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-rw-r--r--theories/Strings/String.v140
1 files changed, 69 insertions, 71 deletions
diff --git a/theories/Strings/String.v b/theories/Strings/String.v
index c26b8818..6294d156 100644
--- a/theories/Strings/String.v
+++ b/theories/Strings/String.v
@@ -1,14 +1,12 @@
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(* $Id: String.v 14641 2011-11-06 11:59:10Z herbelin $ *)
-
(** Contributed by Laurent Théry (INRIA);
Adapted to Coq V8 by the Coq Development Team *)
@@ -26,7 +24,7 @@ Inductive string : Set :=
Delimit Scope string_scope with string.
Bind Scope string_scope with string.
-Open Local Scope string_scope.
+Local Open Scope string_scope.
(** Equality is decidable *)
@@ -74,14 +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)).
@@ -96,9 +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.
@@ -109,10 +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.
@@ -137,16 +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 *)
@@ -154,14 +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.
@@ -190,11 +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.
@@ -236,28 +234,28 @@ 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; intros H2; injection H2; 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; intros H2; injection H2; auto.
+intros x H H1; apply H; injection H1; auto.
intros; discriminate.
Qed.
@@ -269,38 +267,38 @@ 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; intros H4; injection H4; 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; intros H3; injection H3; auto.
+injection H0; auto.
apply Le.le_S_n; auto.
apply Lt.lt_S_n; auto.
intros; discriminate.
@@ -314,33 +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.
@@ -355,13 +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.