Library Coq.Strings.String
Contributed by Laurent Théry (INRIA);
Adapted to Coq V8 by the Coq Development Team
Require Import Arith.
Require Import Ascii.
Implementation of string as list of ascii characters
Inductive string : Set :=
| EmptyString : string
| String : ascii -> string -> string.
Delimit Scope string_scope with string.
Bind Scope string_scope with string.
Open Local Scope string_scope.
Equality is decidable
Definition string_dec : forall s1 s2 : string, {s1 = s2} + {s1 <> s2}.
decide equality; apply ascii_dec.
Defined.
Reserved Notation "x ++ y" (right associativity, at level 60).
Fixpoint append (s1 s2 : string) {struct s1} : string :=
match s1 with
| EmptyString => s2
| String c s1' => String c (s1' ++ s2)
end
where "s1 ++ s2" := (append s1 s2) : string_scope.
Length
Fixpoint length (s : string) : nat :=
match s with
| EmptyString => 0
| String c s' => S (length s')
end.
Nth character of a string
Fixpoint get (n : nat) (s : string) {struct s} : option ascii :=
match s with
| EmptyString => None
| String c s' => match n with
| O => Some c
| S n' => get n' s'
end
end.
Two lists that are identical through get are syntactically equal
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 H; generalize (H 0); intros H1; inversion H1.
intros; discriminate.
intros a s1' Rec s2; case s2; simpl in |- *; split; auto.
intros H; generalize (H 0); intros H1; inversion H1.
intros; discriminate.
intros H; generalize (H 0); simpl in |- *; intros H1; inversion H1.
case (Rec s).
intros H0; rewrite H0; auto.
intros n; exact (H (S n)).
intros H; injection H; intros H1 H2 n; case n; auto.
rewrite H2; trivial.
rewrite H1; auto.
Qed.
The first elements of
s1 ++ s2 are the ones of s1
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 s2 n H; inversion H.
intros a s1' Rec s2 n; case n; simpl in |- *; auto.
intros n0 H; apply Rec; auto.
apply lt_S_n; auto.
Qed.
The last elements of
s1 ++ s2 are the ones of s2
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 n0; rewrite <- Plus.plus_Snm_nSm; auto.
Qed.
substring n m s returns the substring of s that starts
at position n and of length m;
if this does not make sense it returns ""
Fixpoint substring (n m : nat) (s : string) {struct s} : string :=
match n, m, s with
| 0, 0, _ => EmptyString
| 0, S m', EmptyString => s
| 0, S m', String c s' => String c (substring 0 m' s')
| S n', _, EmptyString => s
| S n', _, String c s' => substring n' m s'
end.
The substring is included in the initial string
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 p H; inversion H.
intros m' p; case p; simpl in |- *; auto.
intros n0 H; apply Rec; simpl in |- *; auto.
apply Lt.lt_S_n; auto.
intros n' m p H; rewrite <- Plus.plus_Snm_nSm; simpl in |- *; auto.
Qed.
The substring has at most
m elements
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 H; inversion H.
intros n0 H; apply Rec; simpl in |- *; auto.
apply Le.le_S_n; auto.
Qed.
Test if
s1 is a prefix of s2
Fixpoint prefix (s1 s2 : string) {struct s2} : bool :=
match s1 with
| EmptyString => true
| String a s1' =>
match s2 with
| EmptyString => false
| String b s2' =>
match ascii_dec a b with
| left _ => prefix s1' s2'
| right _ => false
end
end
end.
If
s1 is a prefix of s2, it is the substring of length
length s1 starting at position O of s2
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.
split; intros; discriminate.
intros b s2'; case (ascii_dec a b); simpl in |- *; auto.
intros e; case (Rec s2'); intros H1 H2; split; intros H3; auto.
rewrite e; rewrite H1; auto.
apply H2; injection H3; auto.
intros n; split; intros; try discriminate.
case n; injection H; auto.
Qed.
Test if, starting at position
n, s1 occurs in s2; if
so it returns the position
Fixpoint index (n : nat) (s1 s2 : string) {struct s2} : option nat :=
match s2, n with
| EmptyString, 0 =>
match s1 with
| EmptyString => Some 0
| String a s1' => None
end
| EmptyString, S n' => None
| String b s2', 0 =>
if prefix s1 s2 then Some 0
else
match index 0 s1 s2' with
| Some n => Some (S n)
| None => None
end
| String b s2', S n' =>
match index n' s1 s2' with
| Some n => Some (S n)
| None => None
end
end.
Opaque prefix.
If the result of
index is Some m, s1 in s2 at position m
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 |- *;
auto.
intros n; case n; simpl in |- *; auto.
intros m s1; case s1; simpl in |- *; 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.
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 (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; discriminate.
intros n'; case m; simpl in |- *; 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; discriminate.
Qed.
If the result of
index is Some m,
s1 does not occur in s2 before m
Theorem index_correct2 :
forall (n m : nat) (s1 s2 : string),
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 |- *;
auto.
intros n; case n; simpl in |- *; auto.
intros m s1; case s1; simpl in |- *; 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.
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 (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 H5 H6; absurd (false = true); auto with bool.
intros n0 H2 H3; apply H; auto.
injection H1; intros H4; injection H4; auto.
apply Le.le_O_n.
apply Lt.lt_S_n; auto.
intros; discriminate.
intros n'; case m; simpl in |- *; 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 H1; inversion H1; auto.
intros n0 H1 H2; apply H; auto.
injection H0; intros H3; injection H3; auto.
apply Le.le_S_n; auto.
apply Lt.lt_S_n; auto.
intros; discriminate.
Qed.
If the result of
index is None, s1 does not occur in s2
after n
Theorem index_correct3 :
forall (n m : nat) (s1 s2 : string),
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 |- *;
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' m; case m; auto.
intros s1; case s1; simpl in |- *; auto.
intros b s2' Rec n m s1.
case n; simpl in |- *; 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.
intros H4 H5; absurd (false = true); auto with bool.
case s1; simpl in |- *; auto.
intros a s n0 H H0 H1 H2;
change (substring n0 (length (String a s)) s2' <> String a s) in |- *;
apply (Rec 0); auto.
generalize H0; case (index 0 (String a s) s2'); simpl in |- *; auto; intros;
discriminate.
apply Le.le_O_n.
intros n'; case m; simpl in |- *; 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;
discriminate.
apply Le.le_S_n; auto.
Qed.
Transparent prefix.
If we are searching for the
Empty string and the answer is no
this means that n is greater than the size of s
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; discriminate.
intros; apply Lt.lt_O_Sn.
intros a s' H n; case n; simpl in |- *; auto.
intros; discriminate.
intros n'; generalize (H n'); case (index n' EmptyString s'); simpl in |- *;
auto.
intros; discriminate.
intros H0 H1; apply Lt.lt_n_S; auto.
Qed.
Same as
index but with no optional type, we return 0 when it
does not occur
Definition findex n s1 s2 :=
match index n s1 s2 with
| Some n => n
| None => 0
end.
The concrete syntax for strings in scope string_scope follows the
Coq convention for strings: all ascii characters of code less than
128 are litteral to the exception of the character `double quote'
which must be doubled.
Strings that involve ascii characters of code >= 128 which are not part of a valid utf8 sequence of characters are not representable using the Coq string notation (use explicitly the String constructor with the ascii codes of the characters).
Strings that involve ascii characters of code >= 128 which are not part of a valid utf8 sequence of characters are not representable using the Coq string notation (use explicitly the String constructor with the ascii codes of the characters).
Example HelloWorld := " ""Hello world!""
".