Require Import FMapInterface.

Set Implicit Arguments.

Module Raw (X:DecidableType).

Module Import PX := KeyDecidableType X.

Definition key := X.t.
Definition t (elt:Type) := list (X.t * elt).

Section Elt.

Variable elt : Type.

Notation eqk := (eqk (elt:=elt)).
Notation eqke := (eqke (elt:=elt)).
Notation MapsTo := (MapsTo (elt:=elt)).
Notation In := (In (elt:=elt)).
Notation NoDupA := (NoDupA eqk).

Definition empty : t elt := nil.

Definition Empty m := forall (a : key)(e:elt), ~ MapsTo a e m.

Lemma empty_1 : Empty empty.

Hint Resolve empty_1.

Lemma empty_NoDup : NoDupA empty.

Definition is_empty (l : t elt) : bool := if l then true else false.

Lemma is_empty_1 :forall m, Empty m -> is_empty m = true.

Lemma is_empty_2 : forall m, is_empty m = true -> Empty m.

Function mem (k : key) (s : t elt) {struct s} : bool :=
  match s with
   | nil => false
   | (k´,_) :: l => if X.eq_dec k k´ then true else mem k l
  end.

Lemma mem_1 : forall m (Hm:NoDupA m) x, In x m -> mem x m = true.

Lemma mem_2 : forall m (Hm:NoDupA m) x, mem x m = true -> In x m.

Function find (k:key) (s: t elt) {struct s} : option elt :=
  match s with
   | nil => None
   | (k´,x)::s´ => if X.eq_dec k k´ then Some x else find k s´
  end.

Lemma find_2 : forall m x e, find x m = Some e -> MapsTo x e m.

Lemma find_1 : forall m (Hm:NoDupA m) x e,
  MapsTo x e m -> find x m = Some e.


Lemma find_eq : forall m (Hm:NoDupA m) x x´,
   X.eq x x´ -> find x m = find x´ m.

Function add (k : key) (x : elt) (s : t elt) {struct s} : t elt :=
  match s with
   | nil => (k,x) :: nil
   | (k´,y) :: l => if X.eq_dec k k´ then (k,x)::l else (k´,y)::add k x l
  end.

Lemma add_1 : forall m x y e, X.eq x y -> MapsTo y e (add x e m).

Lemma add_2 : forall m x y e e´,
  ~ X.eq x y -> MapsTo y e m -> MapsTo y e (add x e´ m).

Lemma add_3 : forall m x y e e´,
  ~ X.eq x y -> MapsTo y e (add x e´ m) -> MapsTo y e m.

Lemma add_3´ : forall m x y e e´,
  ~ X.eq x y -> InA eqk (y,e) (add x e´ m) -> InA eqk (y,e) m.

Lemma add_NoDup : forall m (Hm:NoDupA m) x e, NoDupA (add x e m).


Lemma add_eq : forall m (Hm:NoDupA m) x a e,
   X.eq x a -> find x (add a e m) = Some e.

Lemma add_not_eq : forall m (Hm:NoDupA m) x a e,
  ~X.eq x a -> find x (add a e m) = find x m.

Function remove (k : key) (s : t elt) {struct s} : t elt :=
  match s with
   | nil => nil
   | (k´,x) :: l => if X.eq_dec k k´ then l else (k´,x) :: remove k l
  end.

Lemma remove_1 : forall m (Hm:NoDupA m) x y, X.eq x y -> ~ In y (remove x m).

Lemma remove_2 : forall m (Hm:NoDupA m) x y e,
  ~ X.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).

Lemma remove_3 : forall m (Hm:NoDupA m) x y e,
  MapsTo y e (remove x m) -> MapsTo y e m.

Lemma remove_3´ : forall m (Hm:NoDupA m) x y e,
  InA eqk (y,e) (remove x m) -> InA eqk (y,e) m.

Lemma remove_NoDup : forall m (Hm:NoDupA m) x, NoDupA (remove x m).

Definition elements (m: t elt) := m.

Lemma elements_1 : forall m x e, MapsTo x e m -> InA eqke (x,e) (elements m).

Lemma elements_2 : forall m x e, InA eqke (x,e) (elements m) -> MapsTo x e m.

Lemma elements_3w : forall m (Hm:NoDupA m), NoDupA (elements m).

