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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
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(* $Id: FSetInterface.v 8820 2006-05-15 11:44:05Z letouzey $ *)

(** * Finite set library *)

(** Set interfaces *)

(* begin hide *)
Require Export Bool.
Require Export OrderedType.
Set Implicit Arguments.
Unset Strict Implicit.
(* end hide *)

(** Compatibility of a  boolean function with respect to an equality. *)
Definition compat_bool (A:Set)(eqA: A->A->Prop)(f: A-> bool) :=
  forall x y : A, eqA x y -> f x = f y.

(** Compatibility of a predicate with respect to an equality. *)
Definition compat_P (A:Set)(eqA: A->A->Prop)(P : A -> Prop) :=
  forall x y : A, eqA x y -> P x -> P y.

Hint Unfold compat_bool compat_P.

(** * Non-dependent signature

    Signature [S] presents sets as purely informative programs 
    together with axioms *)

Module Type S.

  Declare Module E : OrderedType.
  Definition elt := E.t.

  Parameter t : Set. (** the abstract type of sets *)

  (** Logical predicates *)
  Parameter In : elt -> t -> Prop.
  Definition Equal s s' := forall a : elt, In a s <-> In a s'.
  Definition Subset s s' := forall a : elt, In a s -> In a s'.
  Definition Empty s := forall a : elt, ~ In a s.
  Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
  Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.
  
  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).
  Notation "s [<=] t" := (Subset s t) (at level 70, no associativity).

  Parameter empty : t.
  (** The empty set. *)

  Parameter is_empty : t -> bool.
  (** Test whether a set is empty or not. *)

  Parameter mem : elt -> t -> bool.
  (** [mem x s] tests whether [x] belongs to the set [s]. *)

  Parameter add : elt -> t -> t.
  (** [add x s] returns a set containing all elements of [s],
  plus [x]. If [x] was already in [s], [s] is returned unchanged. *)

  Parameter singleton : elt -> t.
  (** [singleton x] returns the one-element set containing only [x]. *)

  Parameter remove : elt -> t -> t.
  (** [remove x s] returns a set containing all elements of [s],
  except [x]. If [x] was not in [s], [s] is returned unchanged. *)

  Parameter union : t -> t -> t.
  (** Set union. *)

  Parameter inter : t -> t -> t.
  (** Set intersection. *)

  Parameter diff : t -> t -> t.
  (** Set difference. *)

  Definition eq : t -> t -> Prop := Equal.
  Parameter lt : t -> t -> Prop.
  Parameter compare : forall s s' : t, Compare lt eq s s'.
  (** Total ordering between sets. Can be used as the ordering function
  for doing sets of sets. *)

  Parameter equal : t -> t -> bool.
  (** [equal s1 s2] tests whether the sets [s1] and [s2] are
  equal, that is, contain equal elements. *)

  Parameter subset : t -> t -> bool.
  (** [subset s1 s2] tests whether the set [s1] is a subset of
  the set [s2]. *)

  (** Coq comment: [iter] is useless in a purely functional world *)
  (**  iter: (elt -> unit) -> set -> unit. i*)
  (** [iter f s] applies [f] in turn to all elements of [s].
  The order in which the elements of [s] are presented to [f]
  is unspecified. *)

  Parameter fold : forall A : Set, (elt -> A -> A) -> t -> A -> A.
  (** [fold f s a] computes [(f xN ... (f x2 (f x1 a))...)],
  where [x1 ... xN] are the elements of [s], in increasing order. *)

  Parameter for_all : (elt -> bool) -> t -> bool.
  (** [for_all p s] checks if all elements of the set
  satisfy the predicate [p]. *)

  Parameter exists_ : (elt -> bool) -> t -> bool.
  (** [exists p s] checks if at least one element of
  the set satisfies the predicate [p]. *)

  Parameter filter : (elt -> bool) -> t -> t.
  (** [filter p s] returns the set of all elements in [s]
  that satisfy predicate [p]. *)

