Set Implicit Arguments.
Unset Strict Implicit.
Require Import Eqdep_dec.
Require Import FSetList.
Require Import OrderedTypeEx.
Require Import Lib_ListFacts.
Require Import Lib_FinSet.
Module MakeRaw (X : UsualOrderedType) <: FSetInterface.S with Module E := X.
Module Raw := Raw X.
Module E := X.
Module OTFacts := OrderedTypeFacts E.
Definition sort_bool (xs : Raw.t) :=
(if (sort_dec E.lt OTFacts.lt_dec xs) then true else false) = true.
Record slist : Set :=
{this :> Raw.t;
sorted : sort_bool this}.
Definition from_sorted : forall (xs : Raw.t), sort E.lt xs -> sort_bool xs.
Proof.
intros xs H. unfold sort_bool.
case (sort_dec E.lt OTFacts.lt_dec xs); tauto.
Defined.
Definition to_sorted : forall xs, sort_bool xs -> sort E.lt xs.
Proof.
unfold sort_bool. intros xs.
case (sort_dec E.lt OTFacts.lt_dec xs); auto.
intros. discriminate.
Defined.
Coercion to_sorted : sort_bool >-> sort.
Definition Build_slist' (xs : Raw.t) (pf : sort E.lt xs) :=
Build_slist (from_sorted pf).
Definition t := slist.
Definition elt := E.t.
Definition In (x : elt) (s : t) : Prop := InA E.eq x s.(this).
Definition Equal (s s':t) : Prop := forall a : elt, In a s <-> In a s'.
Definition Subset (s s':t) : Prop := forall a : elt, In a s -> In a s'.
Definition Empty (s:t) : Prop := forall a : elt, ~ In a s.
Definition For_all (P : elt -> Prop)(s:t) : Prop := forall x, In x s -> P x.
Definition Exists (P : elt -> Prop)(s:t) : Prop := exists x, In x s /\ P x.
Definition mem (x : elt) (s : t) : bool := Raw.mem x s.
Definition add (x : elt)(s : t) : t := Build_slist' (Raw.add_sort (sorted s) x).
Definition remove (x : elt)(s : t) : t := Build_slist' (Raw.remove_sort (sorted s) x).
Definition singleton (x : elt) : t := Build_slist' (Raw.singleton_sort x).
Definition union (s s' : t) : t :=
Build_slist' (Raw.union_sort (sorted s) (sorted s')).
Definition inter (s s' : t) : t :=
Build_slist' (Raw.inter_sort (sorted s) (sorted s')).
Definition diff (s s' : t) : t :=
Build_slist' (Raw.diff_sort (sorted s) (sorted s')).
Definition equal (s s' : t) : bool := Raw.equal s s'.
Definition subset (s s' : t) : bool := Raw.subset s s'.
Definition empty : t := Build_slist' Raw.empty_sort.
Definition is_empty (s : t) : bool := Raw.is_empty s.
Definition elements (s : t) : list elt := Raw.elements s.
Definition min_elt (s : t) : option elt := Raw.min_elt s.
Definition max_elt (s : t) : option elt := Raw.max_elt s.
Definition choose (s : t) : option elt := Raw.choose s.
Definition fold (B : Set) (f : elt -> B -> B) (s : t) : B -> B := Raw.fold (B:=B) f s.
Definition cardinal (s : t) : nat := Raw.cardinal s.
Definition filter (f : elt -> bool) (s : t) : t :=
Build_slist' (Raw.filter_sort (sorted s) f).
Definition for_all (f : elt -> bool) (s : t) : bool := Raw.for_all f s.
Definition exists_ (f : elt -> bool) (s : t) : bool := Raw.exists_ f s.
Definition partition (f : elt -> bool) (s : t) : t * t :=
let p := Raw.partition f s in
(@Build_slist' (fst p) (Raw.partition_sort_1 (sorted s) f),
@Build_slist' (snd p) (Raw.partition_sort_2 (sorted s) f)).
Definition eq (s s' : t) : Prop := Raw.eq s s'.
Definition lt (s s' : t) : Prop := Raw.lt s s'.
Section Spec.
Variable s s' s'': t.
Variable x y : elt.
Lemma In_1 : E.eq x y -> In x s -> In y s.
Proof. exact (fun H H' => Raw.MX.In_eq H H'). Qed.
Lemma mem_1 : In x s -> mem x s = true.
Proof. exact (fun H => Raw.mem_1 s.(sorted) H). Qed.
Lemma mem_2 : mem x s = true -> In x s.
Proof. exact (fun H => Raw.mem_2 H). Qed.
