 1) * (2 * S n))); apply Rinv_0_lt_compat. apply lt_INR_0. replace ((2 * S n + 1) * (2 * S n))%nat with @@ -342,7 +342,7 @@ Defined. Lemma sin_no_R0 : forall n:nat, sin_n n <> 0. Proof. - intro; unfold sin_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0. + intro; unfold sin_n; unfold Rdiv; apply prod_neq_R0. apply pow_nonzero; discrR. apply Rinv_neq_0_compat; apply INR_fact_neq_0. Qed. @@ -355,7 +355,7 @@ Definition sin_in (x l:R) : Prop := Lemma exist_sin : forall x:R, { l:R | sin_in x l }. Proof. intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin). - unfold Pser, sin_n in |- *; trivial. + unfold Pser, sin_n; trivial. Defined. (***********************) @@ -368,40 +368,40 @@ Definition sin (x:R) : R := let (a,_) := exist_sin (Rsqr x) in x * a. Lemma cos_sym : forall x:R, cos x = cos (- x). Proof. - intros; unfold cos in |- *; replace (Rsqr (- x)) with (Rsqr x). + intros; unfold cos; replace (Rsqr (- x)) with (Rsqr x). reflexivity. apply Rsqr_neg. Qed. Lemma sin_antisym : forall x:R, sin (- x) = - sin x. Proof. - intro; unfold sin in |- *; replace (Rsqr (- x)) with (Rsqr x); + intro; unfold sin; replace (Rsqr (- x)) with (Rsqr x); [ idtac | apply Rsqr_neg ]. case (exist_sin (Rsqr x)); intros; ring. Qed. Lemma sin_0 : sin 0 = 0. Proof. - unfold sin in |- *; case (exist_sin (Rsqr 0)). + unfold sin; case (exist_sin (Rsqr 0)). intros; ring. Qed. Lemma exist_cos0 : { l:R | cos_in 0 l }. Proof. exists 1. - unfold cos_in in |- *; unfold infinite_sum in |- *; intros; exists 0%nat. + unfold cos_in; unfold infinite_sum; intros; exists 0%nat. intros. - unfold R_dist in |- *. + unfold R_dist. induction n as [| n Hrecn]. - unfold cos_n in |- *; simpl in |- *. - unfold Rdiv in |- *; rewrite Rinv_1. + unfold cos_n; simpl. + unfold Rdiv; rewrite Rinv_1. do 2 rewrite Rmult_1_r. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. rewrite tech5. replace (cos_n (S n) * 0 ^ S n) with 0. rewrite Rplus_0_r. - apply Hrecn; unfold ge in |- *; apply le_O_n. - simpl in |- *; ring. + apply Hrecn; unfold ge; apply le_O_n. + simpl; ring. Defined. (* Value of [cos 0] *) @@ -409,10 +409,10 @@ Lemma cos_0 : cos 0 = 1. Proof. cut (cos_in 0 (cos 0)). cut (cos_in 0 1). - unfold cos_in in |- *; intros; eapply uniqueness_sum. + unfold cos_in; intros; eapply uniqueness_sum. apply H0. apply H. exact (proj2_sig exist_cos0). - assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos in |- *; - pattern 0 at 1 in |- *; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ]. + assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos; + pattern 0 at 1; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ]. Qed. diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index a946bc462..ca3cb0108 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -20,8 +20,8 @@ Local Open Scope R_scope. Lemma Alembert_exp : Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0. Proof. - unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro. - split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *; + unfold Un_cv; intros; elim (Rgt_dec eps 1); intro. + split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). intro; rewrite (Rabs_pos_eq (/ INR (S n))). @@ -47,11 +47,11 @@ Proof. rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; assumption. unfold Rgt in H1; apply Rlt_le; assumption. - unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. (**) cut (0 <= up (/ eps - 1))%Z. intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; - rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (simpl_fact n); unfold R_dist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). intro; rewrite (Rabs_pos_eq (/ INR (S n))). @@ -80,10 +80,10 @@ Proof. elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. unfold Rgt in H1; apply Rlt_le; assumption. - unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. apply (le_O_IZR (up (/ eps - 1))); apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). - generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0; + generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle; intro; elim H0; clear H0; intro. left; unfold Rgt in H; generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); @@ -91,8 +91,8 @@ Proof. (Rinv_l eps (not_eq_sym (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); - intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; - unfold Rgt in |- *; assumption. + intro; fold (/ eps - 1 > 0); apply Rgt_minus; + unfold Rgt; assumption. right; rewrite H0; rewrite Rinv_1; symmetry; apply Rminus_diag_eq; auto. elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; assumption. diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v index a369d5ae2..93a889202 100644 --- a/theories/Reals/Rtrigo_reg.v +++ b/theories/Reals/Rtrigo_reg.v @@ -19,13 +19,13 @@ Local Open Scope R_scope. (**********) Lemma continuity_sin : continuity sin. Proof. - unfold continuity in |- *; intro. + unfold continuity; intro. assert (H0 := continuity_cos (PI / 2 - x)). unfold continuity_pt in H0; unfold continue_in in H0; unfold limit1_in in H0; unfold limit_in in H0; simpl in H0; unfold R_dist in H0; - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros. elim (H0 _ H); intros. exists x0; intros. elim H1; intros. @@ -34,9 +34,9 @@ Proof. intros; rewrite <- (cos_shift x); rewrite <- (cos_shift x1); apply H3. elim H4; intros. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. - red in |- *; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8; + red; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8; rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive x1); apply Ropp_eq_compat; apply Rplus_eq_reg_l with (PI / 2); apply H7. @@ -50,7 +50,7 @@ Lemma CVN_R_sin : (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) -> CVN_R fn. Proof. - unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r. + unfold CVN_R; unfold CVN_r; intros fn H r. exists (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)). cut { l:R | @@ -63,7 +63,7 @@ Proof. exists x. split. apply p. - intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult; + intros; rewrite H; unfold Rdiv; do 2 rewrite Rabs_mult; rewrite pow_1_abs; rewrite Rmult_1_l. cut (0 < / INR (fact (2 * n + 1))). intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))). @@ -80,11 +80,11 @@ Proof. apply Rinv_neq_0_compat; apply INR_fact_neq_0. apply pow_nonzero; assumption. assert (H1 := Alembert_sin). - unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv in |- *; intros. + unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv; intros. cut (0 < eps / Rsqr r). intro; elim (H1 _ H3); intros N0 H4. exists N0; intros. - unfold R_dist in |- *; assert (H6 := H4 _ H5). + unfold R_dist; assert (H6 := H4 _ H5). unfold R_dist in H5; replace (Rabs @@ -96,15 +96,15 @@ Proof. ((-1) ^ n / INR (fact (2 * n + 1))))). apply Rmult_lt_reg_l with (/ Rsqr r). apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. - pattern (/ Rsqr r) at 1 in |- *; rewrite <- (Rabs_right (/ Rsqr r)). + pattern (/ Rsqr r) at 1; rewrite <- (Rabs_right (/ Rsqr r)). rewrite <- Rabs_mult. rewrite Rmult_minus_distr_l. rewrite Rmult_0_r; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). apply H6. - unfold Rsqr in |- *; apply prod_neq_R0; assumption. + unfold Rsqr; apply prod_neq_R0; assumption. apply Rle_ge; left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. - unfold Rdiv in |- *; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult; + unfold Rdiv; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult; rewrite Rabs_Rabsolu; rewrite pow_1_abs. rewrite Rmult_1_l. repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l. @@ -126,10 +126,10 @@ Proof. replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r). do 2 rewrite <- Rmult_assoc. rewrite <- Rinv_l_sym. - unfold Rsqr in |- *; ring. + unfold Rsqr; ring. apply pow_nonzero; assumption. replace (2 * S n)%nat with (S (S (2 * n))). - simpl in |- *; ring. + simpl; ring. ring. apply Rle_ge; apply pow_le; left; apply (cond_pos r). apply Rle_ge; apply pow_le; left; apply (cond_pos r). @@ -142,16 +142,16 @@ Proof. apply INR_fact_neq_0. apply pow_nonzero; discrR. apply Rinv_neq_0_compat; apply INR_fact_neq_0. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption ]. - assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0; + assert (H0 := cond_pos r); red; intro; rewrite H1 in H0; elim (Rlt_irrefl _ H0). Qed. (** (sin h)/h -> 1 when h -> 0 *) Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1. Proof. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)). cut (CVN_R fn). @@ -167,58 +167,58 @@ Proof. elim (H2 _ H); intros alp H3. elim H3; intros. exists (mkposreal _ H4). - simpl in |- *; intros. - rewrite sin_0; rewrite Rplus_0_l; unfold Rminus in |- *; rewrite Ropp_0; + simpl; intros. + rewrite sin_0; rewrite Rplus_0_l; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r. cut (Rabs (SFL fn cv h - SFL fn cv 0) < eps). intro; cut (SFL fn cv 0 = 1). intro; cut (SFL fn cv h = sin h / h). intro; rewrite H9 in H8; rewrite H10 in H8. apply H8. - unfold SFL, sin in |- *. + unfold SFL, sin. case (cv h); intros. case (exist_sin (Rsqr h)); intros. - unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6). + unfold Rdiv; rewrite (Rinv_r_simpl_m h x0 H6). eapply UL_sequence. apply u. unfold sin_in in s; unfold sin_n, infinite_sum in s; - unfold SP, fn, Un_cv in |- *; intros. + unfold SP, fn, Un_cv; intros. elim (s _ H10); intros N0 H11. exists N0; intros. - unfold R_dist in |- *; unfold R_dist in H11. + unfold R_dist; unfold R_dist in H11. replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * h ^ (2 * k)) n) with (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * Rsqr h ^ i) n). apply H11; assumption. - apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr in |- *; + apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr; rewrite pow_sqr; reflexivity. - unfold SFL, sin in |- *. + unfold SFL, sin. case (cv 0); intros. eapply UL_sequence. apply u. - unfold SP, fn in |- *; unfold Un_cv in |- *; intros; exists 1%nat; intros. - unfold R_dist in |- *; + unfold SP, fn; unfold Un_cv; intros; exists 1%nat; intros. + unfold R_dist; replace (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k)) n) with 1. - unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. + unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. rewrite decomp_sum. - simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite Rinv_1; - rewrite Rmult_1_r; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + simpl; rewrite Rmult_1_r; unfold Rdiv; rewrite Rinv_1; + rewrite Rmult_1_r; pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_eq_compat_l. - symmetry in |- *; apply sum_eq_R0; intros. + symmetry ; apply sum_eq_R0; intros. rewrite Rmult_0_l; rewrite Rmult_0_r; reflexivity. unfold ge in H10; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H10 ]. apply H5. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. apply (not_eq_sym (A:=R)); apply H6. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply H7. - unfold Boule in |- *; unfold Rminus in |- *; rewrite Ropp_0; + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply H7. + unfold Boule; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_R0; apply (cond_pos r). - intros; unfold fn in |- *; + intros; unfold fn; replace (fun x:R => (-1) ^ n / INR (fact (2 * n + 1)) * x ^ (2 * n)) with (fct_cte ((-1) ^ n / INR (fact (2 * n + 1))) * pow_fct (2 * n))%F; [ idtac | reflexivity ]. @@ -229,13 +229,13 @@ Proof. apply (derivable_pt_pow (2 * n) y). apply (X r). apply (CVN_R_CVS _ X). - apply CVN_R_sin; unfold fn in |- *; reflexivity. + apply CVN_R_sin; unfold fn; reflexivity. Qed. (** ((cos h)-1)/h -> 0 when h -> 0 *) Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0. Proof. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. assert (H0 := derivable_pt_lim_sin_0). unfold derivable_pt_lim in H0. cut (0 < eps / 2). @@ -250,8 +250,8 @@ Proof. intro; set (delta := mkposreal _ H6). exists delta; intros. rewrite Rplus_0_l; replace (cos h - cos 0) with (-2 * Rsqr (sin (h / 2))). - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. - unfold Rdiv in |- *; do 2 rewrite Ropp_mult_distr_l_reverse. + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r. + unfold Rdiv; do 2 rewrite Ropp_mult_distr_l_reverse. rewrite Rabs_Ropp. replace (2 * Rsqr (sin (h * / 2)) * / h) with (sin (h / 2) * (sin (h / 2) / (h / 2) - 1) + sin (h / 2)). @@ -261,12 +261,12 @@ Proof. rewrite (double_var eps); apply Rplus_lt_compat. apply Rle_lt_trans with (Rabs (sin (h / 2) / (h / 2) - 1)). rewrite Rabs_mult; rewrite Rmult_comm; - pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2 in |- *; + pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2; rewrite <- Rmult_1_r; apply Rmult_le_compat_l. apply Rabs_pos. assert (H9 := SIN_bound (h / 2)). - unfold Rabs in |- *; case (Rcase_abs (sin (h / 2))); intro. - pattern 1 at 3 in |- *; rewrite <- (Ropp_involutive 1). + unfold Rabs; case (Rcase_abs (sin (h / 2))); intro. + pattern 1 at 3; rewrite <- (Ropp_involutive 1). apply Ropp_le_contravar. elim H9; intros; assumption. elim H9; intros; assumption. @@ -275,50 +275,50 @@ Proof. intro; assert (H11 := H2 _ H10 H9). rewrite Rplus_0_l in H11; rewrite sin_0 in H11. rewrite Rminus_0_r in H11; apply H11. - unfold Rdiv in |- *; apply prod_neq_R0. + unfold Rdiv; apply prod_neq_R0. apply H7. apply Rinv_neq_0_compat; discrR. apply Rlt_trans with (del / 2). - unfold Rdiv in |- *; rewrite Rabs_mult. + unfold Rdiv; rewrite Rabs_mult. rewrite (Rabs_right (/ 2)). do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. apply Rinv_0_lt_compat; prove_sup0. apply Rlt_le_trans with (pos delta). apply H8. - unfold delta in |- *; simpl in |- *; apply Rmin_l. + unfold delta; simpl; apply Rmin_l. apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. - rewrite <- (Rplus_0_r (del / 2)); pattern del at 1 in |- *; + rewrite <- (Rplus_0_r (del / 2)); pattern del at 1; rewrite (double_var del); apply Rplus_lt_compat_l; - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply (cond_pos del). apply Rinv_0_lt_compat; prove_sup0. elim H5; intros; assert (H11 := H10 (h / 2)). rewrite sin_0 in H11; do 2 rewrite Rminus_0_r in H11. apply H11. split. - unfold D_x, no_cond in |- *; split. + unfold D_x, no_cond; split. trivial. - apply (not_eq_sym (A:=R)); unfold Rdiv in |- *; apply prod_neq_R0. + apply (not_eq_sym (A:=R)); unfold Rdiv; apply prod_neq_R0. apply H7. apply Rinv_neq_0_compat; discrR. apply Rlt_trans with (del_c / 2). - unfold Rdiv in |- *; rewrite Rabs_mult. + unfold Rdiv; rewrite Rabs_mult. rewrite (Rabs_right (/ 2)). do 2 rewrite <- (Rmult_comm (/ 2)). apply Rmult_lt_compat_l. apply Rinv_0_lt_compat; prove_sup0. apply Rlt_le_trans with (pos delta). apply H8. - unfold delta in |- *; simpl in |- *; apply Rmin_r. + unfold delta; simpl; apply Rmin_r. apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. - rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2 in |- *; + rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2; rewrite (double_var del_c); apply Rplus_lt_compat_l. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply H9. apply Rinv_0_lt_compat; prove_sup0. - rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; - rewrite (Rmult_comm 2); unfold Rdiv, Rsqr in |- *. + rewrite (Rmult_comm 2); unfold Rdiv, Rsqr. repeat rewrite Rmult_assoc. repeat apply Rmult_eq_compat_l. rewrite Rinv_mult_distr. @@ -327,16 +327,16 @@ Proof. discrR. apply H7. apply Rinv_neq_0_compat; discrR. - pattern h at 2 in |- *; replace h with (2 * (h / 2)). + pattern h at 2; replace h with (2 * (h / 2)). rewrite (cos_2a_sin (h / 2)). - rewrite cos_0; unfold Rsqr in |- *; ring. - unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. + rewrite cos_0; unfold Rsqr; ring. + unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. discrR. - unfold Rmin in |- *; case (Rle_dec del del_c); intro. + unfold Rmin; case (Rle_dec del del_c); intro. apply (cond_pos del). elim H5; intros; assumption. apply continuity_sin. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. @@ -346,10 +346,10 @@ Proof. intro; assert (H0 := derivable_pt_lim_sin_0). assert (H := derivable_pt_lim_cos_0). unfold derivable_pt_lim in H0, H. