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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 *)
(***********************************************************************)
(* $Id$ *)
(** * Finite sets library *)
(** Set interfaces for types with only a decidable equality, but no ordering *)
Require Export Bool.
Require Export DecidableType.
Set Implicit Arguments.
Unset Strict Implicit.
(** 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 : DecidableType.
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. *)
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].
The order in which elements of [s] are presented to [f] is
unspecified. *)
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, in any order. *)
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.
Equal sets could return different elements. *)
(** Coq comment: [Not_found] is represented by the option type *)
Section Spec.
Variable 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 [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 : Equal s s' -> equal s s' = true.
Parameter equal_2 : equal s s' = true -> Equal s s'.
(** Specification of [subset] *)
Parameter subset_1 : Subset s s' -> subset s s' = true.
Parameter subset_2 : subset s s' = true -> Subset 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 -> Equal (fst (partition f s)) (filter f s).
Parameter partition_2 :
compat_bool E.eq f ->
Equal (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 : NoDupA E.eq (elements s).
(** Specification of [choose] *)
Parameter choose_1 : choose s = Some x -> In x s.
Parameter choose_2 : choose s = None -> Empty s.
End Spec.
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.
End S.
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