Lemma equal_1 : Equal s s' -> equal s s' = true.
Proof. exact (Raw.equal_1 s.(sorted) s'.(sorted)). Qed.
Lemma equal_2 : equal s s' = true -> Equal s s'.
Proof. exact (fun H => Raw.equal_2 H). Qed.
Lemma subset_1 : Subset s s' -> subset s s' = true.
Proof. exact (Raw.subset_1 s.(sorted) s'.(sorted)). Qed.
Lemma subset_2 : subset s s' = true -> Subset s s'.
Proof. exact (fun H => Raw.subset_2 H). Qed.
Lemma empty_1 : Empty empty.
Proof. exact Raw.empty_1. Qed.
Lemma is_empty_1 : Empty s -> is_empty s = true.
Proof. exact (fun H => Raw.is_empty_1 H). Qed.
Lemma is_empty_2 : is_empty s = true -> Empty s.
Proof. exact (fun H => Raw.is_empty_2 H). Qed.
Lemma add_1 : E.eq x y -> In y (add x s).
Proof. exact (fun H => Raw.add_1 s.(sorted) H). Qed.
Lemma add_2 : In y s -> In y (add x s).
Proof. exact (fun H => Raw.add_2 s.(sorted) x H). Qed.
Lemma add_3 : ~ E.eq x y -> In y (add x s) -> In y s.
Proof. exact (fun H => Raw.add_3 s.(sorted) H). Qed.
Lemma remove_1 : E.eq x y -> ~ In y (remove x s).
Proof. exact (fun H => Raw.remove_1 s.(sorted) H). Qed.
Lemma remove_2 : ~ E.eq x y -> In y s -> In y (remove x s).
Proof. exact (fun H H' => Raw.remove_2 s.(sorted) H H'). Qed.
Lemma remove_3 : In y (remove x s) -> In y s.
Proof. exact (fun H => Raw.remove_3 s.(sorted) H). Qed.
Lemma singleton_1 : In y (singleton x) -> E.eq x y.
Proof. exact (fun H => Raw.singleton_1 H). Qed.
Lemma singleton_2 : E.eq x y -> In y (singleton x).
Proof. exact (fun H => Raw.singleton_2 H). Qed.
Lemma union_1 : In x (union s s') -> In x s \/ In x s'.
Proof. exact (fun H => Raw.union_1 s.(sorted) s'.(sorted) H). Qed.
Lemma union_2 : In x s -> In x (union s s').
Proof. exact (fun H => Raw.union_2 s.(sorted) s'.(sorted) H). Qed.
Lemma union_3 : In x s' -> In x (union s s').
Proof. exact (fun H => Raw.union_3 s.(sorted) s'.(sorted) H). Qed.
Lemma inter_1 : In x (inter s s') -> In x s.
Proof. exact (fun H => Raw.inter_1 s.(sorted) s'.(sorted) H). Qed.
Lemma inter_2 : In x (inter s s') -> In x s'.
Proof. exact (fun H => Raw.inter_2 s.(sorted) s'.(sorted) H). Qed.
Lemma inter_3 : In x s -> In x s' -> In x (inter s s').
Proof. exact (fun H => Raw.inter_3 s.(sorted) s'.(sorted) H). Qed.
Lemma diff_1 : In x (diff s s') -> In x s.
Proof. exact (fun H => Raw.diff_1 s.(sorted) s'.(sorted) H). Qed.
Lemma diff_2 : In x (diff s s') -> ~ In x s'.
Proof. exact (fun H => Raw.diff_2 s.(sorted) s'.(sorted) H). Qed.
Lemma diff_3 : In x s -> ~ In x s' -> In x (diff s s').
Proof. exact (fun H => Raw.diff_3 s.(sorted) s'.(sorted) H). Qed.
Lemma 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.
Proof. exact (Raw.fold_1 s.(sorted)). Qed.
Lemma cardinal_1 : cardinal s = length (elements s).
Proof. exact (Raw.cardinal_1 s.(sorted)). Qed.
Section Filter.
Variable f : elt -> bool.
Lemma filter_1 : compat_bool E.eq f -> In x (filter f s) -> In x s.
Proof. exact (@Raw.filter_1 s x f). Qed.
Lemma filter_2 : compat_bool E.eq f -> In x (filter f s) -> f x = true.
Proof. exact (@Raw.filter_2 s x f). Qed.
Lemma filter_3 :
compat_bool E.eq f -> In x s -> f x = true -> In x (filter f s).
Proof. exact (@Raw.filter_3 s x f). Qed.
Lemma for_all_1 :
compat_bool E.eq f ->
For_all (fun x => f x = true) s -> for_all f s = true.