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. cut (0 < eps / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ apply H1 | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H0 _ H2); intros alp1 H3. elim (H _ H2); intros alp2 H4. @@ -364,11 +364,11 @@ Proof. rewrite (double_var eps); apply Rplus_lt_compat. apply Rle_lt_trans with (Rabs ((cos h - 1) / h)). rewrite Rabs_mult; rewrite Rmult_comm; - pattern (Rabs ((cos h - 1) / h)) at 2 in |- *; rewrite <- Rmult_1_r; + pattern (Rabs ((cos h - 1) / h)) at 2; rewrite <- Rmult_1_r; apply Rmult_le_compat_l. apply Rabs_pos. assert (H8 := SIN_bound x); elim H8; intros. - unfold Rabs in |- *; case (Rcase_abs (sin x)); intro. + unfold Rabs; case (Rcase_abs (sin x)); intro. rewrite <- (Ropp_involutive 1). apply Ropp_le_contravar; assumption. assumption. @@ -378,14 +378,14 @@ Proof. apply H9. apply Rlt_le_trans with alp. apply H7. - unfold alp in |- *; apply Rmin_r. + unfold alp; apply Rmin_r. apply Rle_lt_trans with (Rabs (sin h / h - 1)). rewrite Rabs_mult; rewrite Rmult_comm; - pattern (Rabs (sin h / h - 1)) at 2 in |- *; rewrite <- Rmult_1_r; + pattern (Rabs (sin h / h - 1)) at 2; rewrite <- Rmult_1_r; apply Rmult_le_compat_l. apply Rabs_pos. assert (H8 := COS_bound x); elim H8; intros. - unfold Rabs in |- *; case (Rcase_abs (cos x)); intro. + unfold Rabs; case (Rcase_abs (cos x)); intro. rewrite <- (Ropp_involutive 1); apply Ropp_le_contravar; assumption. assumption. cut (Rabs h < alp1). @@ -394,8 +394,8 @@ Proof. apply H9. apply Rlt_le_trans with alp. apply H7. - unfold alp in |- *; apply Rmin_l. - rewrite sin_plus; unfold Rminus, Rdiv in |- *; + unfold alp; apply Rmin_l. + rewrite sin_plus; unfold Rminus, Rdiv; repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; repeat rewrite Rmult_assoc; repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. @@ -404,7 +404,7 @@ Proof. rewrite Ropp_mult_distr_r_reverse; rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_r; rewrite Rmult_1_l; rewrite Ropp_mult_distr_r_reverse; rewrite <- Ropp_mult_distr_l_reverse; apply Rplus_comm. - unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec alp1 alp2); intro. + unfold alp; unfold Rmin; case (Rle_dec alp1 alp2); intro. apply (cond_pos alp1). apply (cond_pos alp2). Qed. @@ -419,7 +419,7 @@ Proof. intros; generalize (H0 _ _ _ H2 H1); replace (comp sin (id + fct_cte (PI / 2))%F) with (fun x:R => sin (x + PI / 2)); [ idtac | reflexivity ]. - unfold derivable_pt_lim in |- *; intros. + unfold derivable_pt_lim; intros. elim (H3 eps H4); intros. exists x0. intros; rewrite <- (H (x + h)); rewrite <- (H x); apply H5; assumption. @@ -433,26 +433,26 @@ Qed. Lemma derivable_pt_sin : forall x:R, derivable_pt sin x. Proof. - unfold derivable_pt in |- *; intro. + unfold derivable_pt; intro. exists (cos x). apply derivable_pt_lim_sin. Qed. Lemma derivable_pt_cos : forall x:R, derivable_pt cos x. Proof. - unfold derivable_pt in |- *; intro. + unfold derivable_pt; intro. exists (- sin x). apply derivable_pt_lim_cos. Qed. Lemma derivable_sin : derivable sin. Proof. - unfold derivable in |- *; intro; apply derivable_pt_sin. + unfold derivable; intro; apply derivable_pt_sin. Qed. Lemma derivable_cos : derivable cos. Proof. - unfold derivable in |- *; intro; apply derivable_pt_cos. + unfold derivable; intro; apply derivable_pt_cos. Qed. Lemma derive_pt_sin : diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v index fb1b81ac3..b7f0d7ae3 100644 --- a/theories/Reals/SeqProp.v +++ b/theories/Reals/SeqProp.v @@ -36,7 +36,7 @@ Lemma decreasing_growing : forall Un:nat -> R, Un_decreasing Un -> Un_growing (opp_seq Un). Proof. intro. - unfold Un_growing, opp_seq, Un_decreasing in |- *. + unfold Un_growing, opp_seq, Un_decreasing. intros. apply Ropp_le_contravar. apply H. @@ -58,8 +58,8 @@ Proof. unfold Un_cv in p. unfold R_dist in p. unfold opp_seq in p. - unfold Un_cv in |- *. - unfold R_dist in |- *. + unfold Un_cv. + unfold R_dist. intros. elim (p eps H1); intros. exists x0; intros. @@ -77,7 +77,7 @@ Proof. apply completeness. assumption. exists (Un 0%nat). - unfold EUn in |- *. + unfold EUn. exists 0%nat; reflexivity. Qed. @@ -114,9 +114,9 @@ Proof. unfold bound in H. elim H; intros. unfold is_upper_bound in H0. - unfold has_ub in |- *. + unfold has_ub. exists x. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. apply H0. elim H1; intros. @@ -132,9 +132,9 @@ Proof. unfold bound in H. elim H; intros. unfold is_upper_bound in H0. - unfold has_lb in |- *. + unfold has_lb. exists x. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. apply H0. elim H1; intros. @@ -155,9 +155,9 @@ Lemma Wn_decreasing : forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_ub Un pr). Proof. intros. - unfold Un_decreasing in |- *. + unfold Un_decreasing. intro. - unfold sequence_ub in |- *. + unfold sequence_ub. assert (H := ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). assert (H0 := ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). elim H; intros. @@ -171,7 +171,7 @@ Proof. elim p; intros. apply H2. elim p0; intros. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. unfold is_upper_bound in H3. apply H3. @@ -190,7 +190,7 @@ Proof. assert (H7 := H3 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4). apply Rle_antisym; assumption. - unfold lub in |- *. + unfold lub. case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). trivial. cut @@ -204,7 +204,7 @@ Proof. (H7 := H3 (lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4). apply Rle_antisym; assumption. - unfold lub in |- *. + unfold lub. case (ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). trivial. Qed. @@ -213,9 +213,9 @@ Lemma Vn_growing : forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_lb Un pr). Proof. intros. - unfold Un_growing in |- *. + unfold Un_growing. intro. - unfold sequence_lb in |- *. + unfold sequence_lb. assert (H := lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)). assert (H0 := lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)). elim H; intros. @@ -230,14 +230,14 @@ Proof. apply Ropp_le_contravar. apply H2. elim p0; intros. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. unfold is_upper_bound in H3. apply H3. elim H5; intros. exists (1 + x2)%nat. unfold opp_seq in H6. - unfold opp_seq in |- *. + unfold opp_seq. replace (n + (1 + x2))%nat with (S n + x2)%nat. assumption. replace (S n) with (1 + n)%nat; [ ring | ring ]. @@ -254,7 +254,7 @@ Proof. (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))) . apply Ropp_eq_compat; apply Rle_antisym; assumption. - unfold glb in |- *. + unfold glb. case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)); simpl. intro; rewrite Ropp_involutive. trivial. @@ -273,7 +273,7 @@ Proof. (glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))) . apply Ropp_eq_compat; apply Rle_antisym; assumption. - unfold glb in |- *. + unfold glb. case (lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)); simpl. intro; rewrite Ropp_involutive. trivial. @@ -286,7 +286,7 @@ Lemma Vn_Un_Wn_order : Proof. intros. split. - unfold sequence_lb in |- *. + unfold sequence_lb. cut { l:R | is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l }. intro X. elim X; intros. @@ -298,7 +298,7 @@ Proof. apply Ropp_le_contravar. apply H. exists 0%nat. - unfold opp_seq in |- *. + unfold opp_seq. replace (n + 0)%nat with n; [ reflexivity | ring ]. cut (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat))) @@ -313,13 +313,13 @@ Proof. (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))) . apply Ropp_eq_compat; apply Rle_antisym; assumption. - unfold glb in |- *. + unfold glb. case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)); simpl. intro; rewrite Ropp_involutive. trivial. apply lb_to_glb. apply min_ss; assumption. - unfold sequence_ub in |- *. + unfold sequence_ub. cut { l:R | is_lub (EUn (fun i:nat => Un (n + i)%nat)) l }. intro X. elim X; intros. @@ -340,7 +340,7 @@ Proof. assert (H5 := H1 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2). apply Rle_antisym; assumption. - unfold lub in |- *. + unfold lub. case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)). intro; trivial. apply ub_to_lub. @@ -353,13 +353,13 @@ Lemma min_maj : Proof. intros. assert (H := Vn_Un_Wn_order Un pr1 pr2). - unfold has_ub in |- *. - unfold bound in |- *. + unfold has_ub. + unfold bound. unfold has_ub in pr1. unfold bound in pr1. elim pr1; intros. exists x. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. unfold is_upper_bound in H0. elim H1; intros. @@ -376,20 +376,20 @@ Lemma maj_min : Proof. intros. assert (H := Vn_Un_Wn_order Un pr1 pr2). - unfold has_lb in |- *. - unfold bound in |- *. + unfold has_lb. + unfold bound. unfold has_lb in pr2. unfold bound in pr2. elim pr2; intros. exists x. - unfold is_upper_bound in |- *. + unfold is_upper_bound. intros. unfold is_upper_bound in H0. elim H1; intros. rewrite H2. apply Rle_trans with (opp_seq Un x1). assert (H3 := H x1); elim H3; intros. - unfold opp_seq in |- *; apply Ropp_le_contravar. + unfold opp_seq; apply Ropp_le_contravar. assumption. apply H0. exists x1; reflexivity. @@ -399,7 +399,7 @@ Qed. Lemma cauchy_maj : forall Un:nat -> R, Cauchy_crit Un -> has_ub Un. Proof. intros. - unfold has_ub in |- *. + unfold has_ub. apply cauchy_bound. assumption. Qed. @@ -409,12 +409,12 @@ Lemma cauchy_opp : forall Un:nat -> R, Cauchy_crit Un -> Cauchy_crit (opp_seq Un). Proof. intro. - unfold Cauchy_crit in |- *. - unfold R_dist in |- *. + unfold Cauchy_crit. + unfold R_dist. intros. elim (H eps H0); intros. exists x; intros. - unfold opp_seq in |- *. + unfold opp_seq. rewrite <- Rabs_Ropp. replace (- (- Un n - - Un m)) with (Un n - Un m); [ apply H1; assumption | ring ]. @@ -424,7 +424,7 @@ Qed. Lemma cauchy_min : forall Un:nat -> R, Cauchy_crit Un -> has_lb Un. Proof. intros. - unfold has_lb in |- *. + unfold has_lb. assert (H0 := cauchy_opp _ H). apply cauchy_bound. assumption. @@ -485,7 +485,7 @@ Qed. Lemma not_Rlt : forall r1 r2:R, ~ r1 < r2 -> r1 >= r2. Proof. - intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge in |- *. + intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge. tauto. Qed. @@ -595,11 +595,11 @@ Qed. Lemma UL_sequence : forall (Un:nat -> R) (l1 l2:R), Un_cv Un l1 -> Un_cv Un l2 -> l1 = l2. Proof. - intros Un l1 l2; unfold Un_cv in |- *; unfold R_dist in |- *; intros. + intros Un l1 l2; unfold Un_cv; unfold R_dist; intros. apply cond_eq. intros; cut (0 < eps / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H (eps / 2) H2); intros. elim (H0 (eps / 2) H2); intros. @@ -609,8 +609,8 @@ Proof. [ apply Rabs_triang | ring ]. rewrite (double_var eps); apply Rplus_lt_compat. rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H3; - unfold ge, N in |- *; apply le_max_l. - apply H4; unfold ge, N in |- *; apply le_max_r. + unfold ge, N; apply le_max_l. + apply H4; unfold ge, N; apply le_max_r. Qed. (**********) @@ -618,10 +618,10 @@ Lemma CV_plus : forall (An Bn:nat -> R) (l1 l2:R), Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i + Bn i) (l1 + l2). Proof. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. cut (0 < eps / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (H (eps / 2) H2); intros. elim (H0 (eps / 2) H2); intros. @@ -632,10 +632,10 @@ Proof. apply Rle_lt_trans with (Rabs (An n - l1) + Rabs (Bn n - l2)). apply Rabs_triang. rewrite (double_var eps); apply Rplus_lt_compat. - apply H3; unfold ge in |- *; apply le_trans with N; - [ unfold N in |- *; apply le_max_l | assumption ]. - apply H4; unfold ge in |- *; apply le_trans with N; - [ unfold N in |- *; apply le_max_r | assumption ]. + apply H3; unfold ge; apply le_trans with N; + [ unfold N; apply le_max_l | assumption ]. + apply H4; unfold ge; apply le_trans with N; + [ unfold N; apply le_max_r | assumption ]. Qed. (**********) @@ -643,7 +643,7 @@ Lemma cv_cvabs : forall (Un:nat -> R) (l:R), Un_cv Un l -> Un_cv (fun i:nat => Rabs (Un i)) (Rabs l). Proof. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H eps H0); intros. exists x; intros. apply Rle_lt_trans with (Rabs (Un n - l)). @@ -656,15 +656,15 @@ Lemma CV_Cauchy : forall Un:nat -> R, { l:R | Un_cv Un l } -> Cauchy_crit Un. Proof. intros Un X; elim X; intros. - unfold Cauchy_crit in |- *; intros. + unfold Cauchy_crit; intros. unfold Un_cv in p; unfold R_dist in p. cut (0 < eps / 2); [ intro - | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + | unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. elim (p (eps / 2) H0); intros. exists x0; intros. - unfold R_dist in |- *; + unfold R_dist; apply Rle_lt_trans with (Rabs (Un n - x) + Rabs (x - Un m)). replace (Un n - Un m) with (Un n - x + (x - Un m)); [ apply Rabs_triang | ring ]. @@ -695,7 +695,7 @@ Proof. unfold is_upper_bound in H1. apply H1. exists n; reflexivity. - pattern x0 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + pattern x0 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply Rlt_0_1. apply Rle_trans with (Rabs (Un 0%nat)). apply Rabs_pos. @@ -717,7 +717,7 @@ Proof. assert (H1 := maj_by_pos An X). elim H1; intros M H2. elim H2; intros. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. cut (0 < eps / (2 * M)). intro. case (Req_dec l2 0); intro. @@ -744,24 +744,24 @@ Proof. rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). apply Rlt_trans with (eps / (2 * M)). apply H8; assumption. - unfold Rdiv in |- *; rewrite Rinv_mult_distr. + unfold Rdiv; rewrite Rinv_mult_distr. apply Rmult_lt_reg_l with 2. prove_sup0. replace (2 * (eps * (/ 2 * / M))) with (2 * / 2 * (eps * / M)); [ idtac | ring ]. rewrite <- Rinv_r_sym. rewrite Rmult_1_l; rewrite double. - pattern (eps * / M) at 1 in |- *; rewrite <- Rplus_0_r. + pattern (eps * / M) at 1; rewrite <- Rplus_0_r. apply Rplus_lt_compat_l; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; assumption ]. discrR. discrR. - red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). - red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). - rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus in |- *; + red; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). + red; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). + rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; reflexivity. replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); [ idtac | ring ]. - symmetry in |- *; apply Rabs_mult. + symmetry ; apply Rabs_mult. cut (0 < eps / (2 * Rabs l2)). intro. unfold Un_cv in H; unfold R_dist in H; unfold Un_cv in H0; @@ -790,36 +790,36 @@ Proof. rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). apply Rlt_le_trans with (eps / (2 * M)). apply H10. - unfold ge in |- *; apply le_trans with N. - unfold N in |- *; apply le_max_r. + unfold ge; apply le_trans with N. + unfold N; apply le_max_r. assumption. - unfold Rdiv in |- *; rewrite Rinv_mult_distr. + unfold Rdiv; rewrite Rinv_mult_distr. right; ring. discrR. - red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). - red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). + red; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). + red; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). apply Rmult_lt_reg_l with (/ Rabs l2). apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l; apply Rlt_le_trans with (eps / (2 * Rabs l2)). apply H9. - unfold ge in |- *; apply le_trans with N. - unfold N in |- *; apply le_max_l. + unfold ge; apply le_trans with N. + unfold N; apply le_max_l. assumption. - unfold Rdiv in |- *; right; rewrite Rinv_mult_distr. + unfold Rdiv; right; rewrite Rinv_mult_distr. ring. discrR. apply Rabs_no_R0; assumption. apply Rabs_no_R0; assumption. replace (An n * l2 - l1 * l2) with (l2 * (An n - l1)); - [ symmetry in |- *; apply Rabs_mult | ring ]. + [ symmetry ; apply Rabs_mult | ring ]. replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); - [ symmetry in |- *; apply Rabs_mult | ring ]. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + [ symmetry ; apply Rabs_mult | ring ]. + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; [ prove_sup0 | apply Rabs_pos_lt; assumption ]. - unfold Rdiv in |- *; apply Rmult_lt_0_compat; + unfold Rdiv; apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ]. @@ -858,15 +858,15 @@ Proof. intros; exists (k + (1 - k) / 2). split. split. - pattern k at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + pattern k at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. + unfold Rdiv; apply Rmult_lt_0_compat. apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; replace (k + (1 - k)) with 1; [ elim H; intros; assumption | ring ]. apply Rinv_0_lt_compat; prove_sup0. apply Rmult_lt_reg_l with 2. prove_sup0. - unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; - pattern 2 at 1 in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + unfold Rdiv; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; + pattern 2 at 1; rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_r; replace (2 * k + (1 - k)) with (1 + k); [ idtac | ring ]. elim H; intros. @@ -885,7 +885,7 @@ Proof. repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l; repeat rewrite Rplus_0_l; apply H4. apply Rle_ge; elim H; intros; assumption. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; elim H; intros; replace (k + (1 - k)) with 1; [ assumption | ring ]. apply Rinv_0_lt_compat; prove_sup0. @@ -910,12 +910,12 @@ Proof. apply Rle_lt_trans with (Rabs (Un N - l)). apply RRle_abs. apply H2. - unfold ge, N in |- *; apply le_max_r. - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + unfold ge, N; apply le_max_r. + unfold Rminus; do 2 rewrite <- (Rplus_comm (- l)); apply Rplus_le_compat_l. apply tech9. assumption. - unfold N in |- *; apply le_max_l. + unfold N; apply le_max_l. apply Rplus_lt_reg_r with l. rewrite Rplus_0_r. replace (l + (Un n - l)) with (Un n); [ assumption | ring ]. @@ -926,10 +926,10 @@ Lemma CV_opp : forall (An:nat -> R) (l:R), Un_cv An l -> Un_cv (opp_seq An) (- l). Proof. intros An l. - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H eps H0); intros. exists x; intros. - unfold opp_seq in |- *; replace (- An n - - l) with (- (An n - l)); + unfold opp_seq; replace (- An n - - l) with (- (An n - l)); [ rewrite Rabs_Ropp | ring ]. apply H1; assumption. Qed. @@ -954,10 +954,10 @@ Lemma CV_minus : Proof. intros. replace (fun i:nat => An i - Bn i) with (fun i:nat => An i + opp_seq Bn i). - unfold Rminus in |- *; apply CV_plus. + unfold Rminus; apply CV_plus. assumption. apply CV_opp; assumption. - unfold Rminus, opp_seq in |- *; reflexivity. + unfold Rminus, opp_seq; reflexivity. Qed. (** Un -> +oo *) @@ -969,10 +969,10 @@ Lemma cv_infty_cv_R0 : forall Un:nat -> R, (forall n:nat, Un n <> 0) -> cv_infty Un -> Un_cv (fun n:nat => / Un n) 0. Proof. - unfold cv_infty, Un_cv in |- *; unfold R_dist in |- *; intros. + unfold cv_infty, Un_cv; unfold R_dist; intros. elim (H0 (/ eps)); intros N0 H2. exists N0; intros. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite (Rabs_Rinv _ (H n)). apply Rmult_lt_reg_l with (Rabs (Un n)). apply Rabs_pos_lt; apply H. @@ -984,7 +984,7 @@ Proof. rewrite Rmult_1_r; apply Rlt_le_trans with (Un n). apply H2; assumption. apply RRle_abs. - red in |- *; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1). + red; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1). apply Rabs_no_R0; apply H. Qed. @@ -993,7 +993,7 @@ Lemma decreasing_prop : forall (Un:nat -> R) (m n:nat), Un_decreasing Un -> (m <= n)%nat -> Un n <= Un m. Proof. - unfold Un_decreasing in |- *; intros. + unfold Un_decreasing; intros. induction n as [| n Hrecn]. induction m as [| m Hrecm]. right; reflexivity. @@ -1016,17 +1016,17 @@ Proof. (Un_cv (fun n:nat => Rabs x ^ n / INR (fact n)) 0 -> Un_cv (fun n:nat => x ^ n / INR (fact n)) 0). intro; apply H. - unfold Un_cv in |- *; unfold R_dist in |- *; intros; case (Req_dec x 0); + unfold Un_cv; unfold R_dist; intros; case (Req_dec x 0); intro. exists 1%nat; intros. - rewrite H1; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite H1; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_R0; rewrite pow_ne_zero; - [ unfold Rdiv in |- *; rewrite Rmult_0_l; rewrite Rabs_R0; assumption - | red in |- *; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ]. + [ unfold Rdiv; rewrite Rmult_0_l; rewrite Rabs_R0; assumption + | red; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ]. assert (H2 := Rabs_pos_lt x H1); set (M := up (Rabs x)); cut (0 <= M)%Z. intro; elim (IZN M H3); intros M_nat H4. set (Un := fun n:nat => Rabs x ^ (M_nat + n) / INR (fact (M_nat + n))). - cut (Un_cv Un 0); unfold Un_cv in |- *; unfold R_dist in |- *; intros. + cut (Un_cv Un 0); unfold Un_cv; unfold R_dist; intros. elim (H5 eps H0); intros N H6. exists (M_nat + N)%nat; intros; cut (exists p : nat, (p >= N)%nat /\ n = (M_nat + p)%nat). @@ -1034,7 +1034,7 @@ Proof. elim H9; intros; rewrite H11; unfold Un in H6; apply H6; assumption. exists (n - M_nat)%nat. split. - unfold ge in |- *; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat; + unfold ge; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat; rewrite <- le_plus_minus. assumption. apply le_trans with (M_nat + N)%nat. @@ -1048,43 +1048,43 @@ Proof. intro; cut (Un_decreasing Un). intro; cut (forall n:nat, Un (S n) <= Vn n). intro; cut (Un_cv Vn 0). - unfold Un_cv in |- *; unfold R_dist in |- *; intros. + unfold Un_cv; unfold R_dist; intros. elim (H10 eps0 H5); intros N1 H11. exists (S N1); intros. cut (forall n:nat, 0 < Vn n). intro; apply Rle_lt_trans with (Rabs (Vn (pred n) - 0)). repeat rewrite Rabs_right. - unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + unfold Rminus; rewrite Ropp_0; do 2 rewrite Rplus_0_r; replace n with (S (pred n)). apply H9. - inversion H12; simpl in |- *; reflexivity. - apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + inversion H12; simpl; reflexivity. + apply Rle_ge; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; left; apply H13. - apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + apply Rle_ge; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; left; apply H7. - apply H11; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n; - [ unfold ge in H12; exact H12 | inversion H12; simpl in |- *; reflexivity ]. + apply H11; unfold ge; apply le_S_n; replace (S (pred n)) with n; + [ unfold ge in H12; exact H12 | inversion H12; simpl; reflexivity ]. intro; apply Rlt_le_trans with (Un (S n0)); [ apply H7 | apply H9 ]. cut (cv_infty (fun n:nat => INR (S n))). intro; cut (Un_cv (fun n:nat => / INR (S n)) 0). - unfold Un_cv, R_dist in |- *; intros; unfold Vn in |- *. + unfold Un_cv, R_dist; intros; unfold Vn. cut (0 < eps1 / (Rabs x * Un 0%nat)). intro; elim (H11 _ H13); intros N H14. exists N; intros; replace (Rabs x * (Un 0%nat / INR (S n)) - 0) with (Rabs x * Un 0%nat * (/ INR (S n) - 0)); - [ idtac | unfold Rdiv in |- *; ring ]. + [ idtac | unfold Rdiv; ring ]. rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs (Rabs x * Un 0%nat)). apply Rinv_0_lt_compat; apply Rabs_pos_lt. apply prod_neq_R0. apply Rabs_no_R0; assumption. - assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + assert (H16 := H7 0%nat); red; intro; rewrite H17 in H16; elim (Rlt_irrefl _ H16). rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_l. replace (/ Rabs (Rabs x * Un 0%nat) * eps1) with (eps1 / (Rabs x * Un 0%nat)). apply H14; assumption. - unfold Rdiv in |- *; rewrite (Rabs_right (Rabs x * Un 0%nat)). + unfold Rdiv; rewrite (Rabs_right (Rabs x * Un 0%nat)). apply Rmult_comm. apply Rle_ge; apply Rmult_le_pos. apply Rabs_pos. @@ -1092,9 +1092,9 @@ Proof. apply Rabs_no_R0. apply prod_neq_R0; [ apply Rabs_no_R0; assumption - | assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + | assert (H16 := H7 0%nat); red; intro; rewrite H17 in H16; elim (Rlt_irrefl _ H16) ]. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. assumption. apply Rinv_0_lt_compat; apply Rmult_lt_0_compat. apply Rabs_pos_lt; assumption. @@ -1102,7 +1102,7 @@ Proof. apply (cv_infty_cv_R0 (fun n:nat => INR (S n))). intro; apply not_O_INR; discriminate. assumption. - unfold cv_infty in |- *; intro; case (total_order_T M0 0); intro. + unfold cv_infty; intro; case (total_order_T M0 0); intro. elim s; intro. exists 0%nat; intros. apply Rlt_trans with 0; [ assumption | apply lt_INR_0; apply lt_O_Sn ]. @@ -1116,13 +1116,13 @@ Proof. elim H10; intros; assumption. rewrite H12; rewrite <- INR_IZR_INZ; apply le_INR. apply le_trans with n; [ assumption | apply le_n_Sn ]. - apply le_IZR; left; simpl in |- *; unfold M0_z in |- *; + apply le_IZR; left; simpl; unfold M0_z; apply Rlt_trans with M0; [ assumption | elim H10; intros; assumption ]. intro; apply Rle_trans with (Rabs x * Un n * / INR (S n)). - unfold Un in |- *; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. + unfold Un; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. rewrite pow_add; replace (Rabs x ^ 1) with (Rabs x); - [ idtac | simpl in |- *; ring ]. - unfold Rdiv in |- *; rewrite <- (Rmult_comm (Rabs x)); + [ idtac | simpl; ring ]. + unfold Rdiv; rewrite <- (Rmult_comm (Rabs x)); repeat rewrite Rmult_assoc; repeat apply Rmult_le_compat_l. apply Rabs_pos. left; apply pow_lt; assumption. @@ -1130,30 +1130,30 @@ Proof. rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; rewrite Rinv_mult_distr. apply Rmult_le_compat_l. - left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H10 := eq_sym H9); elim (fact_neq_0 _ H10). left; apply Rinv_lt_contravar. apply Rmult_lt_0_compat; apply lt_INR_0; apply lt_O_Sn. apply lt_INR; apply lt_n_S. - pattern n at 1 in |- *; replace n with (0 + n)%nat; [ idtac | reflexivity ]. + pattern n at 1; replace n with (0 + n)%nat; [ idtac | reflexivity ]. apply plus_lt_compat_r. apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. apply INR_fact_neq_0. apply not_O_INR; discriminate. ring. ring. - unfold Vn in |- *; rewrite Rmult_assoc; unfold Rdiv in |- *; + unfold Vn; rewrite Rmult_assoc; unfold Rdiv; rewrite (Rmult_comm (Un 0%nat)); rewrite (Rmult_comm (Un n)). repeat apply Rmult_le_compat_l. apply Rabs_pos. left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. apply decreasing_prop; [ assumption | apply le_O_n ]. - unfold Un_decreasing in |- *; intro; unfold Un in |- *. + unfold Un_decreasing; intro; unfold Un. replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. - rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc; + rewrite pow_add; unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l. left; apply pow_lt; assumption. - replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl in |- *; ring ]. + replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl; ring ]. replace (M_nat + n + 1)%nat with (S (M_nat + n)). apply Rmult_le_reg_l with (INR (fact (S (M_nat + n)))). apply lt_INR_0; apply neq_O_lt; red; intro; assert (H9 := eq_sym H8); @@ -1170,37 +1170,37 @@ Proof. apply INR_fact_neq_0. ring. ring. - intro; unfold Un in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat. + intro; unfold Un; unfold Rdiv; apply Rmult_lt_0_compat. apply pow_lt; assumption. - apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H8 := eq_sym H7); elim (fact_neq_0 _ H8). - clear Un Vn; apply INR_le; simpl in |- *. + clear Un Vn; apply INR_le; simpl. induction M_nat as [| M_nat HrecM_nat]. assert (H6 := archimed (Rabs x)); fold M in H6; elim H6; intros. rewrite H4 in H7; rewrite <- INR_IZR_INZ in H7. simpl in H7; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H2 H7)). replace 1 with (INR 1); [ apply le_INR | reflexivity ]; apply le_n_S; apply le_O_n. - apply le_IZR; simpl in |- *; left; apply Rlt_trans with (Rabs x). + apply le_IZR; simpl; left; apply Rlt_trans with (Rabs x). assumption. elim (archimed (Rabs x)); intros; assumption. - unfold Un_cv in |- *; unfold R_dist in |- *; intros; elim (H eps H0); intros. + unfold Un_cv; unfold R_dist; intros; elim (H eps H0); intros. exists x0; intros; apply Rle_lt_trans with (Rabs (Rabs x ^ n / INR (fact n) - 0)). - unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + unfold Rminus; rewrite Ropp_0; do 2 rewrite Rplus_0_r; rewrite (Rabs_right (Rabs x ^ n / INR (fact n))). - unfold Rdiv in |- *; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))). + unfold Rdiv; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))). rewrite RPow_abs; right; reflexivity. apply Rle_ge; left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H4 := eq_sym H3); elim (fact_neq_0 _ H4). - apply Rle_ge; unfold Rdiv in |- *; apply Rmult_le_pos. + apply Rle_ge; unfold Rdiv; apply Rmult_le_pos. case (Req_dec x 0); intro. rewrite H3; rewrite Rabs_R0. induction n as [| n Hrecn]; - [ simpl in |- *; left; apply Rlt_0_1 - | simpl in |- *; rewrite Rmult_0_l; right; reflexivity ]. + [ simpl; left; apply Rlt_0_1 + | simpl; rewrite Rmult_0_l; right; reflexivity ]. left; apply pow_lt; apply Rabs_pos_lt; assumption. - left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red; intro; assert (H4 := eq_sym H3); elim (fact_neq_0 _ H4). apply H1; assumption. Qed. diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v index 3085d0200..08bf54d6f 100644 --- a/theories/Reals/SeqSeries.v +++ b/theories/Reals/SeqSeries.v @@ -41,21 +41,21 @@ Proof. intro; rewrite H4; rewrite H5. apply sum_cv_maj with (fun l:nat => An (S N + l)%nat) (fun (l:nat) (x:R) => fn (S N + l)%nat x) x. - unfold SP in |- *; apply H2. + unfold SP; apply H2. apply H3. intros; apply H1. - symmetry in |- *; eapply UL_sequence. + symmetry ; eapply UL_sequence. apply H3. - unfold Un_cv in H0; unfold Un_cv in |- *; intros; elim (H0 eps H5); + unfold Un_cv in H0; unfold Un_cv; intros; elim (H0 eps H5); intros N0 H6. unfold R_dist in H6; exists N0; intros. - unfold R_dist in |- *; + unfold R_dist; replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); [ idtac | ring ]. replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with (sum_f_R0 An (S (N + n))). - apply H6; unfold ge in |- *; apply le_trans with n. + apply H6; unfold ge; apply le_trans with n. apply H7. apply le_trans with (N + n)%nat. apply le_plus_r. @@ -80,12 +80,12 @@ Proof. reflexivity. apply le_lt_n_Sm; apply le_plus_l. apply le_O_n. - symmetry in |- *; eapply UL_sequence. + symmetry ; eapply UL_sequence. apply H2. - unfold Un_cv in H; unfold Un_cv in |- *; intros. + unfold Un_cv in H; unfold Un_cv; intros. elim (H eps H4); intros N0 H5. unfold R_dist in H5; exists N0; intros. - unfold R_dist, SP in |- *; + unfold R_dist, SP; replace (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with @@ -96,7 +96,7 @@ Proof. (sum_f_R0 (fun k:nat => fn k x) N + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). - unfold SP in H5; apply H5; unfold ge in |- *; apply le_trans with n. + unfold SP in H5; apply H5; unfold ge; apply le_trans with n. apply H6. apply le_trans with (N + n)%nat. apply le_plus_r. @@ -124,16 +124,16 @@ Proof. apply le_plus_l. apply le_O_n. exists (l2 - sum_f_R0 An N). - unfold Un_cv in H0; unfold Un_cv in |- *; intros. + unfold Un_cv in H0; unfold Un_cv; intros. elim (H0 eps H2); intros N0 H3. unfold R_dist in H3; exists N0; intros. - unfold R_dist in |- *; + unfold R_dist; replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); [ idtac | ring ]. replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with (sum_f_R0 An (S (N + n))). - apply H3; unfold ge in |- *; apply le_trans with n. + apply H3; unfold ge; apply le_trans with n. apply H4. apply le_trans with (N + n)%nat. apply le_plus_r. @@ -160,10 +160,10 @@ Proof. apply le_plus_l. apply le_O_n. exists (l1 - SP fn N x). - unfold Un_cv in H; unfold Un_cv in |- *; intros. + unfold Un_cv in H; unfold Un_cv; intros. elim (H eps H2); intros N0 H3. unfold R_dist in H3; exists N0; intros. - unfold R_dist, SP in |- *. + unfold R_dist, SP. replace (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with @@ -175,7 +175,7 @@ Proof. sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). unfold SP in H3; apply H3. - unfold ge in |- *; apply le_trans with n. + unfold ge; apply le_trans with n. apply H4. apply le_trans with (N + n)%nat. apply le_plus_r. @@ -213,7 +213,7 @@ Lemma Rseries_CV_comp : Proof. intros An Bn H X; apply cv_cauchy_2. assert (H0 := cv_cauchy_1 _ X). - unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. + unfold Cauchy_crit_series; unfold Cauchy_crit. intros; elim (H0 eps H1); intros. exists x; intros. cut @@ -227,7 +227,7 @@ Proof. elim a; intro. rewrite (tech2 An n m); [ idtac | assumption ]. rewrite (tech2 Bn n m); [ idtac | assumption ]. - unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Ropp_plus_distr; + unfold R_dist; unfold Rminus; do 2 rewrite Ropp_plus_distr; do 2 rewrite <- Rplus_assoc; do 2 rewrite Rplus_opp_r; do 2 rewrite Rplus_0_l; do 2 rewrite Rabs_Ropp; repeat rewrite Rabs_right. apply sum_Rle; intros. @@ -238,12 +238,12 @@ Proof. apply Rle_trans with (An (S n + n0)%nat); assumption. apply Rle_ge; apply cond_pos_sum; intro. elim (H (S n + n0)%nat); intros; assumption. - rewrite b; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite b; unfold R_dist; unfold Rminus; do 2 rewrite Rplus_opp_r; rewrite Rabs_R0; right; reflexivity. rewrite (tech2 An m n); [ idtac | assumption ]. rewrite (tech2 Bn m n); [ idtac | assumption ]. - unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Rplus_assoc; + unfold R_dist; unfold Rminus; do 2 rewrite Rplus_assoc; rewrite (Rplus_comm (sum_f_R0 An m)); rewrite (Rplus_comm (sum_f_R0 Bn m)); do 2 rewrite Rplus_assoc; do 2 rewrite Rplus_opp_l; do 2 rewrite Rplus_0_r; repeat rewrite Rabs_right. @@ -266,13 +266,13 @@ Lemma Cesaro : Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n) l. Proof with trivial. - unfold Un_cv in |- *; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)... + unfold Un_cv; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)... intro; apply tech1... assert (H4 : forall n:nat, sum_f_R0 An n <> 0)... - intro; red in |- *; intro; assert (H5 := H3 n); rewrite H4 in H5; + intro; red; intro; assert (H5 := H3 n); rewrite H4 in H5; elim (Rlt_irrefl _ H5)... assert (H5 := cv_infty_cv_R0 _ H4 H1); assert (H6 : 0 < eps / 2)... - unfold Rdiv in |- *; apply Rmult_lt_0_compat... + unfold Rdiv; apply Rmult_lt_0_compat... apply Rinv_0_lt_compat; prove_sup... elim (H _ H6); clear H; intros N1 H; set (C := Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1)); @@ -282,10 +282,10 @@ Proof with trivial. (forall n:nat, (N <= n)%nat -> C / sum_f_R0 An n < eps / 2))... case (Req_dec C 0); intro... exists 0%nat; intros... - rewrite H7; unfold Rdiv in |- *; rewrite Rmult_0_l; apply Rmult_lt_0_compat... + rewrite H7; unfold Rdiv; rewrite Rmult_0_l; apply Rmult_lt_0_compat... apply Rinv_0_lt_compat; prove_sup... assert (H8 : 0 < eps / (2 * Rabs C))... - unfold Rdiv in |- *; apply Rmult_lt_0_compat... + unfold Rdiv; apply Rmult_lt_0_compat... apply Rinv_0_lt_compat; apply Rmult_lt_0_compat... prove_sup... apply Rabs_pos_lt... @@ -294,23 +294,23 @@ Proof with trivial. rewrite Rplus_0_r in H11... apply Rle_lt_trans with (Rabs (C / sum_f_R0 An n))... apply RRle_abs... - unfold Rdiv in |- *; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)... + unfold Rdiv; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)... apply Rinv_0_lt_compat; apply Rabs_pos_lt... rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... rewrite Rmult_1_l; replace (/ Rabs C * (eps * / 2)) with (eps / (2 * Rabs C))... - unfold Rdiv in |- *; rewrite Rinv_mult_distr... + unfold Rdiv; rewrite Rinv_mult_distr... ring... discrR... apply Rabs_no_R0... apply Rabs_no_R0... elim H7; clear H7; intros N2 H7; set (N := max N1 N2); exists (S N); intros; - unfold R_dist in |- *; + unfold R_dist; replace (sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n - l) with (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n / sum_f_R0 An n)... assert (H9 : (N1 < n)%nat)... apply lt_le_trans with (S N)... - apply le_lt_n_Sm; unfold N in |- *; apply le_max_l... - rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv in |- *; + apply le_lt_n_Sm; unfold N; apply le_max_l... + rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv; rewrite Rmult_plus_distr_r; apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1 / sum_f_R0 An n) + @@ -319,12 +319,12 @@ Proof with trivial. (n - S N1) / sum_f_R0 An n))... apply Rabs_triang... rewrite (double_var eps); apply Rplus_lt_compat... - unfold Rdiv in |- *; rewrite Rabs_mult; fold C in |- *; rewrite Rabs_right... + unfold Rdiv; rewrite Rabs_mult; fold C; rewrite Rabs_right... apply (H7 n); apply le_trans with (S N)... - apply le_trans with N; [ unfold N in |- *; apply le_max_r | apply le_n_Sn ]... + apply le_trans with N; [ unfold N; apply le_max_r | apply le_n_Sn ]... apply Rle_ge; left; apply Rinv_0_lt_compat... - unfold R_dist in H; unfold Rdiv in |- *; rewrite Rabs_mult; + unfold R_dist in H; unfold Rdiv; rewrite Rabs_mult; rewrite (Rabs_right (/ sum_f_R0 An n))... apply Rle_lt_trans with (sum_f_R0 (fun i:nat => Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))) @@ -340,22 +340,22 @@ Proof with trivial. do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l... left; apply Rinv_0_lt_compat... apply sum_Rle; intros; rewrite Rabs_mult; - pattern (An (S N1 + n0)%nat) at 2 in |- *; + pattern (An (S N1 + n0)%nat) at 2; rewrite <- (Rabs_right (An (S N1 + n0)%nat))... apply Rmult_le_compat_l... apply Rabs_pos... - left; apply H; unfold ge in |- *; apply le_trans with (S N1); + left; apply H; unfold ge; apply le_trans with (S N1); [ apply le_n_Sn | apply le_plus_l ]... apply Rle_ge; left... rewrite <- (scal_sum (fun i:nat => An (S N1 + i)%nat) (n - S N1) (eps / 2)); - unfold Rdiv in |- *; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l... - pattern (/ 2) at 2 in |- *; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l... + unfold Rdiv; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l... + pattern (/ 2) at 2; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l... apply Rinv_0_lt_compat; prove_sup... rewrite Rmult_comm; apply Rmult_lt_reg_l with (sum_f_R0 An n)... rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... rewrite Rmult_1_l; rewrite Rmult_1_r; rewrite (tech2 An N1 n)... rewrite Rplus_comm; - pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1 in |- *; + pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l... apply Rle_ge; left; apply Rinv_0_lt_compat... replace (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n) with @@ -371,41 +371,41 @@ Lemma Cesaro_1 : Proof with trivial. intros Bn l H; set (An := fun _:nat => 1)... assert (H0 : forall n:nat, 0 < An n)... - intro; unfold An in |- *; apply Rlt_0_1... + intro; unfold An; apply Rlt_0_1... assert (H1 : forall n:nat, 0 < sum_f_R0 An n)... intro; apply tech1... assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))... - unfold cv_infty in |- *; intro; case (Rle_dec M 0); intro... + unfold cv_infty; intro; case (Rle_dec M 0); intro... exists 0%nat; intros; apply Rle_lt_trans with 0... assert (H2 : 0 < M)... auto with real... clear n; set (m := up M); elim (archimed M); intros; assert (H5 : (0 <= m)%Z)... - apply le_IZR; unfold m in |- *; simpl in |- *; left; apply Rlt_trans with M... - elim (IZN _ H5); intros; exists x; intros; unfold An in |- *; rewrite sum_cte; + apply le_IZR; unfold m; simpl; left; apply Rlt_trans with M... + elim (IZN _ H5); intros; exists x; intros; unfold An; rewrite sum_cte; rewrite Rmult_1_l; apply Rlt_trans with (IZR (up M))... apply Rle_lt_trans with (INR x)... - rewrite INR_IZR_INZ; fold m in |- *; rewrite <- H6; right... + rewrite INR_IZR_INZ; fold m; rewrite <- H6; right... apply lt_INR; apply le_lt_n_Sm... assert (H3 := Cesaro _ _ _ H H0 H2)... - unfold Un_cv in |- *; unfold Un_cv in H3; intros; elim (H3 _ H4); intros; - exists (S x); intros; unfold R_dist in |- *; unfold R_dist in H5; + unfold Un_cv; unfold Un_cv in H3; intros; elim (H3 _ H4); intros; + exists (S x); intros; unfold R_dist; unfold R_dist in H5; apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l))... right; replace (sum_f_R0 Bn (pred n) / INR n - l) with (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l)... - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + unfold Rminus; do 2 rewrite <- (Rplus_comm (- l)); apply Rplus_eq_compat_l... - unfold An in |- *; + unfold An; replace (sum_f_R0 (fun k:nat => 1 * Bn k) (pred n)) with (sum_f_R0 Bn (pred n))... rewrite sum_cte; rewrite Rmult_1_l; replace (S (pred n)) with n... apply S_pred with 0%nat; apply lt_le_trans with (S x)... apply lt_O_Sn... apply sum_eq; intros; ring... - apply H5; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n... + apply H5; unfold ge; apply le_S_n; replace (S (pred n)) with n... apply S_pred with 0%nat; apply lt_le_trans with (S x)... apply lt_O_Sn... Qed. diff --git a/theories/Reals/SplitAbsolu.v b/theories/Reals/SplitAbsolu.v index 819606c46..f6ddf03a0 100644 --- a/theories/Reals/SplitAbsolu.v +++ b/theories/Reals/SplitAbsolu.v @@ -19,5 +19,5 @@ Ltac split_Rabs := match goal with | id:context [(Rabs _)] |- _ => generalize id; clear id; try split_Rabs | |- context [(Rabs ?X1)] => - unfold Rabs in |- *; try split_case_Rabs; intros + unfold Rabs; try split_case_Rabs; intros end. diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v index c429567fe..a2e746c2f 100644 --- a/theories/Reals/Sqrt_reg.v +++ b/theories/Reals/Sqrt_reg.v @@ -21,67 +21,67 @@ Proof. case (total_order_T h 0); intro. elim s; intro. repeat rewrite Rabs_left. - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)). + unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)). do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive; apply Rplus_le_compat_l. apply Ropp_le_contravar; apply sqrt_le_1. apply Rle_0_sqr. apply H0. - pattern (1 + h) at 2 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + pattern (1 + h) at 2; rewrite <- Rmult_1_r; unfold Rsqr; apply Rmult_le_compat_l. apply H0. - pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; assumption. apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. - pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. + pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1. apply Rle_0_sqr. left; apply Rlt_0_1. - pattern 1 at 2 in |- *; rewrite <- Rsqr_1; apply Rsqr_incrst_1. - pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern 1 at 2; rewrite <- Rsqr_1; apply Rsqr_incrst_1. + pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. apply H0. left; apply Rlt_0_1. apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; - unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. - pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. + pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1. apply H0. left; apply Rlt_0_1. - pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. rewrite b; rewrite Rplus_0_r; rewrite Rsqr_1; rewrite sqrt_1; right; reflexivity. repeat rewrite Rabs_right. - unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)); + unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)); apply Rplus_le_compat_l. apply sqrt_le_1. apply H0. apply Rle_0_sqr. - pattern (1 + h) at 1 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + pattern (1 + h) at 1; rewrite <- Rmult_1_r; unfold Rsqr; apply Rmult_le_compat_l. apply H0. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; assumption. apply Rle_ge; apply Rplus_le_reg_l with 1. - rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. - pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_le_1. + pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_le_1. left; apply Rlt_0_1. apply Rle_0_sqr. - pattern 1 at 1 in |- *; rewrite <- Rsqr_1; apply Rsqr_incr_1. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + pattern 1 at 1; rewrite <- Rsqr_1; apply Rsqr_incr_1. + pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; assumption. left; apply Rlt_0_1. apply H0. apply Rle_ge; left; apply Rplus_lt_reg_r with 1. - rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. - pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. + pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_lt_1. left; apply Rlt_0_1. apply H0. - pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; assumption. rewrite sqrt_Rsqr. replace (1 + h - 1) with h; [ right; reflexivity | ring ]. @@ -101,14 +101,14 @@ Qed. (** sqrt is continuous in 1 *) Lemma sqrt_continuity_pt_R1 : continuity_pt sqrt 1. Proof. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + unfold dist; simpl; unfold R_dist; intros. set (alpha := Rmin eps 1). exists alpha; intros. split. - unfold alpha in |- *; unfold Rmin in |- *; case (Rle_dec eps 1); intro. + unfold alpha; unfold Rmin; case (Rle_dec eps 1); intro. assumption. apply Rlt_0_1. intros; elim H0; intros. @@ -117,18 +117,18 @@ Proof. apply sqrt_var_maj. apply Rle_trans with alpha. left; apply H2. - unfold alpha in |- *; apply Rmin_r. + unfold alpha; apply Rmin_r. apply Rlt_le_trans with alpha; - [ apply H2 | unfold alpha in |- *; apply Rmin_l ]. + [ apply H2 | unfold alpha; apply Rmin_l ]. Qed. (** sqrt is continuous forall x>0 *) Lemma sqrt_continuity_pt : forall x:R, 0 < x -> continuity_pt sqrt x. Proof. intros; generalize sqrt_continuity_pt_R1. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + unfold dist; simpl; unfold R_dist; intros. cut (0 < eps / sqrt x). intro; elim (H0 _ H2); intros alp_1 H3. @@ -136,9 +136,9 @@ Proof. set (alpha := alp_1 * x). exists (Rmin alpha x); intros. split. - change (0 < Rmin alpha x) in |- *; unfold Rmin in |- *; + change (0 < Rmin alpha x); unfold Rmin; case (Rle_dec alpha x); intro. - unfold alpha in |- *; apply Rmult_lt_0_compat; assumption. + unfold alpha; apply Rmult_lt_0_compat; assumption. apply H. intros; replace x0 with (x + (x0 - x)); [ idtac | ring ]; replace (sqrt (x + (x0 - x)) - sqrt x) with @@ -150,7 +150,7 @@ Proof. rewrite Rmult_1_l; rewrite Rmult_comm. unfold Rdiv in H5. case (Req_dec x x0); intro. - rewrite H7; unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; + rewrite H7; unfold Rminus, Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_0_r; rewrite Rplus_opp_r; rewrite Rabs_R0. apply Rmult_lt_0_compat. @@ -158,10 +158,10 @@ Proof. apply Rinv_0_lt_compat; rewrite <- H7; apply sqrt_lt_R0; assumption. apply H5. split. - unfold D_x, no_cond in |- *. + unfold D_x, no_cond. split. trivial. - red in |- *; intro. + red; intro. cut ((x0 - x) * / x = 0). intro. elim (Rmult_integral _ _ H9); intro. @@ -170,35 +170,35 @@ Proof. assert (H11 := Rmult_eq_0_compat_r _ x H10). rewrite <- Rinv_l_sym in H11. elim R1_neq_R0; exact H11. - red in |- *; intro; rewrite H12 in H; elim (Rlt_irrefl _ H). - symmetry in |- *; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r; + red; intro; rewrite H12 in H; elim (Rlt_irrefl _ H). + symmetry ; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r; unfold Rdiv in H8; exact H8. - unfold Rminus in |- *; rewrite Rplus_comm; rewrite <- Rplus_assoc; + unfold Rminus; rewrite Rplus_comm; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; elim H6; intros. - unfold Rdiv in |- *; rewrite Rabs_mult. + unfold Rdiv; rewrite Rabs_mult. rewrite Rabs_Rinv. rewrite (Rabs_right x). rewrite Rmult_comm; apply Rmult_lt_reg_l with x. apply H. rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. - rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha in |- *. + rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha. apply Rlt_le_trans with (Rmin alpha x). apply H9. apply Rmin_l. - red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). apply Rle_ge; left; apply H. - red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). assert (H7 := sqrt_lt_R0 x H). - red in |- *; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7). + red; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7). apply Rle_ge; apply sqrt_positivity. left; apply H. - unfold Rminus in |- *; rewrite Rmult_plus_distr_l; + unfold Rminus; rewrite Rmult_plus_distr_l; rewrite Ropp_mult_distr_r_reverse; repeat rewrite <- sqrt_mult. rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; - unfold Rdiv in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + unfold Rdiv; rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. rewrite Rmult_1_r; reflexivity. - red in |- *; intro; rewrite H7 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H7 in H; elim (Rlt_irrefl _ H). left; apply H. left; apply Rlt_0_1. left; apply H. @@ -208,7 +208,7 @@ Proof. rewrite Rplus_comm. apply Rplus_le_reg_l with (- ((x0 - x) / x)). rewrite Rplus_0_r; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; - rewrite Rplus_0_l; unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. + rewrite Rplus_0_l; unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse. apply Rmult_le_reg_l with x. apply H. rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; @@ -216,13 +216,13 @@ Proof. rewrite Rmult_1_r; left; apply Rlt_le_trans with (Rmin alpha x). apply H8. apply Rmin_r. - red in |- *; intro; rewrite H9 in H; elim (Rlt_irrefl _ H). + red; intro; rewrite H9 in H; elim (Rlt_irrefl _ H). apply Rplus_le_le_0_compat. left; apply Rlt_0_1. - unfold Rdiv in |- *; apply Rmult_le_pos. + unfold Rdiv; apply Rmult_le_pos. apply Rge_le; exact r. left; apply Rinv_0_lt_compat; apply H. - unfold Rdiv in |- *; apply Rmult_lt_0_compat. + unfold Rdiv; apply Rmult_lt_0_compat. apply H1. apply Rinv_0_lt_compat; apply sqrt_lt_R0; apply H. Qed. @@ -235,7 +235,7 @@ Proof. cut (continuity_pt g 0). intro; cut (g 0 <> 0). intro; assert (H2 := continuity_pt_inv g 0 H0 H1). - unfold derivable_pt_lim in |- *; intros; unfold continuity_pt in H2; + unfold derivable_pt_lim; intros; unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2; unfold limit_in in H2; simpl in H2; unfold R_dist in H2. elim (H2 eps H3); intros alpha H4. @@ -247,29 +247,29 @@ Proof. unfold inv_fct, g in H6; replace (2 * sqrt x) with (sqrt x + sqrt (x + 0)). apply H6. split. - unfold D_x, no_cond in |- *. + unfold D_x, no_cond. split. trivial. apply (not_eq_sym (A:=R)); exact H8. - unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rlt_le_trans with alpha1. exact H9. - unfold alpha1 in |- *; apply Rmin_l. + unfold alpha1; apply Rmin_l. rewrite Rplus_0_r; ring. cut (0 <= x + h). intro; cut (0 < sqrt x + sqrt (x + h)). intro; apply Rmult_eq_reg_l with (sqrt x + sqrt (x + h)). rewrite <- Rinv_r_sym. - rewrite Rplus_comm; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rplus_comm; unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rsqr_plus_minus; repeat rewrite Rsqr_sqrt. - rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym. reflexivity. apply H8. left; apply H. assumption. - red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). - red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). + red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). + red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). apply Rplus_lt_le_0_compat. apply sqrt_lt_R0; apply H. apply sqrt_positivity; apply H10. @@ -279,35 +279,35 @@ Proof. rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_r; left; apply Rlt_le_trans with alpha1. apply H9. - unfold alpha1 in |- *; apply Rmin_r. + unfold alpha1; apply Rmin_r. apply Rplus_le_le_0_compat. left; assumption. apply Rge_le; apply r. - unfold alpha1 in |- *; unfold Rmin in |- *; case (Rle_dec alpha x); intro. + unfold alpha1; unfold Rmin; case (Rle_dec alpha x); intro. apply H5. apply H. - unfold g in |- *; rewrite Rplus_0_r. + unfold g; rewrite Rplus_0_r. cut (0 < sqrt x + sqrt x). - intro; red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). + intro; red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). apply Rplus_lt_0_compat; apply sqrt_lt_R0; apply H. replace g with (fct_cte (sqrt x) + comp sqrt (fct_cte x + id))%F; [ idtac | reflexivity ]. apply continuity_pt_plus. - apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + apply continuity_pt_const; unfold constant, fct_cte; intro; reflexivity. apply continuity_pt_comp. apply continuity_pt_plus. - apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + apply continuity_pt_const; unfold constant, fct_cte; intro; reflexivity. apply derivable_continuous_pt; apply derivable_pt_id. apply sqrt_continuity_pt. - unfold plus_fct, fct_cte, id in |- *; rewrite Rplus_0_r; apply H. + unfold plus_fct, fct_cte, id; rewrite Rplus_0_r; apply H. Qed. (**********) Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x. Proof. - unfold derivable_pt in |- *; intros. + unfold derivable_pt; intros. exists (/ (2 * sqrt x)). apply derivable_pt_lim_sqrt; assumption. Qed. @@ -330,19 +330,19 @@ Proof. intros; case (Rtotal_order 0 x); intro. apply (sqrt_continuity_pt x H0). elim H0; intro. - unfold continuity_pt in |- *; unfold continue_in in |- *; - unfold limit1_in in |- *; unfold limit_in in |- *; - simpl in |- *; unfold R_dist in |- *; intros. + unfold continuity_pt; unfold continue_in; + unfold limit1_in; unfold limit_in; + simpl; unfold R_dist; intros. exists (Rsqr eps); intros. split. - change (0 < Rsqr eps) in |- *; apply Rsqr_pos_lt. - red in |- *; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2). + change (0 < Rsqr eps); apply Rsqr_pos_lt. + red; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2). intros; elim H3; intros. - rewrite <- H1; rewrite sqrt_0; unfold Rminus in |- *; rewrite Ropp_0; + rewrite <- H1; rewrite sqrt_0; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5. case (Rcase_abs x0); intro. - unfold sqrt in |- *; case (Rcase_abs x0); intro. + unfold sqrt; case (Rcase_abs x0); intro. rewrite Rabs_R0; apply H2. assert (H6 := Rge_le _ _ r0); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 r)). rewrite Rabs_right. diff --git a/theories/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v index f7f5512e7..3f3810083 100644 --- a/theories/Relations/Operators_Properties.v +++ b/theories/Relations/Operators_Properties.v @@ -50,7 +50,7 @@ Section Properties. Lemma clos_rt_idempotent : inclusion (R*)* R*. Proof. - red in |- *. + red. induction 1; auto with sets. intros. apply rt_trans with y; auto with sets. @@ -66,7 +66,7 @@ Section Properties. Lemma clos_rt_clos_rst : inclusion (clos_refl_trans R) (clos_refl_sym_trans R). Proof. - red in |- *. + red. induction 1; auto with sets. apply rst_trans with y; auto with sets. Qed. @@ -87,7 +87,7 @@ Section Properties. inclusion (clos_refl_sym_trans (clos_refl_sym_trans R)) (clos_refl_sym_trans R). Proof. - red in |- *. + red. induction 1; auto with sets. apply rst_trans with y; auto with sets. Qed. diff --git a/theories/Relations/Relations.v b/theories/Relations/Relations.v index f9fb2c442..ed2567396 100644 --- a/theories/Relations/Relations.v +++ b/theories/Relations/Relations.v @@ -14,16 +14,16 @@ Lemma inverse_image_of_equivalence : forall (A B:Type) (f:A -> B) (r:relation B), equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)). Proof. - intros; split; elim H; red in |- *; auto. + intros; split; elim H; red; auto. intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption. Qed. Lemma inverse_image_of_eq : forall (A B:Type) (f:A -> B), equivalence A (fun x y:A => f x = f y). Proof. - split; red in |- *; + split; red; [ (* reflexivity *) reflexivity | (* transitivity *) intros; transitivity (f y); assumption - | (* symmetry *) intros; symmetry in |- *; assumption ]. + | (* symmetry *) intros; symmetry ; assumption ]. Qed. diff --git a/theories/Sets/Classical_sets.v b/theories/Sets/Classical_sets.v index f93631c7e..bf7b52b22 100644 --- a/theories/Sets/Classical_sets.v +++ b/theories/Sets/Classical_sets.v @@ -38,8 +38,8 @@ Section Ensembles_classical. elim (not_all_ex_not U (fun x:U => ~ In U A x)). intros x H; apply Inhabited_intro with x. apply NNPP; auto with sets. - red in |- *; intro. - apply NI; red in |- *. + red; intro. + apply