  Parameter partition : (elt -> bool) -> t -> t * t.
  (** [partition p s] returns a pair of sets [(s1, s2)], where
  [s1] is the set of all the elements of [s] that satisfy the
  predicate [p], and [s2] is the set of all the elements of
  [s] that do not satisfy [p]. *)

  Parameter cardinal : t -> nat.
  (** Return the number of elements of a set. *)
  (** Coq comment: nat instead of int ... *)

  Parameter elements : t -> list elt.
  (** Return the list of all elements of the given set.
  The returned list is sorted in increasing order with respect
  to the ordering [Ord.compare], where [Ord] is the argument
  given to {!Set.Make}. *)

  Parameter min_elt : t -> option elt.
  (** Return the smallest element of the given set
  (with respect to the [Ord.compare] ordering), or raise
  [Not_found] if the set is empty. *)
  (** Coq comment: [Not_found] is represented by the option type *)

  Parameter max_elt : t -> option elt.
  (** Same as {!Set.S.min_elt}, but returns the largest element of the
  given set. *)
  (** Coq comment: [Not_found] is represented by the option type *)

  Parameter choose : t -> option elt.
  (** Return one element of the given set, or raise [Not_found] if
  the set is empty. Which element is chosen is unspecified,
  but equal elements will be chosen for equal sets. *)
  (** Coq comment: [Not_found] is represented by the option type *)

  Section Spec. 

  Variable s s' s'' : t.
  Variable x y : elt.

  (** Specification of [In] *)
  Parameter In_1 : E.eq x y -> In x s -> In y s.
 
  (** Specification of [eq] *)
  Parameter eq_refl : eq s s. 
  Parameter eq_sym : eq s s' -> eq s' s.
  Parameter eq_trans : eq s s' -> eq s' s'' -> eq s s''.
 
  (** Specification of [lt] *)
  Parameter lt_trans : lt s s' -> lt s' s'' -> lt s s''.
  Parameter lt_not_eq : lt s s' -> ~ eq s s'.

  (** Specification of [mem] *)
  Parameter mem_1 : In x s -> mem x s = true.
  Parameter mem_2 : mem x s = true -> In x s. 
 
  (** Specification of [equal] *) 
  Parameter equal_1 : s[=]s' -> equal s s' = true.
  Parameter equal_2 : equal s s' = true ->s[=]s'.

  (** Specification of [subset] *)
  Parameter subset_1 : s[<=]s' -> subset s s' = true.
  Parameter subset_2 : subset s s' = true -> s[<=]s'.

  (** Specification of [empty] *)
  Parameter empty_1 : Empty empty.

  (** Specification of [is_empty] *)
  Parameter is_empty_1 : Empty s -> is_empty s = true. 
  Parameter is_empty_2 : is_empty s = true -> Empty s.
 
  (** Specification of [add] *)
  Parameter add_1 : E.eq x y -> In y (add x s).
  Parameter add_2 : In y s -> In y (add x s).
  Parameter add_3 : ~ E.eq x y -> In y (add x s) -> In y s. 

  (** Specification of [remove] *)
  Parameter remove_1 : E.eq x y -> ~ In y (remove x s).
  Parameter remove_2 : ~ E.eq x y -> In y s -> In y (remove x s).
  Parameter remove_3 : In y (remove x s) -> In y s.

  (** Specification of [singleton] *)
  Parameter singleton_1 : In y (singleton x) -> E.eq x y. 
  Parameter singleton_2 : E.eq x y -> In y (singleton x). 

  (** Specification of [union] *)
  Parameter union_1 : In x (union s s') -> In x s \/ In x s'.
  Parameter union_2 : In x s -> In x (union s s'). 
  Parameter union_3 : In x s' -> In x (union s s').

  (** Specification of [inter] *)
  Parameter inter_1 : In x (inter s s') -> In x s.
  Parameter inter_2 : In x (inter s s') -> In x s'.
  Parameter inter_3 : In x s -> In x s' -> In x (inter s s').

  (** Specification of [diff] *)
  Parameter diff_1 : In x (diff s s') -> In x s. 
  Parameter diff_2 : In x (diff s s') -> ~ In x s'.
  Parameter diff_3 : In x s -> ~ In x s' -> In x (diff s s').
 