Proof. exact (@Raw.for_all_1 s f). Qed.
Lemma for_all_2 :
compat_bool E.eq f ->
for_all f s = true -> For_all (fun x => f x = true) s.
Proof. exact (@Raw.for_all_2 s f). Qed.
Lemma exists_1 :
compat_bool E.eq f ->
Exists (fun x => f x = true) s -> exists_ f s = true.
Proof. exact (@Raw.exists_1 s f). Qed.
Lemma exists_2 :
compat_bool E.eq f ->
exists_ f s = true -> Exists (fun x => f x = true) s.
Proof. exact (@Raw.exists_2 s f). Qed.
Lemma partition_1 :
compat_bool E.eq f -> Equal (fst (partition f s)) (filter f s).
Proof. exact (@Raw.partition_1 s f). Qed.
Lemma partition_2 :
compat_bool E.eq f ->
Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
Proof. exact (@Raw.partition_2 s f). Qed.
End Filter.
Lemma elements_1 : In x s -> InA E.eq x (elements s).
Proof. exact (fun H => Raw.elements_1 H). Qed.
Lemma elements_2 : InA E.eq x (elements s) -> In x s.
Proof. exact (fun H => Raw.elements_2 H). Qed.
Lemma elements_3 : sort E.lt (elements s).
Proof. exact (Raw.elements_3 s.(sorted)). Qed.
Lemma min_elt_1 : min_elt s = Some x -> In x s.
Proof. exact (fun H => Raw.min_elt_1 H). Qed.
Lemma min_elt_2 : min_elt s = Some x -> In y s -> ~ E.lt y x.
Proof. exact (fun H => Raw.min_elt_2 s.(sorted) H). Qed.
Lemma min_elt_3 : min_elt s = None -> Empty s.
Proof. exact (fun H => Raw.min_elt_3 H). Qed.
Lemma max_elt_1 : max_elt s = Some x -> In x s.
Proof. exact (fun H => Raw.max_elt_1 H). Qed.
Lemma max_elt_2 : max_elt s = Some x -> In y s -> ~ E.lt x y.
Proof. exact (fun H => Raw.max_elt_2 s.(sorted) H). Qed.
Lemma max_elt_3 : max_elt s = None -> Empty s.
Proof. exact (fun H => Raw.max_elt_3 H). Qed.
Lemma choose_1 : choose s = Some x -> In x s.
Proof. exact (fun H => Raw.choose_1 H). Qed.
Lemma choose_2 : choose s = None -> Empty s.
Proof. exact (fun H => Raw.choose_2 H). Qed.
Lemma eq_refl : eq s s.
Proof. exact (Raw.eq_refl s). Qed.
Lemma eq_sym : eq s s' -> eq s' s.
Proof. exact (@Raw.eq_sym s s'). Qed.
Lemma eq_trans : eq s s' -> eq s' s'' -> eq s s''.
Proof. exact (@Raw.eq_trans s s' s''). Qed.
Lemma lt_trans : lt s s' -> lt s' s'' -> lt s s''.
Proof. exact (@Raw.lt_trans s s' s''). Qed.
Lemma lt_not_eq : lt s s' -> ~ eq s s'.
Proof. exact (Raw.lt_not_eq s.(sorted) s'.(sorted)). Qed.
Definition compare : Compare lt eq s s'.
Proof.
elim (Raw.compare s.(sorted) s'.(sorted));
[ constructor 1 | constructor 2 | constructor 3 ];
auto.
Defined.
End Spec.
End MakeRaw.
Module Make (X : UsualOrderedType) <: FinSet with Module E := X.
Module Import E := X.
Module Import S := MakeRaw X.
Definition fset := S.t.
Definition elt := S.elt.
Lemma bool_dec : forall x y : bool,
x = y \/ x <> y.
Proof.
induction x; induction y; intuition.
Qed.
Theorem eq_ext : forall s s' : t, (forall a, In a s <-> In a s') -> s = s'.
Proof.
intros [s H] [s' J] K.
assert (s = s').
unfold Raw.t in *. eapply Sort_InA_eq_ext; eauto using to_sorted.
eexact E.lt_trans.
intros. eapply OTFacts.lt_eq; eauto.
intros. eapply OTFacts.eq_lt; eauto.
intros. subst.
rewrite (eq_proofs_unicity bool_dec H J).
reflexivity.
Qed.
Theorem eq_if_Equal : forall s s' : t, Equal s s' -> s = s'.
Proof.
unfold Equal. intros s s'.
auto using eq_ext.
Qed.
End Make.