# Finite set library

Set interfaces, inspired by the one of Ocaml. When compared with Ocaml, the main differences are:
• the lack of iter function, useless since Coq is purely functional
• the use of option types instead of Not_found exceptions
• the use of nat instead of int for the cardinal function
Several variants of the set interfaces are available:
• WSfun : functorial signature for weak sets, non-dependent style
• WS : self-contained version of WSfun
• Sfun : functorial signature for ordered sets, non-dependent style
• S : self-contained version of Sfun
• Sdep : analog of S written using dependent style
If unsure, S is probably what you're looking for: other signatures are subsets of it, apart from Sdep which is isomorphic to S (see FSetBridge).

Require Export Bool OrderedType DecidableType.
Set Implicit Arguments.

# Non-dependent signatures

The following signatures presents sets as purely informative programs together with axioms

## Functorial signature for weak sets

Weak sets are sets without ordering on base elements, only a decidable equality.

Module Type WSfun (E : DecidableType).

Definition elt := E.t.

Parameter t : Type.
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 eq_dec : forall s s', { eq s s' } + { ~ eq s s' }.

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.

Parameter fold : forall A : Type, (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.

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 None if the set is empty. Which element is chosen is unspecified. Equal sets could return different elements.

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 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.

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 : Type) (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.
When compared with ordered sets, here comes the only property that is really weaker:
Parameter elements_3w : 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.

#[global]
Hint Transparent elt : core.
#[global]
Hint Resolve mem_1 equal_1 subset_1 empty_1
remove_2 singleton_2 union_1 union_2 union_3
inter_3 diff_3 fold_1 filter_3 for_all_1 exists_1
partition_1 partition_2 elements_1 elements_3w
: set.
#[global]
Hint Immediate In_1 mem_2 equal_2 subset_2 is_empty_2 add_3
remove_3 singleton_1 inter_1 inter_2 diff_1 diff_2
filter_1 filter_2 for_all_2 exists_2 elements_2
: set.

End WSfun.

## Static signature for weak sets

Similar to the functorial signature SW, except that the module E of base elements is incorporated in the signature.

Module Type WS.
Declare Module E : DecidableType.
Include WSfun E.
End WS.

## Functorial signature for sets on ordered elements

Based on WSfun, plus ordering on sets and min_elt and max_elt and some stronger specifications for other functions.

Module Type Sfun (E : OrderedType).
Include WSfun E.

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 min_elt : t -> option elt.
Return the smallest element of the given set (with respect to the E.compare ordering), or None if the set is empty.

Parameter max_elt : t -> option elt.
Same as min_elt, but returns the largest element of the given set.

Section Spec.

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

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'.

Parameter elements_3 : sort E.lt (elements s).

Remark: since fold is specified via elements, this stronger specification of elements has an indirect impact on fold, which can now be proved to receive elements in increasing order.
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.

Parameter choose_3 : choose s = Some x -> choose s' = Some y ->
Equal s s' -> E.eq x y.

End Spec.

#[global]
Hint Resolve elements_3 : set.
#[global]
Hint Immediate
min_elt_1 min_elt_2 min_elt_3 max_elt_1 max_elt_2 max_elt_3 : set.

End Sfun.

## Static signature for sets on ordered elements

Similar to the functorial signature Sfun, except that the module E of base elements is incorporated in the signature.

Module Type S.
Declare Module E : OrderedType.
Include Sfun E.
End S.

## Some subtyping tests

```WSfun ---> WS
|         |
|         |
V         V
Sfun  ---> S

Module S_WS (M : S) <: WS := M.
Module Sfun_WSfun (E:OrderedType)(M : Sfun E) <: WSfun E := M.
Module S_Sfun (M : S) <: Sfun M.E := M.
Module WS_WSfun (M : WS) <: WSfun M.E := M.
```

# Dependent signature

Signature Sdep presents ordered sets using dependent types

Module Type Sdep.

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

Parameter t : Type.

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 : Type) (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}.

The choose_3 specification of S cannot be packed in the dependent version of choose, so we leave it separate.
Parameter choose_equal : forall s s', Equal s s' ->
match choose s, choose s' with
| inleft (exist _ x _), inleft (exist _ x' _) => E.eq x x'
| inright _, inright _ => True
| _, _ => False
end.

End Sdep.