(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(** Contributed by Laurent Théry (INRIA); Adapted to Coq V8 by the Coq Development Team *)
(** *** Definition of strings *)
(** Implementation of string as list of ascii characters *)
Inductive string : Set :=
| EmptyString : string
| String : ascii -> string -> string.
Declare Scope string_scope.
Delimit Scope string_scope with string.
Bind Scope string_scope with string.
Local Open Scope string_scope.
Register EmptyString as plugins.syntax.EmptyString.
Register String as plugins.syntax.String.
(** Equality is decidable *)
Definition string_dec : forall s1 s2 : string, {s1 = s2} + {s1 <> s2}.
Local Ltac t_eqb :=
repeat first [ congruence
| progress subst
| apply conj
| match goal with
| [ |- context[eqb ?x ?y] ] => destruct (eqb_spec x y)
end
| intro ].
(** The first elements of [s1 ++ s2] are the ones of [s1] *)
(** The last elements of [s1 ++ s2] are the ones of [s2] *)
(** *** 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
| O, O, _ => EmptyString
| O, S m', EmptyString => s
| O, 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.