(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(** Contributed by Laurent Théry (INRIA);
    Adapted to Coq V8 by the Coq Development Team *)

Require Import Arith.
Require Import Ascii.
Declare ML Module "string_syntax_plugin".

(** *** Definition of strings *)

(** 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.
Local Open Scope string_scope.

(** Equality is decidable *)

Definition string_dec : forall s1 s2 : string, {s1 = s2} + {s1 <> s2}.
Proof.
 decide equality; apply ascii_dec.
Defined.

(** *** Concatenation of strings *)

Reserved Notation "x ++ y" (right associativity, at level 60).

Fixpoint append (s1 s2 : string) : 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.
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; split; auto.
intros H; generalize (H 0); intros H1; inversion H1.
intros; discriminate.
intros H; generalize (H 0); simpl; intros H1; inversion H1.
case (Rec s).
intros H0; rewrite H0; auto.
intros n; exact (H (S n)).
intros H; injection H as H1 H2.
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; auto.
intros s2 n H; inversion H.
intros a s1' Rec s2 n; case n; simpl; 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; 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.

(** *** Substrings *)

(** [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) : 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; 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; 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; 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; 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; auto.
apply Le.le_S_n; auto.
Qed.

(** *** Test functions *)

(** 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; 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; 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) : 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.

(* Dirty trick to avoid locally that prefix reduces itself *)
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;
 auto.
intros n; case n; simpl; auto.
intros m s1; case s1; simpl; auto.
intros H; injection H as <-; auto.
intros; discriminate.
intros; discriminate.
intros b s2' Rec n m s1.
case n; simpl; auto.
generalize (prefix_correct s1 (String b s2'));
 case (prefix s1 (String b s2')).
intros H0 H; injection H as <-; 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; 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.
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;
 auto.
intros n; case n; simpl; auto.
intros m s1; case s1; simpl; auto.
intros H; injection H as <-.
intros p H0 H2; inversion H2.
intros; discriminate.
intros; discriminate.
intros b s2' Rec n m s1.
case n; simpl; auto.
generalize (prefix_correct s1 (String b s2'));
 case (prefix s1 (String b s2')).
intros H0 H; injection H as <-; auto.
intros p H2 H3; inversion H3.
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; 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; 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; auto.
intros H1; inversion H1; auto.
intros n0 H1 H2; apply H; auto.
injection H0; 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;
 auto.
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; auto.
intros b s2' Rec n m s1.
case n; simpl; auto.
generalize (prefix_correct s1 (String b s2'));
 case (prefix s1 (String b s2')).
intros; discriminate.
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; auto.
intros a s n0 H H0 H1 H2;
 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; auto; intros;
 discriminate.
apply Le.le_O_n.
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; auto; intros;
 discriminate.
apply Le.le_S_n; auto.
Qed.

(* Back to normal for prefix *)
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; auto.
intros n; case n; simpl; auto.
intros; discriminate.
intros; apply Lt.lt_O_Sn.
intros a s' H n; case n; simpl; auto.
intros; discriminate.
intros n'; generalize (H n'); case (index n' EmptyString s'); simpl;
 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.

(** *** Concrete syntax *)

(**
  The concrete syntax for strings in scope string_scope follows the
  Coq convention for strings: all ascii characters of code less than
  128 are literals 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).
*)

Example HelloWorld := "	""Hello world!""
".