  (** Specification of [fold] *)  
  Parameter fold_1 : forall (A : Set) (i : A) (f : elt -> A -> A),
      fold f s i = fold_left (fun a e => f e a) (elements s) i.

  (** Specification of [cardinal] *)  
  Parameter cardinal_1 : cardinal s = length (elements s).

  Section Filter.
  
  Variable f : elt -> bool.

  (** Specification of [filter] *)
  Parameter filter_1 : compat_bool E.eq f -> In x (filter f s) -> In x s. 
  Parameter filter_2 : compat_bool E.eq f -> In x (filter f s) -> f x = true. 
  Parameter filter_3 :
      compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).

  (** Specification of [for_all] *)
  Parameter for_all_1 :
      compat_bool E.eq f ->
      For_all (fun x => f x = true) s -> for_all f s = true.
  Parameter for_all_2 :
      compat_bool E.eq f ->
      for_all f s = true -> For_all (fun x => f x = true) s.

  (** Specification of [exists] *)
  Parameter exists_1 :
      compat_bool E.eq f ->
      Exists (fun x => f x = true) s -> exists_ f s = true.
  Parameter exists_2 :
      compat_bool E.eq f ->
      exists_ f s = true -> Exists (fun x => f x = true) s.

  (** Specification of [partition] *)
  Parameter partition_1 : compat_bool E.eq f -> 
      fst (partition f s) [=] filter f s.
  Parameter partition_2 : compat_bool E.eq f -> 
      snd (partition f s) [=] filter (fun x => negb (f x)) s.

  End Filter.

  (** Specification of [elements] *)
  Parameter elements_1 : In x s -> InA E.eq x (elements s).
  Parameter elements_2 : InA E.eq x (elements s) -> In x s.
  Parameter elements_3 : sort E.lt (elements s).  

  (** Specification of [min_elt] *)
  Parameter min_elt_1 : min_elt s = Some x -> In x s. 
  Parameter min_elt_2 : min_elt s = Some x -> In y s -> ~ E.lt y x. 
  Parameter min_elt_3 : min_elt s = None -> Empty s.

  (** Specification of [max_elt] *)  
  Parameter max_elt_1 : max_elt s = Some x -> In x s. 
  Parameter max_elt_2 : max_elt s = Some x -> In y s -> ~ E.lt x y. 
  Parameter max_elt_3 : max_elt s = None -> Empty s.

  (** Specification of [choose] *)
  Parameter choose_1 : choose s = Some x -> In x s.
  Parameter choose_2 : choose s = None -> Empty s.
(*  Parameter choose_equal: 
      (equal s s')=true -> E.eq (choose s) (choose s'). *)

  End Spec.

  (* begin hide *)
  Hint Immediate In_1.
  
  Hint Resolve mem_1 mem_2 equal_1 equal_2 subset_1 subset_2 empty_1
    is_empty_1 is_empty_2 choose_1 choose_2 add_1 add_2 add_3 remove_1
    remove_2 remove_3 singleton_1 singleton_2 union_1 union_2 union_3 inter_1
    inter_2 inter_3 diff_1 diff_2 diff_3 filter_1 filter_2 filter_3 for_all_1
    for_all_2 exists_1 exists_2 partition_1 partition_2 elements_1 elements_2
    elements_3 min_elt_1 min_elt_2 min_elt_3 max_elt_1 max_elt_2 max_elt_3.
  (* end hide *)

End S.

(** * Dependent signature 

    Signature [Sdep] presents sets using dependent types *)

Module Type Sdep.

  Declare Module E : OrderedType.
  Definition elt := E.t.

  Parameter t : Set.

  Parameter In : elt -> t -> Prop.
  Definition Equal s s' := forall a : elt, In a s <-> In a s'.
  Definition Subset s s' := forall a : elt, In a s -> In a s'.
  Definition Add x s s' := forall y, In y s' <-> E.eq x y \/ In y s.
  Definition Empty s := forall a : elt, ~ In a s.
  Definition For_all (P : elt -> Prop) s := forall x, In x s -> P x.
  Definition Exists (P : elt -> Prop) s := exists x, In x s /\ P x.