NI; red. intros x H'; elim (H x); trivial with sets. Qed. @@ -47,7 +47,7 @@ Section Ensembles_classical. forall A:Ensemble U, A <> Empty_set U -> Inhabited U A. Proof. intros; apply not_included_empty_Inhabited. - red in |- *; auto with sets. + red; auto with sets. Qed. Lemma Inhabited_Setminus : @@ -73,7 +73,7 @@ Section Ensembles_classical. Lemma Subtract_intro : forall (A:Ensemble U) (x y:U), In U A y -> x <> y -> In U (Subtract U A x) y. Proof. - unfold Subtract at 1 in |- *; auto with sets. + unfold Subtract at 1; auto with sets. Qed. Hint Resolve Subtract_intro : sets. @@ -103,7 +103,7 @@ Section Ensembles_classical. Lemma not_SIncl_empty : forall X:Ensemble U, ~ Strict_Included U X (Empty_set U). Proof. - intro X; red in |- *; intro H'; try exact H'. + intro X; red; intro H'; try exact H'. lapply (Strict_Included_inv X (Empty_set U)); auto with sets. intro H'0; elim H'0; intros H'1 H'2; elim H'2; clear H'0. intros x H'0; elim H'0. @@ -113,10 +113,10 @@ Section Ensembles_classical. Lemma Complement_Complement : forall A:Ensemble U, Complement U (Complement U A) = A. Proof. - unfold Complement in |- *; intros; apply Extensionality_Ensembles; + unfold Complement; intros; apply Extensionality_Ensembles; auto with sets. - red in |- *; split; auto with sets. - red in |- *; intros; apply NNPP; auto with sets. + red; split; auto with sets. + red; intros; apply NNPP; auto with sets. Qed. End Ensembles_classical. diff --git a/theories/Sets/Constructive_sets.v b/theories/Sets/Constructive_sets.v index e6dd83810..324255f6d 100644 --- a/theories/Sets/Constructive_sets.v +++ b/theories/Sets/Constructive_sets.v @@ -36,24 +36,24 @@ Section Ensembles_facts. Lemma Noone_in_empty : forall x:U, ~ In U (Empty_set U) x. Proof. - red in |- *; destruct 1. + red; destruct 1. Qed. Lemma Included_Empty : forall A:Ensemble U, Included U (Empty_set U) A. Proof. - intro; red in |- *. + intro; red. intros x H; elim (Noone_in_empty x); auto with sets. Qed. Lemma Add_intro1 : forall (A:Ensemble U) (x y:U), In U A y -> In U (Add U A x) y. Proof. - unfold Add at 1 in |- *; auto with sets. + unfold Add at 1; auto with sets. Qed. Lemma Add_intro2 : forall (A:Ensemble U) (x:U), In U (Add U A x) x. Proof. - unfold Add at 1 in |- *; auto with sets. + unfold Add at 1; auto with sets. Qed. Lemma Inhabited_add : forall (A:Ensemble U) (x:U), Inhabited U (Add U A x). @@ -66,7 +66,7 @@ Section Ensembles_facts. forall X:Ensemble U, Inhabited U X -> X <> Empty_set U. Proof. intros X H'; elim H'. - intros x H'0; red in |- *; intro H'1. + intros x H'0; red; intro H'1. absurd (In U X x); auto with sets. rewrite H'1; auto using Noone_in_empty with sets. Qed. @@ -78,7 +78,7 @@ Section Ensembles_facts. Lemma not_Empty_Add : forall (A:Ensemble U) (x:U), Empty_set U <> Add U A x. Proof. - intros; red in |- *; intro H; generalize (Add_not_Empty A x); auto with sets. + intros; red; intro H; generalize (Add_not_Empty A x); auto with sets. Qed. Lemma Singleton_inv : forall x y:U, In U (Singleton U x) y -> x = y. @@ -121,7 +121,7 @@ Section Ensembles_facts. forall (A B:Ensemble U) (x:U), In U A x -> ~ In U B x -> In U (Setminus U A B) x. Proof. - unfold Setminus at 1 in |- *; red in |- *; auto with sets. + unfold Setminus at 1; red; auto with sets. Qed. Lemma Strict_Included_intro : @@ -132,7 +132,7 @@ Section Ensembles_facts. Lemma Strict_Included_strict : forall X:Ensemble U, ~ Strict_Included U X X. Proof. - intro X; red in |- *; intro H'; elim H'. + intro X; red; intro H'; elim H'. intros H'0 H'1; elim H'1; auto with sets. Qed. diff --git a/theories/Sets/Finite_sets.v b/theories/Sets/Finite_sets.v index f08436754..07543276b 100644 --- a/theories/Sets/Finite_sets.v +++ b/theories/Sets/Finite_sets.v @@ -61,7 +61,7 @@ Section Ensembles_finis_facts. (exists x : _, X = Add U A x /\ ~ In U A x /\ cardinal U A n) end. Proof. - induction 1; simpl in |- *; auto. + induction 1; simpl; auto. exists A; exists x; auto. Qed. @@ -73,7 +73,7 @@ Section Ensembles_finis_facts. | S n => Inhabited U X end. Proof. - intros X p C; elim C; simpl in |- *; trivial with sets. + intros X p C; elim C; simpl; trivial with sets. Qed. End Ensembles_finis_facts. diff --git a/theories/Sets/Finite_sets_facts.v b/theories/Sets/Finite_sets_facts.v index 350cd783c..8895e4569 100644 --- a/theories/Sets/Finite_sets_facts.v +++ b/theories/Sets/Finite_sets_facts.v @@ -62,7 +62,7 @@ Section Finite_sets_facts. Theorem Singleton_is_finite : forall x:U, Finite U (Singleton U x). Proof. intro x; rewrite <- (Empty_set_zero U (Singleton U x)). - change (Finite U (Add U (Empty_set U) x)) in |- *; auto with sets. + change (Finite U (Add U (Empty_set U) x)); auto with sets. Qed. Theorem Union_preserves_Finite : @@ -134,15 +134,15 @@ Section Finite_sets_facts. cut (S (pred n) = pred (S n)). intro H'5; rewrite <- H'5. apply card_add; auto with sets. - red in |- *; intro H'6; elim H'6. + red; intro H'6; elim H'6. intros H'7 H'8; try assumption. elim H'1; auto with sets. - unfold pred at 2 in |- *; symmetry in |- *. + unfold pred at 2; symmetry . apply S_pred with (m := 0). - change (n > 0) in |- *. + change (n > 0). apply inh_card_gt_O with (X := X); auto with sets. apply Inhabited_intro with (x := x0); auto with sets. - red in |- *; intro H'3. + red; intro H'3. apply H'1. elim H'3; auto with sets. rewrite H'3; auto with sets. @@ -152,7 +152,7 @@ Section Finite_sets_facts. intro H'4; rewrite H'4; auto with sets. intros H'3 H'4; try assumption. absurd (In U (Add U X x) x0); auto with sets. - red in |- *; intro H'5; try exact H'5. + red; intro H'5; try exact H'5. lapply (Add_inv U X x x0); tauto. Qed. @@ -183,11 +183,11 @@ Section Finite_sets_facts. intros H'6 H'7; apply f_equal. apply H'0 with (Y := X0); auto with sets. apply Simplify_add with (x := x); auto with sets. - pattern x at 2 in |- *; rewrite H'6; auto with sets. + pattern x at 2; rewrite H'6; auto with sets. intros H'6 H'7. absurd (Add U X x = Add U X0 x0); auto with sets. clear H'0 H' H'3 n H'5 H'4 H'2 H'1 c2. - red in |- *; intro H'. + red; intro H'. lapply (Extension U (Add U X x) (Add U X0 x0)); auto with sets. clear H'. intro H'; red in H'. @@ -254,7 +254,7 @@ Section Finite_sets_facts. apply H'0 with (Y := X0); auto with sets arith. apply sincl_add_x with (x := x0). rewrite <- H'6; auto with sets arith. - pattern x0 at 1 in |- *; rewrite <- H'6; trivial with sets arith. + pattern x0 at 1; rewrite <- H'6; trivial with sets arith. intros H'6 H'7; red in H'5. elim H'5; intros H'8 H'9; try exact H'8; clear H'5. red in H'8. diff --git a/theories/Sets/Image.v b/theories/Sets/Image.v index 24facb6f6..440e636cb 100644 --- a/theories/Sets/Image.v +++ b/theories/Sets/Image.v @@ -55,7 +55,7 @@ Section Image. Proof. intros X x f. apply Extensionality_Ensembles. - split; red in |- *; intros x0 H'. + split; red; intros x0 H'. elim H'; intros. rewrite H0. elim Add_inv with U X x x1; auto using Im_def with sets. @@ -72,7 +72,7 @@ Section Image. intro f; try assumption. apply Extensionality_Ensembles. split; auto with sets. - red in |- *. + red. intros x H'; elim H'. intros x0 H'0; elim H'0; auto with sets. Qed. @@ -102,7 +102,7 @@ Section Image. forall f:U -> V, ~ injective f -> exists x : _, (exists y : _, f x = f y /\ x <> y). Proof. - unfold injective in |- *; intros f H. + unfold injective; intros f H. cut (exists x : _, ~ (forall y:U, f x = f y -> x = y)). 2: apply not_all_ex_not with (P := fun x:U => forall y:U, f x = f y -> x = y); trivial with sets. @@ -153,7 +153,7 @@ Section Image. apply cardinal_unicity with V (Add _ (Im A f) (f x)); trivial with sets. apply card_add; auto with sets. rewrite <- H1; trivial with sets. - red in |- *; intro; apply H'2. + red; intro; apply H'2. apply In_Image_elim with f; trivial with sets. Qed. @@ -180,7 +180,7 @@ Section Image. cardinal U A n -> forall n':nat, cardinal V (Im A f) n' -> n' < n -> ~ injective f. Proof. - unfold not in |- *; intros A f n CAn n' CIfn' ltn'n I. + unfold not; intros A f n CAn n' CIfn' ltn'n I. cut (n' = n). intro E; generalize ltn'n; rewrite E; exact (lt_irrefl n). apply injective_preserves_cardinal with (A := A) (f := f) (n := n); diff --git a/theories/Sets/Infinite_sets.v b/theories/Sets/Infinite_sets.v index a21fe880c..f2862e14e 100644 --- a/theories/Sets/Infinite_sets.v +++ b/theories/Sets/Infinite_sets.v @@ -56,7 +56,7 @@ Section Infinite_sets. intros A X H' H'0. elim H'0; intros H'1 H'2. apply Strict_super_set_contains_new_element; auto with sets. - red in |- *; intro H'3; apply H'. + red; intro H'3; apply H'. rewrite <- H'3; auto with sets. Qed. @@ -76,7 +76,7 @@ Section Infinite_sets. split. apply card_add; auto with sets. cut (In U A x). - intro H'4; red in |- *; auto with sets. + intro H'4; red; auto with sets. intros x0 H'5; elim H'5; auto with sets. intros x1 H'6; elim H'6; auto with sets. elim H'3; auto with sets. @@ -91,7 +91,7 @@ Section Infinite_sets. split. apply card_add; auto with sets. elim H'2; auto with sets. - red in |- *. + red. intros x2 H'9; elim H'9; auto with sets. intros x3 H'10; elim H'10; auto with sets. elim H'2; auto with sets. @@ -167,11 +167,11 @@ Section Infinite_sets. apply ex_intro with (x := Add U x0 x1). split; [ split; [ try assumption | idtac ] | idtac ]. apply card_add; auto with sets. - red in |- *; intro H'9; try exact H'9. + red; intro H'9; try exact H'9. apply H'1. elim H'4; intros H'10 H'11; rewrite <- H'11; clear H'4; auto with sets. elim H'4; intros H'9 H'10; try exact H'9; clear H'4; auto with sets. - red in |- *; auto with sets. + red; auto with sets. intros x2 H'4; elim H'4; auto with sets. intros x3 H'11; elim H'11; auto with sets. elim H'4; intros H'9 H'10; rewrite <- H'10; clear H'4; auto with sets. @@ -235,7 +235,7 @@ Section Infinite_sets. Proof. intros A f H' H'0 H'1. apply NNPP. - red in |- *; intro H'2. + red; intro H'2. elim (Pigeonhole_bis A f); auto with sets. Qed. diff --git a/theories/Sets/Integers.v b/theories/Sets/Integers.v index 2c94a2e16..5ba7856eb 100644 --- a/theories/Sets/Integers.v +++ b/theories/Sets/Integers.v @@ -49,17 +49,17 @@ Section Integers_sect. Lemma le_reflexive : Reflexive nat le. Proof. - red in |- *; auto with arith. + red; auto with arith. Qed. Lemma le_antisym : Antisymmetric nat le. Proof. - red in |- *; intros x y H H'; rewrite (le_antisym x y); auto. + red; intros x y H H'; rewrite (le_antisym x y); auto. Qed. Lemma le_trans : Transitive nat le. Proof. - red in |- *; intros; apply le_trans with y; auto. + red; intros; apply le_trans with y; auto. Qed. Lemma le_Order : Order nat le. @@ -83,7 +83,7 @@ Section Integers_sect. Lemma le_total_order : Totally_ordered nat nat_po Integers. Proof. apply Totally_ordered_definition. - simpl in |- *. + simpl. intros H' x y H'0. elim le_or_lt with (n := x) (m := y). intro H'1; left; auto with sets arith. @@ -103,7 +103,7 @@ Section Integers_sect. intros A H'0 H'1 x H'2; try assumption. elim H'1; intros x0 H'3; clear H'1. elim le_total_order. - simpl in |- *. + simpl. intro H'1; try assumption. lapply H'1; [ intro H'4; idtac | try assumption ]; auto with sets arith. generalize (H'4 x0 x). @@ -114,28 +114,28 @@ Section Integers_sect. [ intro H'5; try exact H'5; clear H'4 H'1 | intro H'5; clear H'4 H'1 ] | clear H'1 ]. exists x. - apply Upper_Bound_definition. simpl in |- *. apply triv_nat. + apply Upper_Bound_definition. simpl. apply triv_nat. intros y H'1; elim H'1. generalize le_trans. intro H'4; red in H'4. intros x1 H'6; try assumption. - apply H'4 with (y := x0). elim H'3; simpl in |- *; auto with sets arith. trivial. + apply H'4 with (y := x0). elim H'3; simpl; auto with sets arith. trivial. intros x1 H'4; elim H'4. unfold nat_po; simpl; trivial. exists x0. apply Upper_Bound_definition. unfold nat_po. simpl. apply triv_nat. intros y H'1; elim H'1. intros x1 H'4; try assumption. - elim H'3; simpl in |- *; auto with sets arith. + elim H'3; simpl; auto with sets arith. intros x1 H'4; elim H'4; auto with sets arith. - red in |- *. + red. intros x1 H'1; elim H'1; apply triv_nat. Qed. Lemma Integers_has_no_ub : ~ (exists m : nat, Upper_Bound nat nat_po Integers m). Proof. - red in |- *; intro H'; elim H'. + red; intro H'; elim H'. intros x H'0. elim H'0; intros H'1 H'2. cut (In nat Integers (S x)). @@ -150,7 +150,7 @@ Section Integers_sect. Lemma Integers_infinite : ~ Finite nat Integers. Proof. generalize Integers_has_no_ub. - intro H'; red in |- *; intro H'0; try exact H'0. + intro H'; red; intro H'0; try exact H'0. apply H'. apply Finite_subset_has_lub; auto with sets arith. Qed. diff --git a/theories/Sets/Multiset.v b/theories/Sets/Multiset.v index 5f21335fd..4159d3877 100644 --- a/theories/Sets/Multiset.v +++ b/theories/Sets/Multiset.v @@ -42,14 +42,14 @@ Section multiset_defs. Lemma meq_trans : forall x y z:multiset, meq x y -> meq y z -> meq x z. Proof. - unfold meq in |- *. + unfold meq. destruct x; destruct y; destruct z. intros; rewrite H; auto. Qed. Lemma meq_sym : forall x y:multiset, meq x y -> meq y x. Proof. - unfold meq in |- *. + unfold meq. destruct x; destruct y; auto. Qed. @@ -59,12 +59,12 @@ Section multiset_defs. Lemma munion_empty_left : forall x:multiset, meq x (munion EmptyBag x). Proof. - unfold meq in |- *; unfold munion in |- *; simpl in |- *; auto. + unfold meq; unfold munion; simpl; auto. Qed. Lemma munion_empty_right : forall x:multiset, meq x (munion x EmptyBag). Proof. - unfold meq in |- *; unfold munion in |- *; simpl in |- *; auto. + unfold meq; unfold munion; simpl; auto. Qed. @@ -72,21 +72,21 @@ Section multiset_defs. Lemma munion_comm : forall x y:multiset, meq (munion x y) (munion y x). Proof. - unfold meq in |- *; unfold multiplicity in |- *; unfold munion in |- *. + unfold meq; unfold multiplicity; unfold munion. destruct x; destruct y; auto with arith. Qed. Lemma munion_ass : forall x y z:multiset, meq (munion (munion x y) z) (munion x (munion y z)). Proof. - unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *. + unfold meq; unfold munion; unfold multiplicity. destruct x; destruct y; destruct z; auto with arith. Qed. Lemma meq_left : forall x y z:multiset, meq x y -> meq (munion x z) (munion y z). Proof. - unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *. + unfold meq; unfold munion; unfold multiplicity. destruct x; destruct y; destruct z. intros; elim H; auto with arith. Qed. @@ -94,7 +94,7 @@ Section multiset_defs. Lemma meq_right : forall x y z:multiset, meq x y -> meq (munion z x) (munion z y). Proof. - unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *. + unfold meq; unfold munion; unfold multiplicity. destruct x; destruct y; destruct z. intros; elim H; auto. Qed. diff --git a/theories/Sets/Partial_Order.v b/theories/Sets/Partial_Order.v index a319b9832..bb1cf7083 100644 --- a/theories/Sets/Partial_Order.v +++ b/theories/Sets/Partial_Order.v @@ -63,13 +63,13 @@ Section Partial_order_facts. forall x y z:U, Strict_Rel_of U D x y -> Rel_of U D y z -> Strict_Rel_of U D x z. Proof. - unfold Strict_Rel_of at 1 in |- *. - red in |- *. - elim D; simpl in |- *. + unfold Strict_Rel_of at 1. + red. + elim D; simpl. intros C R H' H'0; elim H'0. intros H'1 H'2 H'3 x y z H'4 H'5; split. apply H'2 with (y := y); tauto. - red in |- *; intro H'6. + red; intro H'6. elim H'4; intros H'7 H'8; apply H'8; clear H'4. apply H'3; auto. rewrite H'6; tauto. @@ -79,20 +79,20 @@ Section Partial_order_facts. forall x y z:U, Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z. Proof. - unfold Strict_Rel_of at 1 in |- *. - red in |- *. - elim D; simpl in |- *. + unfold Strict_Rel_of at 1. + red. + elim D; simpl. intros C R H' H'0; elim H'0. intros H'1 H'2 H'3 x y z H'4 H'5; split. apply H'2 with (y := y); tauto. - red in |- *; intro H'6. + red; intro H'6. elim H'5; intros H'7 H'8; apply H'8; clear H'5. apply H'3; auto. rewrite <- H'6; auto. Qed. Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D). - red in |- *. + red. intros x y z H' H'0. apply Strict_Rel_Transitive_with_Rel with (y := y); [ intuition | unfold Strict_Rel_of in H', H'0; intuition ]. diff --git a/theories/Sets/Powerset.v b/theories/Sets/Powerset.v index f8b24e747..f3b7c4deb 100644 --- a/theories/Sets/Powerset.v +++ b/theories/Sets/Powerset.v @@ -39,7 +39,7 @@ Inductive Power_set (A:Ensemble U) : Ensemble (Ensemble U) := Hint Resolve Definition_of_Power_set. Theorem Empty_set_minimal : forall X:Ensemble U, Included U (Empty_set U) X. -intro X; red in |- *. +intro X; red. intros x H'; elim H'. Qed. Hint Resolve Empty_set_minimal. @@ -79,7 +79,7 @@ Lemma Strict_inclusion_is_transitive_with_inclusion : Strict_Included U x y -> Included U y z -> Strict_Included U x z. intros x y z H' H'0; try assumption. elim Strict_Rel_is_Strict_Included. -unfold contains in |- *. +unfold contains. intros H'1 H'2; try assumption. apply H'1. apply Strict_Rel_Transitive_with_Rel with (y := y); auto with sets. @@ -90,7 +90,7 @@ Lemma Strict_inclusion_is_transitive_with_inclusion_left : Included U x y -> Strict_Included U y z -> Strict_Included U x z. intros x y z H' H'0; try assumption. elim Strict_Rel_is_Strict_Included. -unfold contains in |- *. +unfold contains. intros H'1 H'2; try assumption. apply H'1. apply Strict_Rel_Transitive_with_Rel_left with (y := y); auto with sets. @@ -105,14 +105,14 @@ Qed. Theorem Empty_set_is_Bottom : forall A:Ensemble U, Bottom (Ensemble U) (Power_set_PO A) (Empty_set U). -intro A; apply Bottom_definition; simpl in |- *; auto with sets. +intro A; apply Bottom_definition; simpl; auto with sets. Qed. Hint Resolve Empty_set_is_Bottom. Theorem Union_minimal : forall a b X:Ensemble U, Included U a X -> Included U b X -> Included U (Union U a b) X. -intros a b X H' H'0; red in |- *. +intros a b X H' H'0; red. intros x H'1; elim H'1; auto with sets. Qed. Hint Resolve Union_minimal. @@ -133,13 +133,13 @@ Qed. Theorem Intersection_decreases_l : forall a b:Ensemble U, Included U (Intersection U a b) a. -intros a b; red in |- *. +intros a b; red. intros x H'; elim H'; auto with sets. Qed. Theorem Intersection_decreases_r : forall a b:Ensemble U, Included U (Intersection U a b) b. -intros a b; red in |- *. +intros a b; red. intros x H'; elim H'; auto with sets. Qed. Hint Resolve Union_increases_l Union_increases_r Intersection_decreases_l @@ -151,10 +151,10 @@ Theorem Union_is_Lub : Included U b A -> Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b). intros A a b H' H'0. -apply Lub_definition; simpl in |- *. -apply Upper_Bound_definition; simpl in |- *; auto with sets. +apply Lub_definition; simpl. +apply Upper_Bound_definition; simpl; auto with sets. intros y H'1; elim H'1; auto with sets. -intros y H'1; elim H'1; simpl in |- *; auto with sets. +intros y H'1; elim H'1; simpl; auto with sets. Qed. Theorem Intersection_is_Glb : @@ -164,13 +164,13 @@ Theorem Intersection_is_Glb : Glb (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Intersection U a b). intros A a b H' H'0. -apply Glb_definition; simpl in |- *. -apply Lower_Bound_definition; simpl in |- *; auto with sets. +apply Glb_definition; simpl. +apply Lower_Bound_definition; simpl; auto with sets. apply Definition_of_Power_set. generalize Inclusion_is_transitive; intro IT; red in IT; apply IT with a; auto with sets. intros y H'1; elim H'1; auto with sets. -intros y H'1; elim H'1; simpl in |- *; auto with sets. +intros y H'1; elim H'1; simpl; auto with sets. Qed. End The_power_set_partial_order. diff --git a/theories/Sets/Powerset_Classical_facts.v b/theories/Sets/Powerset_Classical_facts.v index 09fc20948..a7601aaaf 100644 --- a/theories/Sets/Powerset_Classical_facts.v +++ b/theories/Sets/Powerset_Classical_facts.v @@ -44,13 +44,13 @@ Section Sets_as_an_algebra. ~ In U A x -> Strict_Included U (Add U A x) (Add U B x) -> Strict_Included U A B. Proof. - intros A B x H' H'0; red in |- *. + intros A B x H' H'0; red. lapply (Strict_Included_inv U (Add U A x) (Add U B x)); auto with sets. clear H'0; intro H'0; split. apply incl_add_x with (x := x); tauto. elim H'0; intros H'1 H'2; elim H'2; clear H'0 H'2. intros x0 H'0. - red in |- *; intro H'2. + red; intro H'2. elim H'0; clear H'0. rewrite <- H'2; auto with sets. Qed. @@ -58,7 +58,7 @@ Section Sets_as_an_algebra. Lemma incl_soustr_in : forall (X:Ensemble U) (x:U), In U X x -> Included U (Subtract U X x) X. Proof. - intros X x H'; red in |- *. + intros X x H'; red. intros x0 H'0; elim H'0; auto with sets. Qed. @@ -66,7 +66,7 @@ Section Sets_as_an_algebra. forall (X Y:Ensemble U) (x:U), Included U X Y -> Included U (Subtract U X x) (Subtract U Y x). Proof. - intros X Y x H'; red in |- *. + intros X Y x H'; red. intros x0 H'0; elim H'0. intros H'1 H'2. apply Subtract_intro; auto with sets. @@ -75,7 +75,7 @@ Section Sets_as_an_algebra. Lemma incl_soustr_add_l : forall (X:Ensemble U) (x:U), Included U (Subtract U (Add U X x) x) X. Proof. - intros X x; red in |- *. + intros X x; red. intros x0 H'; elim H'; auto with sets. intro H'0; elim H'0; auto with sets. intros t H'1 H'2; elim H'2; auto with sets. @@ -85,10 +85,10 @@ Section Sets_as_an_algebra. forall (X:Ensemble U) (x:U), ~ In U X x -> Included U X (Subtract U (Add U X x) x). Proof. - intros X x H'; red in |- *. + intros X x H'; red. intros x0 H'0; try assumption. apply Subtract_intro; auto with sets. - red in |- *; intro H'1; apply H'; rewrite H'1; auto with sets. + red; intro H'1; apply H'; rewrite H'1; auto with sets. Qed. Hint Resolve incl_soustr_add_r: sets v62. @@ -96,7 +96,7 @@ Section Sets_as_an_algebra. forall (X:Ensemble U) (x:U), In U X x -> Included U X (Add U (Subtract U X x) x). Proof. - intros X x H'; red in |- *. + intros X x H'; red. intros x0 H'0; try assumption. elim (classic (x = x0)); intro K; auto with sets. elim K; auto with sets. @@ -106,7 +106,7 @@ Section Sets_as_an_algebra. forall (X:Ensemble U) (x:U), In U X x -> Included U (Add U (Subtract U X x) x) X. Proof. - intros X x H'; red in |- *. + intros X x H'; red. intros x0 H'0; elim H'0; auto with sets. intros y H'1; elim H'1; auto with sets. intros t H'1; try assumption. @@ -118,7 +118,7 @@ Section Sets_as_an_algebra. x <> y -> Subtract U (Add U X x) y = Add U (Subtract U X y) x. Proof. intros X x y H'; apply Extensionality_Ensembles. - split; red in |- *. + split; red. intros x0 H'0; elim H'0; auto with sets. intro H'1; elim H'1. intros u H'2 H'3; try assumption. @@ -146,7 +146,7 @@ Section Sets_as_an_algebra. apply H'4 with (y := Y); auto using add_soustr_2 with sets. red in H'0. elim H'0; intros H'1 H'2; try exact H'1; clear H'0. (* PB *) - red in |- *; intro H'0; apply H'2. + red; intro H'0; apply H'2. rewrite H'0; auto 8 using add_soustr_xy, add_soustr_1, add_soustr_2 with sets. Qed. @@ -177,7 +177,7 @@ Section Sets_as_an_algebra. exists (Subtract U X x). split; auto using incl_soustr_in, add_soustr_xy, add_soustr_1, add_soustr_2 with sets. red in H'0. - red in |- *. + red. intros x0 H'2; try assumption. lapply (Subtract_inv U X x x0); auto with sets. intro H'3; elim H'3; intros K K'; clear H'3. @@ -189,7 +189,7 @@ Section Sets_as_an_algebra. elim K'; auto with sets. intro H'1; left; try assumption. red in H'0. - red in |- *. + red. intros x0 H'2; try assumption. lapply (H'0 x0); auto with sets. intro H'3; try assumption. @@ -207,7 +207,7 @@ Section Sets_as_an_algebra. (forall z:Ensemble U, Included U x z -> Included U z y -> x = z \/ z = y). Proof. intros A x y H'; elim H'. - unfold Strict_Rel_of in |- *; simpl in |- *. + unfold Strict_Rel_of; simpl. intros H'0 H'1; split; [ auto with sets | idtac ]. intros z H'2 H'3; try assumption. elim (classic (x = z)); auto with sets. @@ -227,11 +227,11 @@ Section Sets_as_an_algebra. Proof. intros A a H' x H'0 H'1; try assumption. apply setcover_intro; auto with sets. - red in |- *. - split; [ idtac | red in |- *; intro H'2; try exact H'2 ]; auto with sets. + red. + split; [ idtac | red; intro H'2; try exact H'2 ]; auto with sets. apply H'1. rewrite H'2; auto with sets. - red in |- *; intro H'2; elim H'2; clear H'2. + red; intro H'2; elim H'2; clear H'2. intros z H'2; elim H'2; intros H'3 H'4; try exact H'3; clear H'2. lapply (Strict_Included_inv U a z); auto with sets; clear H'3. intro H'2; elim H'2; intros H'3 H'5; elim H'5; clear H'2 H'5. @@ -249,7 +249,7 @@ Section Sets_as_an_algebra. red in K. elim K; intros H'11 H'12; apply H'12; clear K; auto with sets. rewrite H'15. - red in |- *. + red. intros x1 H'10; elim H'10; auto with sets. intros x2 H'11; elim H'11; auto with sets. Qed. @@ -275,11 +275,11 @@ Section Sets_as_an_algebra. elim (H'7 (Add U a x)); auto with sets. intro H'1. absurd (a = Add U a x); auto with sets. - red in |- *; intro H'8; try exact H'8. + red; intro H'8; try exact H'8. apply H'3. rewrite H'8; auto with sets. auto with sets. - red in |- *. + red. intros x0 H'1; elim H'1; auto with sets. intros x1 H'8; elim H'8; auto with sets. split; [ idtac | try assumption ]. diff --git a/theories/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v index f756f9854..14a2d25cc 100644 --- a/theories/Sets/Powerset_facts.v +++ b/theories/Sets/Powerset_facts.v @@ -42,7 +42,7 @@ Section Sets_as_an_algebra. Theorem Empty_set_zero' : forall x:U, Add U (Empty_set U) x = Singleton U x. Proof. - unfold Add at 1 in |- *; auto using Empty_set_zero with sets. + unfold Add at 1; auto using Empty_set_zero with sets. Qed. Lemma less_than_empty : @@ -76,7 +76,7 @@ Section Sets_as_an_algebra. Theorem Couple_as_union : forall x y:U, Union U (Singleton U x) (Singleton U y) = Couple U x y. Proof. - intros x y; apply Extensionality_Ensembles; split; red in |- *. + intros x y; apply Extensionality_Ensembles; split; red. intros x0 H'; elim H'; (intros x1 H'0; elim H'0; auto with sets). intros x0 H'; elim H'; auto with sets. Qed. @@ -86,7 +86,7 @@ Section Sets_as_an_algebra. Union U (Union U (Singleton U x) (Singleton U y)) (Singleton U z) = Triple U x y z. Proof. - intros x y z; apply Extensionality_Ensembles; split; red in |- *. + intros x y z; apply Extensionality_Ensembles; split; red. intros x0 H'; elim H'. intros x1 H'0; elim H'0; (intros x2 H'1; elim H'1; auto with sets). intros x1 H'0; elim H'0; auto with sets. @@ -114,7 +114,7 @@ Section Sets_as_an_algebra. Proof. intros A B. apply Extensionality_Ensembles. - split; red in |- *; intros x H'; elim H'; auto with sets. + split; red; intros x H'; elim H'; auto with sets. Qed. Theorem Distributivity : @@ -124,7 +124,7 @@ Section Sets_as_an_algebra. Proof. intros A B C. apply Extensionality_Ensembles. - split; red in |- *; intros x H'. + split; red; intros x H'. elim H'. intros x0 H'0 H'1; generalize H'0. elim H'1; auto with sets. @@ -138,7 +138,7 @@ Section Sets_as_an_algebra. Proof. intros A B C. apply Extensionality_Ensembles. - split; red in |- *; intros x H'. + split; red; intros x H'. elim H'; auto with sets. intros x0 H'0; elim H'0; auto with sets. elim H'. @@ -151,15 +151,15 @@ Section Sets_as_an_algebra. Theorem Union_add : forall (A B:Ensemble U) (x:U), Add U (Union U A B) x = Union U A (Add U B x). Proof. - unfold Add in |- *; auto using Union_associative with sets. + unfold Add; auto using Union_associative with sets. Qed. Theorem Non_disjoint_union : forall (X:Ensemble U) (x:U), In U X x -> Add U X x = X. Proof. - intros X x H'; unfold Add in |- *. - apply Extensionality_Ensembles; red in |- *. - split; red in |- *; auto with sets. + intros X x H'; unfold Add. + apply Extensionality_Ensembles; red. + split; red; auto with sets. intros x0 H'0; elim H'0; auto with sets. intros t H'1; elim H'1; auto with sets. Qed. @@ -167,12 +167,12 @@ Section Sets_as_an_algebra. Theorem Non_disjoint_union' : forall (X:Ensemble U) (x:U), ~ In U X x -> Subtract U X x = X. Proof. - intros X x H'; unfold Subtract in |- *. + intros X x H'; unfold Subtract. apply Extensionality_Ensembles. - split; red in |- *; auto with sets. + split; red; auto with sets. intros x0 H'0; elim H'0; auto with sets. intros x0 H'0; apply Setminus_intro; auto with sets. - red in |- *; intro H'1; elim H'1. + red; intro H'1; elim H'1. lapply (Singleton_inv U x x0); auto with sets. intro H'4; apply H'; rewrite H'4; auto with sets. Qed. @@ -186,7 +186,7 @@ Section Sets_as_an_algebra. forall (A B:Ensemble U) (x:U), Included U A B -> Included U (Add U A x) (Add U B x). Proof. - intros A B x H'; red in |- *; auto with sets. + intros A B x H'; red; auto with sets. intros x0 H'0. lapply (Add_inv U A x x0); auto with sets. intro H'1; elim H'1; @@ -198,7 +198,7 @@ Section Sets_as_an_algebra. forall (A B:Ensemble U) (x:U), ~ In U A x -> Included U (Add U A x) (Add U B x) -> Included U A B. Proof. - unfold Included in |- *. + unfold Included. intros A B x H' H'0 x0 H'1. lapply (H'0 x0); auto with sets. intro H'2; lapply (Add_inv U B x x0); auto with sets. @@ -212,7 +212,7 @@ Section Sets_as_an_algebra. forall (A:Ensemble U) (x y:U), Add U (Add U A x) y = Add U (Add U A y) x. Proof. intros A x y. - unfold Add in |- *. + unfold Add. rewrite (Union_associative A (Singleton U x) (Singleton U y)). rewrite (Union_commutative (Singleton U x) (Singleton U y)). rewrite <- (Union_associative A (Singleton U y) (Singleton U x)); @@ -234,7 +234,7 @@ Section Sets_as_an_algebra. Proof. intros A B x y H'; try assumption. rewrite <- (Union_add (Add U A x) B y). - unfold Add at 4 in |- *. + unfold Add at 4. rewrite (Union_commutative A (Singleton U x)). rewrite Union_associative. rewrite (Union_absorbs A B H'). diff --git a/theories/Sets/Relations_1_facts.v b/theories/Sets/Relations_1_facts.v index 0c8329dd0..efd895e2c 100644 --- a/theories/Sets/Relations_1_facts.v +++ b/theories/Sets/Relations_1_facts.v @@ -33,8 +33,8 @@ Theorem Rsym_imp_notRsym : forall (U:Type) (R:Relation U), Symmetric U R -> Symmetric U (Complement U R). Proof. -unfold Symmetric, Complement in |- *. -intros U R H' x y H'0; red in |- *; intro H'1; apply H'0; auto with sets. +unfold Symmetric, Complement. +intros U R H' x y H'0; red; intro H'1; apply H'0; auto with sets. Qed. Theorem Equiv_from_preorder : @@ -44,8 +44,8 @@ Proof. intros U R H'; elim H'; intros H'0 H'1. apply Definition_of_equivalence. red in H'0; auto 10 with sets. -2: red in |- *; intros x y h; elim h; intros H'3 H'4; auto 10 with sets. -red in H'1; red in |- *; auto 10 with sets. +2: red; intros x y h; elim h; intros H'3 H'4; auto 10 with sets. +red in H'1; red; auto 10 with sets. intros x y z h; elim h; intros H'3 H'4; clear h. intro h; elim h; intros H'5 H'6; clear h. split; apply H'1 with y; auto 10 with sets. @@ -70,7 +70,7 @@ Hint Resolve contains_is_preorder. Theorem same_relation_is_equivalence : forall U:Type, Equivalence (Relation U) (same_relation U). Proof. -unfold same_relation at 1 in |- *; auto 10 with sets. +unfold same_relation at 1; auto 10 with sets. Qed. Hint Resolve same_relation_is_equivalence. @@ -78,14 +78,14 @@ Theorem cong_reflexive_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Reflexive U R -> Reflexive U R'. Proof. -unfold same_relation in |- *; intuition. +unfold same_relation; intuition. Qed. Theorem cong_symmetric_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Symmetric U R -> Symmetric U R'. Proof. - compute in |- *; intros; elim H; intros; clear H; + compute; intros; elim H; intros; clear H; apply (H3 y x (H0 x y (H2 x y H1))). (*Intuition.*) Qed. @@ -94,7 +94,7 @@ Theorem cong_antisymmetric_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Antisymmetric U R -> Antisymmetric U R'. Proof. - compute in |- *; intros; elim H; intros; clear H; + compute; intros; elim H; intros; clear H; apply (H0 x y (H3 x y H1) (H3 y x H2)). (*Intuition.*) Qed. @@ -103,7 +103,7 @@ Theorem cong_transitive_same_relation : forall (U:Type) (R R':Relation U), same_relation U R R' -> Transitive U R -> Transitive U R'. Proof. -intros U R R' H' H'0; red in |- *. +intros U R R' H' H'0; red. elim H'. intros H'1 H'2 x y z H'3 H'4; apply H'2. apply H'0 with y; auto with sets. diff --git a/theories/Sets/Relations_2_facts.v b/theories/Sets/Relations_2_facts.v index 89b98c1f5..bc5960ebe 100644 --- a/theories/Sets/Relations_2_facts.v +++ b/theories/Sets/Relations_2_facts.v @@ -43,13 +43,13 @@ Qed. Theorem Rstar_contains_R : forall (U:Type) (R:Relation U), contains U (Rstar U R) R. Proof. -intros U R; red in |- *; intros x y H'; apply Rstar_n with y; auto with sets. +intros U R; red; intros x y H'; apply Rstar_n with y; auto with sets. Qed. Theorem Rstar_contains_Rplus : forall (U:Type) (R:Relation U), contains U (Rstar U R) (Rplus U R). Proof. -intros U R; red in |- *. +intros U R; red. intros x y H'; elim H'. generalize Rstar_contains_R; intro T; red in T; auto with sets. intros x0 y0 z H'0 H'1 H'2; apply Rstar_n with y0; auto with sets. @@ -58,7 +58,7 @@ Qed. Theorem Rstar_transitive : forall (U:Type) (R:Relation U), Transitive U (Rstar U R). Proof. -intros U R; red in |- *. +intros U R; red. intros x y z H'; elim H'; auto with sets. intros x0 y0 z0 H'0 H'1 H'2 H'3; apply Rstar_n with y0; auto with sets. Qed. @@ -75,7 +75,7 @@ Theorem Rstar_equiv_Rstar1 : forall (U:Type) (R:Relation U), same_relation U (Rstar U R) (Rstar1 U R). Proof. generalize Rstar_contains_R; intro T; red in T. -intros U R; unfold same_relation, contains in |- *. +intros U R; unfold same_relation, contains. split; intros x y H'; elim H'; auto with sets. generalize Rstar_transitive; intro T1; red in T1. intros x0 y0 z H'0 H'1 H'2 H'3; apply T1 with y0; auto with sets. @@ -85,7 +85,7 @@ Qed. Theorem Rsym_imp_Rstarsym : forall (U:Type) (R:Relation U), Symmetric U R -> Symmetric U (Rstar U R). Proof. -intros U R H'; red in |- *. +intros U R H'; red. intros x y H'0; elim H'0; auto with sets. intros x0 y0 z H'1 H'2 H'3. generalize Rstar_transitive; intro T1; red in T1. @@ -97,7 +97,7 @@ Theorem Sstar_contains_Rstar : forall (U:Type) (R S:Relation U), contains U (Rstar U S) R -> contains U (Rstar U S) (Rstar U R). Proof. -unfold contains in |- *. +unfold contains. intros U R S H' x y H'0; elim H'0; auto with sets. generalize Rstar_transitive; intro T1; red in T1. intros x0 y0 z H'1 H'2 H'3; apply T1 with y0; auto with sets. diff --git a/theories/Sets/Relations_3_facts.v b/theories/Sets/Relations_3_facts.v index 8ac6e7fb4..455ad5843 100644 --- a/theories/Sets/Relations_3_facts.v +++ b/theories/Sets/Relations_3_facts.v @@ -33,7 +33,7 @@ Require Export Relations_3. Theorem Rstar_imp_coherent : forall (U:Type) (R:Relation U) (x y:U), Rstar U R x y -> coherent U R x y. Proof. -intros U R x y H'; red in |- *. +intros U R x y H'; red. exists y; auto with sets. Qed. Hint Resolve Rstar_imp_coherent. @@ -41,8 +41,8 @@ Hint Resolve Rstar_imp_coherent. Theorem coherent_symmetric : forall (U:Type) (R:Relation U), Symmetric U (coherent U R). Proof. -unfold coherent at 1 in |- *. -intros U R; red in |- *. +unfold coherent at 1. +intros U R; red. intros x y H'; elim H'. intros z H'0; exists z; tauto. Qed. @@ -50,9 +50,9 @@ Qed. Theorem Strong_confluence : forall (U:Type) (R:Relation U), Strongly_confluent U R -> Confluent U R. Proof. -intros U R H'; red in |- *. -intro x; red in |- *; intros a b H'0. -unfold coherent at 1 in |- *. +intros U R H'; red. +intro x; red; intros a b H'0. +unfold coherent at 1. generalize b; clear b. elim H'0; clear H'0. intros x0 b H'1; exists b; auto with sets. @@ -75,9 +75,9 @@ Qed. Theorem Strong_confluence_direct : forall (U:Type) (R:Relation U), Strongly_confluent U R -> Confluent U R. Proof. -intros U R H'; red in |- *. -intro x; red in |- *; intros a b H'0. -unfold coherent at 1 in |- *. +intros U R H'; red. +intro x; red; intros a b H'0. +unfold coherent at 1. generalize b; clear b. elim H'0; clear H'0. intros x0 b H'1; exists b; auto with sets. @@ -111,7 +111,7 @@ Theorem Noetherian_contains_Noetherian : forall (U:Type) (R R':Relation U), Noetherian U R -> contains U R R' -> Noetherian U R'. Proof. -unfold Noetherian at 2 in |- *. +unfold Noetherian at 2. intros U R R' H' H'0 x. elim (H' x); auto with sets. Qed. @@ -120,8 +120,8 @@ Theorem Newman : forall (U:Type) (R:Relation U), Noetherian U R -> Locally_confluent U R -> Confluent U R. Proof. -intros U R H' H'0; red