  Notation "s [=] t" := (Equal s t) (at level 70, no associativity).

  Definition eq : t -> t -> Prop := Equal.
  Parameter lt : t -> t -> Prop.
  Parameter compare : forall s s' : t, Compare lt eq s s'.

  Parameter eq_refl : forall s : t, eq s s. 
  Parameter eq_sym : forall s s' : t, eq s s' -> eq s' s.
  Parameter eq_trans : forall s s' s'' : t, eq s s' -> eq s' s'' -> eq s s''.
  Parameter lt_trans : forall s s' s'' : t, lt s s' -> lt s' s'' -> lt s s''.
  Parameter lt_not_eq : forall s s' : t, lt s s' -> ~ eq s s'.

  Parameter eq_In : forall (s : t) (x y : elt), E.eq x y -> In x s -> In y s.

  Parameter empty : {s : t | Empty s}.

  Parameter is_empty : forall s : t, {Empty s} + {~ Empty s}.

  Parameter mem : forall (x : elt) (s : t), {In x s} + {~ In x s}.

  Parameter add : forall (x : elt) (s : t), {s' : t | Add x s s'}.

  Parameter
    singleton : forall x : elt, {s : t | forall y : elt, In y s <-> E.eq x y}.
  
  Parameter
    remove :
      forall (x : elt) (s : t),
      {s' : t | forall y : elt, In y s' <-> ~ E.eq x y /\ In y s}.

  Parameter
    union :
      forall s s' : t,
      {s'' : t | forall x : elt, In x s'' <-> In x s \/ In x s'}.

  Parameter
    inter :
      forall s s' : t,
      {s'' : t | forall x : elt, In x s'' <-> In x s /\ In x s'}.

  Parameter
    diff :
      forall s s' : t,
      {s'' : t | forall x : elt, In x s'' <-> In x s /\ ~ In x s'}.

  Parameter equal : forall s s' : t, {s[=]s'} + {~ s[=]s'}.
 
  Parameter subset : forall s s' : t, {Subset s s'} + {~ Subset s s'}.

  Parameter
    filter :
      forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
        (s : t),
      {s' : t | compat_P E.eq P -> forall x : elt, In x s' <-> In x s /\ P x}.

  Parameter
    for_all :
      forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
        (s : t),
      {compat_P E.eq P -> For_all P s} + {compat_P E.eq P -> ~ For_all P s}.
  
  Parameter
    exists_ :
      forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
        (s : t),
      {compat_P E.eq P -> Exists P s} + {compat_P E.eq P -> ~ Exists P s}.

  Parameter
    partition :
      forall (P : elt -> Prop) (Pdec : forall x : elt, {P x} + {~ P x})
        (s : t),
      {partition : t * t |
      let (s1, s2) := partition in
      compat_P E.eq P ->
      For_all P s1 /\
      For_all (fun x => ~ P x) s2 /\
      (forall x : elt, In x s <-> In x s1 \/ In x s2)}.

  Parameter
    elements :
      forall s : t,
      {l : list elt |
      sort E.lt l /\ (forall x : elt, In x s <-> InA E.eq x l)}.

  Parameter
    fold :
      forall (A : Set) (f : elt -> A -> A) (s : t) (i : A),
      {r : A | let (l,_) := elements s in 
                  r = fold_left (fun a e => f e a) l i}.

  Parameter
    cardinal :
      forall s : t,
      {r : nat | let (l,_) := elements s in r = length l }.

  Parameter
    min_elt :
      forall s : t,
      {x : elt | In x s /\ For_all (fun y => ~ E.lt y x) s} + {Empty s}.

  Parameter
    max_elt :
      forall s : t,
      {x : elt | In x s /\ For_all (fun y => ~ E.lt x y) s} + {Empty s}.

  Parameter choose : forall s : t, {x : elt | In x s} + {Empty s}.

End Sdep.