in |- *; intro x. -elim (H' x); unfold confluent in |- *. +intros U R H' H'0; red; intro x. +elim (H' x); unfold confluent. intros x0 H'1 H'2 y z H'3 H'4. generalize (Rstar_cases U R x0 y); intro h; lapply h; [ intro h0; elim h0; @@ -163,7 +163,7 @@ generalize (H'2 v); intro h; lapply h; | clear h h0 ] | clear h h0 ] | clear h ]; auto with sets. -red in |- *; (exists z1; split); auto with sets. +red; (exists z1; split); auto with sets. apply T with y1; auto with sets. apply T with t; auto with sets. Qed. diff --git a/theories/Sets/Uniset.v b/theories/Sets/Uniset.v index bf1aaf8db..ca4519ee4 100644 --- a/theories/Sets/Uniset.v +++ b/theories/Sets/Uniset.v @@ -51,37 +51,37 @@ Hint Unfold seq. Lemma leb_refl : forall b:bool, leb b b. Proof. -destruct b; simpl in |- *; auto. +destruct b; simpl; auto. Qed. Hint Resolve leb_refl. Lemma incl_left : forall s1 s2:uniset, seq s1 s2 -> incl s1 s2. Proof. -unfold incl in |- *; intros s1 s2 E a; elim (E a); auto. +unfold incl; intros s1 s2 E a; elim (E a); auto. Qed. Lemma incl_right : forall s1 s2:uniset, seq s1 s2 -> incl s2 s1. Proof. -unfold incl in |- *; intros s1 s2 E a; elim (E a); auto. +unfold incl; intros s1 s2 E a; elim (E a); auto. Qed. Lemma seq_refl : forall x:uniset, seq x x. Proof. -destruct x; unfold seq in |- *; auto. +destruct x; unfold seq; auto. Qed. Hint Resolve seq_refl. Lemma seq_trans : forall x y z:uniset, seq x y -> seq y z -> seq x z. Proof. -unfold seq in |- *. -destruct x; destruct y; destruct z; simpl in |- *; intros. +unfold seq. +destruct x; destruct y; destruct z; simpl; intros. rewrite H; auto. Qed. Lemma seq_sym : forall x y:uniset, seq x y -> seq y x. Proof. -unfold seq in |- *. -destruct x; destruct y; simpl in |- *; auto. +unfold seq. +destruct x; destruct y; simpl; auto. Qed. (** uniset union *) @@ -90,20 +90,20 @@ Definition union (m1 m2:uniset) := Lemma union_empty_left : forall x:uniset, seq x (union Emptyset x). Proof. -unfold seq in |- *; unfold union in |- *; simpl in |- *; auto. +unfold seq; unfold union; simpl; auto. Qed. Hint Resolve union_empty_left. Lemma union_empty_right : forall x:uniset, seq x (union x Emptyset). Proof. -unfold seq in |- *; unfold union in |- *; simpl in |- *. +unfold seq; unfold union; simpl. intros x a; rewrite (orb_b_false (charac x a)); auto. Qed. Hint Resolve union_empty_right. Lemma union_comm : forall x y:uniset, seq (union x y) (union y x). Proof. -unfold seq in |- *; unfold charac in |- *; unfold union in |- *. +unfold seq; unfold charac; unfold union. destruct x; destruct y; auto with bool. Qed. Hint Resolve union_comm. @@ -111,14 +111,14 @@ Hint Resolve union_comm. Lemma union_ass : forall x y z:uniset, seq (union (union x y) z) (union x (union y z)). Proof. -unfold seq in |- *; unfold union in |- *; unfold charac in |- *. +unfold seq; unfold union; unfold charac. destruct x; destruct y; destruct z; auto with bool. Qed. Hint Resolve union_ass. Lemma seq_left : forall x y z:uniset, seq x y -> seq (union x z) (union y z). Proof. -unfold seq in |- *; unfold union in |- *; unfold charac in |- *. +unfold seq; unfold union; unfold charac. destruct x; destruct y; destruct z. intros; elim H; auto. Qed. @@ -126,7 +126,7 @@ Hint Resolve seq_left. Lemma seq_right : forall x y z:uniset, seq x y -> seq (union z x) (union z y). Proof. -unfold seq in |- *; unfold union in |- *; unfold charac in |- *. +unfold seq; unfold union; unfold charac. destruct x; destruct y; destruct z. intros; elim H; auto. Qed. diff --git a/theories/Sorting/Heap.v b/theories/Sorting/Heap.v index 60bb50cec..8653640d3 100644 --- a/theories/Sorting/Heap.v +++ b/theories/Sorting/Heap.v @@ -55,13 +55,13 @@ Section defs. Lemma leA_Tree_Leaf : forall a:A, leA_Tree a Tree_Leaf. Proof. - simpl in |- *; auto with datatypes. + simpl; auto with datatypes. Qed. Lemma leA_Tree_Node : forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D). Proof. - simpl in |- *; auto with datatypes. + simpl; auto with datatypes. Qed. @@ -121,7 +121,7 @@ Section defs. forall (T:Tree) (a b:A), leA a b -> leA_Tree b T -> leA_Tree a T. Proof. simple induction T; auto with datatypes. - intros; simpl in |- *; apply leA_trans with b; auto with datatypes. + intros; simpl; apply leA_trans with b; auto with datatypes. Qed. (** ** Merging two sorted lists *) @@ -213,12 +213,12 @@ Section defs. simple induction 1; intros. apply insert_exist with (Tree_Node a Tree_Leaf Tree_Leaf); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. - simpl in |- *; unfold meq, munion in |- *; auto using node_is_heap with datatypes. + simpl; unfold meq, munion; auto using node_is_heap with datatypes. elim (leA_dec a a0); intros. elim (X a0); intros. apply insert_exist with (Tree_Node a T2 T0); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. - simpl in |- *; apply treesort_twist1; trivial with datatypes. + simpl; apply treesort_twist1; trivial with datatypes. elim (X a); intros T3 HeapT3 ConT3 LeA. apply insert_exist with (Tree_Node a0 T2 T3); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. @@ -226,7 +226,7 @@ Section defs. apply low_trans with a; auto with datatypes. apply LeA; auto with datatypes. apply low_trans with a; auto with datatypes. - simpl in |- *; apply treesort_twist2; trivial with datatypes. + simpl; apply treesort_twist2; trivial with datatypes. Qed. @@ -242,10 +242,10 @@ Section defs. Proof. simple induction l. apply (heap_exist nil Tree_Leaf); auto with datatypes. - simpl in |- *; unfold meq in |- *; exact nil_is_heap. + simpl; unfold meq; exact nil_is_heap. simple induction 1. intros T i m; elim (insert T i a). - intros; apply heap_exist with T1; simpl in |- *; auto with datatypes. + intros; apply heap_exist with T1; simpl; auto with datatypes. apply meq_trans with (munion (contents T) (singletonBag a)). apply meq_trans with (munion (singletonBag a) (contents T)). apply meq_right; trivial with datatypes. @@ -269,7 +269,7 @@ Section defs. apply flat_exist with (nil (A:=A)); auto with datatypes. elim X; intros l1 s1 i1 m1; elim X0; intros l2 s2 i2 m2. elim (merge _ s1 _ s2); intros. - apply flat_exist with (a :: l); simpl in |- *; auto with datatypes. + apply flat_exist with (a :: l); simpl; auto with datatypes. apply meq_trans with (munion (list_contents _ eqA_dec l1) (munion (list_contents _ eqA_dec l2) (singletonBag a))). @@ -288,7 +288,7 @@ Section defs. forall l:list A, {m : list A | Sorted leA m & permutation _ eqA_dec l m}. Proof. - intro l; unfold permutation in |- *. + intro l; unfold permutation. elim (list_to_heap l). intros. elim (heap_to_list T); auto with datatypes. diff --git a/theories/Sorting/PermutSetoid.v b/theories/Sorting/PermutSetoid.v index b2b15c705..aed7150c8 100644 --- a/theories/Sorting/PermutSetoid.v +++ b/theories/Sorting/PermutSetoid.v @@ -52,7 +52,7 @@ Lemma list_contents_app : forall l m:list A, meq (list_contents (l ++ m)) (munion (list_contents l) (list_contents m)). Proof. - simple induction l; simpl in |- *; auto with datatypes. + simple induction l; simpl; auto with datatypes. intros. apply meq_trans with (munion (singletonBag a) (munion (list_contents l0) (list_contents m))); @@ -65,7 +65,7 @@ Definition permutation (l m:list A) := meq (list_contents l) (list_contents m). Lemma permut_refl : forall l:list A, permutation l l. Proof. - unfold permutation in |- *; auto with datatypes. + unfold permutation; auto with datatypes. Qed. Lemma permut_sym : @@ -77,7 +77,7 @@ Qed. Lemma permut_trans : forall l m n:list A, permutation l m -> permutation m n -> permutation l n. Proof. - unfold permutation in |- *; intros. + unfold permutation; intros. apply meq_trans with (list_contents m); auto with datatypes. Qed. @@ -102,7 +102,7 @@ Lemma permut_app : forall l l' m m':list A, permutation l l' -> permutation m m' -> permutation (l ++ m) (l' ++ m'). Proof. - unfold permutation in |- *; intros. + unfold permutation; intros. apply meq_trans with (munion (list_contents l) (list_contents m)); auto using permut_cons, list_contents_app with datatypes. apply meq_trans with (munion (list_contents l') (list_contents m')); diff --git a/theories/Strings/String.v b/theories/Strings/String.v index 289ffab31..1e824e2ea 100644 --- a/theories/Strings/String.v +++ b/theories/Strings/String.v @@ -72,14 +72,14 @@ Fixpoint get (n : nat) (s : string) {struct s} : option ascii := Theorem get_correct : forall s1 s2 : string, (forall n : nat, get n s1 = get n s2) <-> s1 = s2. Proof. -intros s1; elim s1; simpl in |- *. -intros s2; case s2; simpl in |- *; split; auto. +intros s1; elim s1; simpl. +intros s2; case s2; simpl; split; auto. intros H; generalize (H 0); intros H1; inversion H1. intros; discriminate. -intros a s1' Rec s2; case s2; simpl in |- *; split; auto. +intros a s1' Rec s2; case s2; simpl; split; auto. intros H; generalize (H 0); intros H1; inversion H1. intros; discriminate. -intros H; generalize (H 0); simpl in |- *; intros H1; inversion H1. +intros H; generalize (H 0); simpl; intros H1; inversion H1. case (Rec s). intros H0; rewrite H0; auto. intros n; exact (H (S n)). @@ -94,9 +94,9 @@ Theorem append_correct1 : forall (s1 s2 : string) (n : nat), n < length s1 -> get n s1 = get n (s1 ++ s2). Proof. -intros s1; elim s1; simpl in |- *; auto. +intros s1; elim s1; simpl; auto. intros s2 n H; inversion H. -intros a s1' Rec s2 n; case n; simpl in |- *; auto. +intros a s1' Rec s2 n; case n; simpl; auto. intros n0 H; apply Rec; auto. apply lt_S_n; auto. Qed. @@ -107,10 +107,10 @@ Theorem append_correct2 : forall (s1 s2 : string) (n : nat), get n s2 = get (n + length s1) (s1 ++ s2). Proof. -intros s1; elim s1; simpl in |- *; auto. -intros s2 n; rewrite plus_comm; simpl in |- *; auto. -intros a s1' Rec s2 n; case n; simpl in |- *; auto. -generalize (Rec s2 0); simpl in |- *; auto. intros. +intros s1; elim s1; simpl; auto. +intros s2 n; rewrite plus_comm; simpl; auto. +intros a s1' Rec s2 n; case n; simpl; auto. +generalize (Rec s2 0); simpl; auto. intros. rewrite <- Plus.plus_Snm_nSm; auto. Qed. @@ -135,16 +135,16 @@ Theorem substring_correct1 : forall (s : string) (n m p : nat), p < m -> get p (substring n m s) = get (p + n) s. Proof. -intros s; elim s; simpl in |- *; auto. -intros n; case n; simpl in |- *; auto. -intros m; case m; simpl in |- *; auto. -intros a s' Rec; intros n; case n; simpl in |- *; auto. -intros m; case m; simpl in |- *; auto. +intros s; elim s; simpl; auto. +intros n; case n; simpl; auto. +intros m; case m; simpl; auto. +intros a s' Rec; intros n; case n; simpl; auto. +intros m; case m; simpl; auto. intros p H; inversion H. -intros m' p; case p; simpl in |- *; auto. -intros n0 H; apply Rec; simpl in |- *; auto. +intros m' p; case p; simpl; auto. +intros n0 H; apply Rec; simpl; auto. apply Lt.lt_S_n; auto. -intros n' m p H; rewrite <- Plus.plus_Snm_nSm; simpl in |- *; auto. +intros n' m p H; rewrite <- Plus.plus_Snm_nSm; simpl; auto. Qed. (** The substring has at most [m] elements *) @@ -152,14 +152,14 @@ Qed. Theorem substring_correct2 : forall (s : string) (n m p : nat), m <= p -> get p (substring n m s) = None. Proof. -intros s; elim s; simpl in |- *; auto. -intros n; case n; simpl in |- *; auto. -intros m; case m; simpl in |- *; auto. -intros a s' Rec; intros n; case n; simpl in |- *; auto. -intros m; case m; simpl in |- *; auto. -intros m' p; case p; simpl in |- *; auto. +intros s; elim s; simpl; auto. +intros n; case n; simpl; auto. +intros m; case m; simpl; auto. +intros a s' Rec; intros n; case n; simpl; auto. +intros m; case m; simpl; auto. +intros m' p; case p; simpl; auto. intros H; inversion H. -intros n0 H; apply Rec; simpl in |- *; auto. +intros n0 H; apply Rec; simpl; auto. apply Le.le_S_n; auto. Qed. @@ -188,11 +188,11 @@ Theorem prefix_correct : forall s1 s2 : string, prefix s1 s2 = true <-> substring 0 (length s1) s2 = s1. Proof. -intros s1; elim s1; simpl in |- *; auto. -intros s2; case s2; simpl in |- *; split; auto. -intros a s1' Rec s2; case s2; simpl in |- *; auto. +intros s1; elim s1; simpl; auto. +intros s2; case s2; simpl; split; auto. +intros a s1' Rec s2; case s2; simpl; auto. split; intros; discriminate. -intros b s2'; case (ascii_dec a b); simpl in |- *; auto. +intros b s2'; case (ascii_dec a b); simpl; auto. intros e; case (Rec s2'); intros H1 H2; split; intros H3; auto. rewrite e; rewrite H1; auto. apply H2; injection H3; auto. @@ -234,25 +234,25 @@ Theorem index_correct1 : forall (n m : nat) (s1 s2 : string), index n s1 s2 = Some m -> substring m (length s1) s2 = s1. Proof. -intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl in |- *; +intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl; auto. -intros n; case n; simpl in |- *; auto. -intros m s1; case s1; simpl in |- *; auto. +intros n; case n; simpl; auto. +intros m s1; case s1; simpl; auto. intros H; injection H; intros H1; rewrite <- H1; auto. intros; discriminate. intros; discriminate. intros b s2' Rec n m s1. -case n; simpl in |- *; auto. +case n; simpl; auto. generalize (prefix_correct s1 (String b s2')); case (prefix s1 (String b s2')). intros H0 H; injection H; intros H1; rewrite <- H1; auto. -case H0; simpl in |- *; auto. -case m; simpl in |- *; auto. +case H0; simpl; auto. +case m; simpl; auto. case (index 0 s1 s2'); intros; discriminate. intros m'; generalize (Rec 0 m' s1); case (index 0 s1 s2'); auto. intros x H H0 H1; apply H; injection H1; auto. intros; discriminate. -intros n'; case m; simpl in |- *; auto. +intros n'; case m; simpl; auto. case (index n' s1 s2'); intros; discriminate. intros m'; generalize (Rec n' m' s1); case (index n' s1 s2'); auto. intros x H H1; apply H; injection H1; auto. @@ -267,35 +267,35 @@ Theorem index_correct2 : index n s1 s2 = Some m -> forall p : nat, n <= p -> p < m -> substring p (length s1) s2 <> s1. Proof. -intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl in |- *; +intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl; auto. -intros n; case n; simpl in |- *; auto. -intros m s1; case s1; simpl in |- *; auto. +intros n; case n; simpl; auto. +intros m s1; case s1; simpl; auto. intros H; injection H; intros H1; rewrite <- H1. intros p H0 H2; inversion H2. intros; discriminate. intros; discriminate. intros b s2' Rec n m s1. -case n; simpl in |- *; auto. +case n; simpl; auto. generalize (prefix_correct s1 (String b s2')); case (prefix s1 (String b s2')). intros H0 H; injection H; intros H1; rewrite <- H1; auto. intros p H2 H3; inversion H3. -case m; simpl in |- *; auto. +case m; simpl; auto. case (index 0 s1 s2'); intros; discriminate. intros m'; generalize (Rec 0 m' s1); case (index 0 s1 s2'); auto. -intros x H H0 H1 p; try case p; simpl in |- *; auto. -intros H2 H3; red in |- *; intros H4; case H0. +intros x H H0 H1 p; try case p; simpl; auto. +intros H2 H3; red; intros H4; case H0. intros H5 H6; absurd (false = true); auto with bool. intros n0 H2 H3; apply H; auto. injection H1; auto. apply Le.le_O_n. apply Lt.lt_S_n; auto. intros; discriminate. -intros n'; case m; simpl in |- *; auto. +intros n'; case m; simpl; auto. case (index n' s1 s2'); intros; discriminate. intros m'; generalize (Rec n' m' s1); case (index n' s1 s2'); auto. -intros x H H0 p; case p; simpl in |- *; auto. +intros x H H0 p; case p; simpl; auto. intros H1; inversion H1; auto. intros n0 H1 H2; apply H; auto. injection H0; auto. @@ -312,33 +312,33 @@ Theorem index_correct3 : index n s1 s2 = None -> s1 <> EmptyString -> n <= m -> substring m (length s1) s2 <> s1. Proof. -intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl in |- *; +intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl; auto. -intros n; case n; simpl in |- *; auto. -intros m s1; case s1; simpl in |- *; auto. -case m; intros; red in |- *; intros; discriminate. +intros n; case n; simpl; auto. +intros m s1; case s1; simpl; auto. +case m; intros; red; intros; discriminate. intros n' m; case m; auto. -intros s1; case s1; simpl in |- *; auto. +intros s1; case s1; simpl; auto. intros b s2' Rec n m s1. -case n; simpl in |- *; auto. +case n; simpl; auto. generalize (prefix_correct s1 (String b s2')); case (prefix s1 (String b s2')). intros; discriminate. -case m; simpl in |- *; auto with bool. -case s1; simpl in |- *; auto. -intros a s H H0 H1 H2; red in |- *; intros H3; case H. +case m; simpl; auto with bool. +case s1; simpl; auto. +intros a s H H0 H1 H2; red; intros H3; case H. intros H4 H5; absurd (false = true); auto with bool. -case s1; simpl in |- *; auto. +case s1; simpl; auto. intros a s n0 H H0 H1 H2; - change (substring n0 (length (String a s)) s2' <> String a s) in |- *; + change (substring n0 (length (String a s)) s2' <> String a s); apply (Rec 0); auto. -generalize H0; case (index 0 (String a s) s2'); simpl in |- *; auto; intros; +generalize H0; case (index 0 (String a s) s2'); simpl; auto; intros; discriminate. apply Le.le_O_n. -intros n'; case m; simpl in |- *; auto. +intros n'; case m; simpl; auto. intros H H0 H1; inversion H1. intros n0 H H0 H1; apply (Rec n'); auto. -generalize H; case (index n' s1 s2'); simpl in |- *; auto; intros; +generalize H; case (index n' s1 s2'); simpl; auto; intros; discriminate. apply Le.le_S_n; auto. Qed. @@ -353,13 +353,13 @@ Theorem index_correct4 : forall (n : nat) (s : string), index n EmptyString s = None -> length s < n. Proof. -intros n s; generalize n; clear n; elim s; simpl in |- *; auto. -intros n; case n; simpl in |- *; auto. +intros n s; generalize n; clear n; elim s; simpl; auto. +intros n; case n; simpl; auto. intros; discriminate. intros; apply Lt.lt_O_Sn. -intros a s' H n; case n; simpl in |- *; auto. +intros a s' H n; case n; simpl; auto. intros; discriminate. -intros n'; generalize (H n'); case (index n' EmptyString s'); simpl in |- *; +intros n'; generalize (H n'); case (index n' EmptyString s'); simpl; auto. intros; discriminate. intros H0 H1; apply Lt.lt_n_S; auto. diff --git a/theories/Wellfounded/Disjoint_Union.v b/theories/Wellfounded/Disjoint_Union.v index f5daa3014..ec0dfeb81 100644 --- a/theories/Wellfounded/Disjoint_Union.v +++ b/theories/Wellfounded/Disjoint_Union.v @@ -41,7 +41,7 @@ Section Wf_Disjoint_Union. well_founded leA -> well_founded leB -> well_founded Le_AsB. Proof. intros. - unfold well_founded in |- *. + unfold well_founded. destruct a as [a| b]. apply (acc_A_sum a). apply (H a). diff --git a/theories/Wellfounded/Inclusion.v b/theories/Wellfounded/Inclusion.v index 1c83c4816..73d66c847 100644 --- a/theories/Wellfounded/Inclusion.v +++ b/theories/Wellfounded/Inclusion.v @@ -24,7 +24,7 @@ Section WfInclusion. Theorem wf_incl : inclusion A R1 R2 -> well_founded R2 -> well_founded R1. Proof. - unfold well_founded in |- *; auto with sets. + unfold well_founded; auto with sets. Qed. End WfInclusion. diff --git a/theories/Wellfounded/Inverse_Image.v b/theories/Wellfounded/Inverse_Image.v index 27a1c3811..db89cb356 100644 --- a/theories/Wellfounded/Inverse_Image.v +++ b/theories/Wellfounded/Inverse_Image.v @@ -31,7 +31,7 @@ Section Inverse_Image. Theorem wf_inverse_image : well_founded R -> well_founded Rof. Proof. - red in |- *; intros; apply Acc_inverse_image; auto. + red; intros; apply Acc_inverse_image; auto. Qed. Variable F : A -> B -> Prop. @@ -49,7 +49,7 @@ Section Inverse_Image. Theorem wf_inverse_rel : well_founded R -> well_founded RoF. Proof. - red in |- *; constructor; intros. + red; constructor; intros. case H0; intros. apply (Acc_inverse_rel x); auto. Qed. diff --git a/theories/Wellfounded/Lexicographic_Exponentiation.v b/theories/Wellfounded/Lexicographic_Exponentiation.v index 6d5b663bd..fe2d83250 100644 --- a/theories/Wellfounded/Lexicographic_Exponentiation.v +++ b/theories/Wellfounded/Lexicographic_Exponentiation.v @@ -36,11 +36,11 @@ Section Wf_Lexicographic_Exponentiation. Proof. simple induction x. simple induction z. - simpl in |- *; intros H. + simpl; intros H. inversion_clear H. - simpl in |- *; intros; apply (Lt_nil A leA). + simpl; intros; apply (Lt_nil A leA). intros a l HInd. - simpl in |- *. + simpl. intros. inversion_clear H. apply (Lt_hd A leA); auto with sets. @@ -54,7 +54,7 @@ Section Wf_Lexicographic_Exponentiation. ltl x (y ++ z) -> ltl x y \/ (exists y' : List, x = y ++ y' /\ ltl y' z). Proof. intros x y; generalize x. - elim y; simpl in |- *. + elim y; simpl. right. exists x0; auto with sets. intros. @@ -196,7 +196,7 @@ Section Wf_Lexicographic_Exponentiation. Descl x0 /\ Descl (l0 ++ Cons x1 Nil)). - simpl in |- *. + simpl. split. generalize (app_inj_tail _ _ _ _ H2); simple induction 1. simple induction 1; auto with sets. @@ -239,7 +239,7 @@ Section Wf_Lexicographic_Exponentiation. Proof. intros a b x. case x. - simpl in |- *. + simpl. simple induction 1. intros. inversion H1; auto with sets. @@ -267,7 +267,7 @@ Section Wf_Lexicographic_Exponentiation. case x. intros; apply (Lt_nil A leA). - simpl in |- *; intros. + simpl; intros. inversion_clear H0. apply (Lt_hd A leA a b); auto with sets. @@ -284,17 +284,17 @@ Section Wf_Lexicographic_Exponentiation. apply (Acc_inv (R:=Lex_Exp) (x:=<< x1 ++ x2, y1 >>)). auto with sets. - unfold lex_exp in |- *; simpl in |- *; auto with sets. + unfold lex_exp; simpl; auto with sets. Qed. Theorem wf_lex_exp : well_founded leA -> well_founded Lex_Exp. Proof. - unfold well_founded at 2 in |- *. + unfold well_founded at 2. simple induction a; intros x y. apply Acc_intro. simple induction y0. - unfold lex_exp at 1 in |- *; simpl in |- *. + unfold lex_exp at 1; simpl. apply rev_ind with (A := A) (P := fun x:List => @@ -335,8 +335,8 @@ Section Wf_Lexicographic_Exponentiation. intro. apply Acc_intro. simple induction y2. - unfold lex_exp at 1 in |- *. - simpl in |- *; intros x4 y3. intros. + unfold lex_exp at 1. + simpl; intros x4 y3. intros. apply (H0 x4 y3); auto with sets. intros. @@ -357,7 +357,7 @@ Section Wf_Lexicographic_Exponentiation. generalize (HInd2 f); intro. apply Acc_intro. simple induction y3. - unfold lex_exp at 1 in |- *; simpl in |- *; intros. + unfold lex_exp at 1; simpl; intros. apply H15; auto with sets. Qed. diff --git a/theories/Wellfounded/Lexicographic_Product.v b/theories/Wellfounded/Lexicographic_Product.v index 9a1d66f43..a0c639e4e 100644 --- a/theories/Wellfounded/Lexicographic_Product.v +++ b/theories/Wellfounded/Lexicographic_Product.v @@ -60,7 +60,7 @@ Section WfLexicographic_Product. well_founded leA -> (forall x:A, well_founded (leB x)) -> well_founded LexProd. Proof. - intros wfA wfB; unfold well_founded in |- *. + intros wfA wfB; unfold well_founded. destruct a. apply acc_A_B_lexprod; auto with sets; intros. red in wfB. @@ -94,7 +94,7 @@ Section Wf_Symmetric_Product. Lemma wf_symprod : well_founded leA -> well_founded leB -> well_founded Symprod. Proof. - red in |- *. + red. destruct a. apply Acc_symprod; auto with sets. Defined. @@ -161,7 +161,7 @@ Section Swap. Lemma wf_swapprod : well_founded R -> well_founded SwapProd. Proof. - red in |- *. + red. destruct a; intros. apply Acc_swapprod; auto with sets. Defined. diff --git a/theories/Wellfounded/Transitive_Closure.v b/theories/Wellfounded/Transitive_Closure.v index e9bc7ccf7..5d06c6c02 100644 --- a/theories/Wellfounded/Transitive_Closure.v +++ b/theories/Wellfounded/Transitive_Closure.v @@ -18,7 +18,7 @@ Section Wf_Transitive_Closure. Notation trans_clos := (clos_trans A R). Lemma incl_clos_trans : inclusion A R trans_clos. - red in |- *; auto with sets. + red; auto with sets. Qed. Lemma Acc_clos_trans : forall x:A, Acc R x -> Acc trans_clos x. @@ -39,7 +39,7 @@ Section Wf_Transitive_Closure. Theorem wf_clos_trans : well_founded R -> well_founded trans_clos. Proof. - unfold well_founded in |- *; auto with sets. + unfold well_founded; auto with sets. Defined. End Wf_Transitive_Closure. diff --git a/theories/Wellfounded/Union.v b/theories/Wellfounded/Union.v index e3fdc4c5e..c0aa1c0db 100644 --- a/theories/Wellfounded/Union.v +++ b/theories/Wellfounded/Union.v @@ -51,7 +51,7 @@ Section WfUnion. elim strip_commut with x x0 y0; auto with sets; intros. apply Acc_inv_trans with x1; auto with sets. - unfold union in |- *. + unfold union. elim H11; auto with sets; intros. apply t_trans with y1; auto with sets. @@ -65,7 +65,7 @@ Section WfUnion. Theorem wf_union : commut A R1 R2 -> well_founded R1 -> well_founded R2 -> well_founded Union. Proof. - unfold well_founded in |- *. + unfold well_founded. intros. apply Acc_union; auto with sets. Qed. diff --git a/theories/Wellfounded/Well_Ordering.v b/theories/Wellfounded/Well_Ordering.v index fc4e2ebce..452da1b2e 100644 --- a/theories/Wellfounded/Well_Ordering.v +++ b/theories/Wellfounded/Well_Ordering.v @@ -25,7 +25,7 @@ Section WellOrdering. Theorem wf_WO : well_founded le_WO. Proof. - unfold well_founded in |- *; intro. + unfold well_founded; intro. apply Acc_intro. elim a. intros. @@ -37,7 +37,7 @@ Section WellOrdering. apply (H v0 y0). cut (f = f1). intros E; rewrite E; auto. - symmetry in |- *. + symmetry . apply (inj_pair2 A (fun a0:A => B a0 -> WO) a0 f1 f H5). Qed. @@ -61,7 +61,7 @@ Section Characterisation_wf_relations. apply (well_founded_induction_type H (fun a:A => WO A B)); auto. intros x H1. apply (sup A B x). - unfold B at 1 in |- *. + unfold B at 1. destruct 1 as [x0]. apply (H1 x0); auto. Qed. diff --git a/theories/ZArith/ZArith_dec.v b/theories/ZArith/ZArith_dec.v index 8d535d509..06c9988a1 100644 --- a/theories/ZArith/ZArith_dec.v +++ b/theories/ZArith/ZArith_dec.v @@ -151,7 +151,7 @@ Proof. intro. apply False_rec. apply H. - symmetry in |- *. + symmetry . assumption. Defined. @@ -174,7 +174,7 @@ Proof. assumption. intro. right. - symmetry in |- *. + symmetry . assumption. Defined. diff --git a/theories/ZArith/Zcomplements.v b/theories/ZArith/Zcomplements.v index 7ae2a67ca..02e3ffe47 100644 --- a/theories/ZArith/Zcomplements.v +++ b/theories/ZArith/Zcomplements.v @@ -56,7 +56,7 @@ Proof. set (Q := fun z => 0 <= z -> P z * P (- z)) in *. cut (Q (Z.abs p)); [ intros | apply (Z_lt_rec Q); auto with zarith ]. elim (Zabs_dec p); intro eq; rewrite eq; elim H; auto with zarith. - unfold Q in |- *; clear Q; intros. + unfold Q; clear Q; intros. split; apply HP. rewrite Z.abs_eq; auto; intros. elim (H (Z.abs m)); intros; auto with zarith. @@ -75,7 +75,7 @@ Proof. set (Q := fun z => 0 <= z -> P z /\ P (- z)) in *. cut (Q (Z.abs p)); [ intros | apply (Z_lt_induction Q); auto with zarith ]. elim (Zabs_dec p); intro eq; rewrite eq; elim H; auto with zarith. - unfold Q in |- *; clear Q; intros. + unfold Q; clear Q; intros. split; apply HP. rewrite Z.abs_eq; auto; intros. elim (H (Z.abs m)); intros; auto with zarith. diff --git a/theories/ZArith/Zdigits.v b/theories/ZArith/Zdigits.v index a9348785a..d252b3e92 100644 --- a/theories/ZArith/Zdigits.v +++ b/theories/ZArith/Zdigits.v @@ -90,13 +90,13 @@ Section ENCODING_VALUE. Lemma Zmod2_twice : forall z:Z, z = (2 * Zmod2 z + bit_value (Z.odd z))%Z. Proof. - destruct z; simpl in |- *. + destruct z; simpl. trivial. - destruct p; simpl in |- *; trivial. + destruct p; simpl; trivial. - destruct p; simpl in |- *. - destruct p as [p| p| ]; simpl in |- *. + destruct p; simpl. + destruct p as [p| p| ]; simpl. rewrite <- (Pos.pred_double_succ p); trivial. trivial. @@ -145,17 +145,17 @@ Section Z_BRIC_A_BRAC. (z >= 0)%Z -> Z_to_binary (S n) (bit_value b + 2 * z) = Bcons b n (Z_to_binary n z). Proof. - destruct b; destruct z; simpl in |- *; auto. + destruct b; destruct z; simpl; auto. intro H; elim H; trivial. Qed. Lemma binary_value_pos : forall (n:nat) (bv:Bvector n), (binary_value n bv >= 0)%Z. Proof. - induction bv as [| a n v IHbv]; simpl in |- *. + induction bv as [| a n v IHbv]; simpl. omega. - destruct a; destruct (binary_value n v); simpl in |- *; auto. + destruct a; destruct (binary_value n v); simpl; auto. auto with zarith. Qed. @@ -174,7 +174,7 @@ Section Z_BRIC_A_BRAC. Proof. destruct b; destruct z as [| p| p]; auto. destruct p as [p| p| ]; auto. - destruct p as [p| p| ]; simpl in |- *; auto. + destruct p as [p| p| ]; simpl; auto. intros; rewrite (Pos.succ_pred_double p); trivial. Qed. @@ -201,7 +201,7 @@ Section Z_BRIC_A_BRAC. auto. destruct p; auto. - simpl in |- *; intros; omega. + simpl; intros; omega. intro H; elim H; trivial. Qed. @@ -233,7 +233,7 @@ Section Z_BRIC_A_BRAC. Lemma Zeven_bit_value : forall z:Z, Zeven.Zeven z -> bit_value (Z.odd z) = 0%Z. Proof. - destruct z; unfold bit_value in |- *; auto. + destruct z; unfold bit_value; auto. destruct p; tauto || (intro H; elim H). destruct p; tauto || (intro H; elim H). Qed. @@ -241,7 +241,7 @@ Section Z_BRIC_A_BRAC. Lemma Zodd_bit_value : forall z:Z, Zeven.Zodd z -> bit_value (Z.odd z) = 1%Z. Proof. - destruct z; unfold bit_value in |- *; auto. + destruct z; unfold bit_value; auto. intros; elim H. destruct p; tauto || (intros; elim H). destruct p; tauto || (intros; elim H). @@ -310,7 +310,7 @@ Section COHERENT_VALUE. (z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z. Proof. induction n as [| n IHn]. - unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros; omega. + unfold two_power_nat, shift_nat; simpl; intros; omega. intros; rewrite Z_to_binary_Sn_z. rewrite binary_value_Sn. @@ -328,7 +328,7 @@ Section COHERENT_VALUE. (z < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z) = z. Proof. induction n as [| n IHn]. - unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros. + unfold two_power_nat, shift_nat; simpl; intros. assert (z = (-1)%Z \/ z = 0%Z). omega. intuition; subst z; trivial. diff --git a/theories/ZArith/Zlogarithm.v b/theories/ZArith/Zlogarithm.v index 3711ea021..09323ebd4 100644 --- a/theories/ZArith/Zlogarithm.v +++ b/theories/ZArith/Zlogarithm.v @@ -77,7 +77,7 @@ Section Log_pos. (* Log of positive integers *) forall x:positive, 0 <= log_inf x /\ two_p (log_inf x) <= Zpos x < two_p (Z.succ (log_inf x)). Proof. - simple induction x; intros; simpl in |- *; + simple induction x; intros; simpl; [ elim H; intros Hp HR; clear H; split; [ auto with zarith | rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial); @@ -90,7 +90,7 @@ Section Log_pos. (* Log of positive integers *) rewrite two_p_S by trivial; rewrite two_p_S in HR by trivial; rewrite (BinInt.Pos2Z.inj_xO p); omega ] - | unfold two_power_pos in |- *; unfold shift_pos in |- *; simpl in |- *; + | unfold two_power_pos; unfold shift_pos; simpl; omega ]. Qed. @@ -103,7 +103,7 @@ Section Log_pos. (* Log of positive integers *) Lemma log_sup_correct1 : forall p:positive, 0 <= log_sup p. Proof. - simple induction p; intros; simpl in |- *; auto with zarith. + simple induction p; intros; simpl; auto with zarith. Qed. (** For every [p], either [p] is a power of two and [(log_inf p)=(log_sup p)] @@ -115,16 +115,16 @@ Section Log_pos. (* Log of positive integers *) else log_sup p = Z.succ (log_inf p). Proof. simple induction p; intros; - [ elim H; right; simpl in |- *; + [ elim H; right; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite BinInt.Pos2Z.inj_xI; unfold Z.succ; omega | elim H; clear H; intro Hif; - [ left; simpl in |- *; + [ left; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite (two_p_S (log_sup p0) (log_sup_correct1 p0)); rewrite <- (proj1 Hif); rewrite <- (proj2 Hif); auto - | right; simpl in |- *; + | right; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite BinInt.Pos2Z.inj_xO; unfold Z.succ; omega ] @@ -146,12 +146,12 @@ Section Log_pos. (* Log of positive integers *) Lemma log_inf_le_log_sup : forall p:positive, log_inf p <= log_sup p. Proof. - simple induction p; simpl in |- *; intros; omega. + simple induction p; simpl; intros; omega. Qed. Lemma log_sup_le_Slog_inf : forall p:positive, log_sup p <= Z.succ (log_inf p). Proof. - simple induction p; simpl in |- *; intros; omega. + simple induction p; simpl; intros; omega. Qed. (** Now it's possible to specify and build the [Log] rounded to the nearest *) @@ -167,7 +167,7 @@ Section Log_pos. (* Log of positive integers *) Theorem log_near_correct1 : forall p:positive, 0 <= log_near p. Proof. - simple induction p; simpl in |- *; intros; + simple induction p; simpl; intros; [ elim p0; auto with zarith | elim p0; auto with zarith | trivial with zarith ]. @@ -182,9 +182,9 @@ Section Log_pos. (* Log of positive integers *) Proof. simple induction p. intros p0 [Einf| Esup]. - simpl in |- *. rewrite Einf. + simpl. rewrite Einf. case p0; [ left | left | right ]; reflexivity. - simpl in |- *; rewrite Esup. + simpl; rewrite Esup. elim (log_sup_log_inf p0). generalize (log_inf_le_log_sup p0). generalize (log_sup_le_Slog_inf p0). @@ -192,10 +192,10 @@ Section Log_pos. (* Log of positive integers *) intros; omega. case p0; intros; auto with zarith. intros p0 [Einf| Esup]. - simpl in |- *. + simpl. repeat rewrite Einf. case p0; intros; auto with zarith. - simpl in |- *. + simpl. repeat rewrite Esup. case p0; intros; auto with zarith. auto. @@ -216,7 +216,7 @@ Section divers. Lemma ZERO_le_N_digits : forall x:Z, 0 <= N_digits x. Proof. - simple induction x; simpl in |- *; + simple induction x; simpl; [ apply Z.le_refl | exact log_inf_correct1 | exact log_inf_correct1 ]. Qed. @@ -245,21 +245,21 @@ Section divers. Proof. split; [ elim p; - [ simpl in |- *; tauto - | simpl in |- *; intros; generalize (H H0); intro H1; elim H1; + [ simpl; tauto + | simpl; intros; generalize (H H0); intro H1; elim H1; intros y0 Hy0; exists (S y0); rewrite Hy0; reflexivity | intro; exists 0%nat; reflexivity ] - | intros; elim H; intros; rewrite H0; elim x; intros; simpl in |- *; trivial ]. + | intros; elim H; intros; rewrite H0; elim x; intros; simpl; trivial ]. Qed. Lemma Is_power_or : forall p:positive, Is_power p \/ ~ Is_power p. Proof. simple induction p; - [ intros; right; simpl in |- *; tauto + [ intros; right; simpl; tauto | intros; elim H; - [ intros; left; simpl in |- *; exact H0 - | intros; right; simpl in |- *; exact H0 ] - | left; simpl in |- *; trivial ]. + [ intros; left; simpl; exact H0 + | intros; right; simpl; exact H0 ] + | left; simpl; trivial ]. Qed. End divers. diff --git a/theories/ZArith/Znumtheory.v b/theories/ZArith/Znumtheory.v index 33f4dc7f4..5d6550f99 100644 --- a/theories/ZArith/Znumtheory.v +++ b/theories/ZArith/Znumtheory.v @@ -305,7 +305,7 @@ Section extended_euclid_algorithm. v1 * a + v2 * b = v3 -> (forall d:Z, Zis_gcd u3 v3 d -> Zis_gcd a b d) -> Euclid. Proof. - intros v3 Hv3; generalize Hv3; pattern v3 in |- *. + intros v3 Hv3; generalize Hv3; pattern v3. apply Zlt_0_rec. clear v3 Hv3; intros. elim (Z_zerop x); intro. @@ -319,8 +319,8 @@ Section extended_euclid_algorithm. apply Z_mod_lt; omega. assert (xpos : x > 0). omega. generalize (Z_div_mod_eq u3 x xpos). - unfold q in |- *. - intro eq; pattern u3 at 2 in |- *; rewrite eq; ring. + unfold q. + intro eq; pattern u3 at 2; rewrite eq; ring. apply (H (u3 - q * x) Hq (proj1 Hq) v1 v2 x (u1 - q * v1) (u2 - q * v2)). tauto. replace ((u1 - q * v1) * a + (u2 - q * v2) * b) with @@ -459,12 +459,12 @@ Proof. apply Gauss with a. rewrite H3. auto with zarith. - red in |- *; auto with zarith. + red; auto with zarith. apply Gauss with c. rewrite Z.mul_comm. rewrite <- H3. auto with zarith. - red in |- *; auto with zarith. + red; auto with zarith. Qed. (** After factorization by a gcd, the original numbers are relatively prime. *) @@ -479,7 +479,7 @@ Proof. elim H1; intros. elim H4; intros. rewrite H2 in H6; subst b; omega. - unfold rel_prime in |- *. + unfold rel_prime. destruct H1. destruct H1 as (a',H1). destruct H3 as (b',H3). diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v index 0a4418671..6f005d01d 100644 --- a/theories/ZArith/Zwf.v +++ b/theories/ZArith/Zwf.v @@ -32,13 +32,13 @@ Section wf_proof. Let f (z:Z) := Z.abs_nat (z - c). Lemma Zwf_well_founded : well_founded (Zwf c). - red in |- *; intros. + red; intros. assert (forall (n:nat) (a:Z), (f a < n)%nat \/ a < c -> Acc (Zwf c) a). clear a; simple induction n; intros. (** n= 0 *) case H; intros. case (lt_n_O (f a)); auto. - apply Acc_intro; unfold Zwf in |- *; intros. + apply Acc_intro; unfold Zwf; intros. assert False; omega || contradiction. (** inductive case *) case H0; clear H0; intro; auto. |