Function fold (A:Type)(f:key->elt->A->A)(m:t elt) (acc : A) {struct m} : A :=
  match m with
   | nil => acc
   | (k,e)::m´ => fold f m´ (f k e acc)
  end.

Lemma fold_1 : forall m (A:Type)(i:A)(f:key->elt->A->A),
  fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.

Definition check (cmp : elt -> elt -> bool)(k:key)(e:elt)(m´: t elt) :=
  match find k m´ with
   | None => false
   | Some e´ => cmp e e´
  end.

Definition submap (cmp : elt -> elt -> bool)(m m´ : t elt) : bool :=
  fold (fun k e b => andb (check cmp k e m´) b) m true.

Definition equal (cmp : elt -> elt -> bool)(m m´ : t elt) : bool :=
  andb (submap cmp m m´) (submap (fun e´ e => cmp e e´) m´ m).

Definition Submap cmp m m´ :=
  (forall k, In k m -> In k m´) /\
  (forall k e e´, MapsTo k e m -> MapsTo k e´ m´ -> cmp e e´ = true).

Definition Equivb cmp m m´ :=
  (forall k, In k m <-> In k m´) /\
  (forall k e e´, MapsTo k e m -> MapsTo k e´ m´ -> cmp e e´ = true).

Lemma submap_1 : forall m (Hm:NoDupA m) m´ (Hm´: NoDupA m´) cmp,
  Submap cmp m m´ -> submap cmp m m´ = true.

Lemma submap_2 : forall m (Hm:NoDupA m) m´ (Hm´: NoDupA m´) cmp,
  submap cmp m m´ = true -> Submap cmp m m´.

Lemma equal_1 : forall m (Hm:NoDupA m) m´ (Hm´: NoDupA m´) cmp,
  Equivb cmp m m´ -> equal cmp m m´ = true.

Lemma equal_2 : forall m (Hm:NoDupA m) m´ (Hm´:NoDupA m´) cmp,
  equal cmp m m´ = true -> Equivb cmp m m´.

Variable elt´:Type.

Fixpoint map (f:elt -> elt´) (m:t elt) : t elt´ :=
  match m with
   | nil => nil
   | (k,e)::m´ => (k,f e) :: map f m´
  end.

Fixpoint mapi (f: key -> elt -> elt´) (m:t elt) : t elt´ :=
  match m with
   | nil => nil
   | (k,e)::m´ => (k,f k e) :: mapi f m´
  end.

End Elt.
Section Elt2.

Variable elt elt´ : Type.

Lemma map_1 : forall (m:t elt)(x:key)(e:elt)(f:elt->elt´),
  MapsTo x e m -> MapsTo x (f e) (map f m).

Lemma map_2 : forall (m:t elt)(x:key)(f:elt->elt´),
  In x (map f m) -> In x m.

Lemma map_NoDup : forall m (Hm : NoDupA (@eqk elt) m)(f:elt->elt´),
  NoDupA (@eqk elt´) (map f m).

Lemma mapi_1 : forall (m:t elt)(x:key)(e:elt)(f:key->elt->elt´),
  MapsTo x e m ->
  exists y, X.eq y x /\ MapsTo x (f y e) (mapi f m).

Lemma mapi_2 : forall (m:t elt)(x:key)(f:key->elt->elt´),
  In x (mapi f m) -> In x m.

Lemma mapi_NoDup : forall m (Hm : NoDupA (@eqk elt) m)(f: key->elt->elt´),
  NoDupA (@eqk elt´) (mapi f m).

End Elt2.
Section Elt3.

Variable elt elt´ elt´´ : Type.

Notation oee´ := (option elt * option elt´)%type.

Definition combine_l (m:t elt)(m´:t elt´) : t oee´ :=
  mapi (fun k e => (Some e, find k m´)) m.

Definition combine_r (m:t elt)(m´:t elt´) : t oee´ :=
  mapi (fun k e´ => (find k m, Some e´)) m´.

Definition fold_right_pair (A B C:Type)(f:A->B->C->C)(l:list (A*B))(i:C) :=
  List.fold_right (fun p => f (fst p) (snd p)) i l.

Definition combine (m:t elt)(m´:t elt´) : t oee´ :=
  let l := combine_l m m´ in
  let r := combine_r m m´ in
  fold_right_pair (add (elt:=oee´)) l r.

Lemma fold_right_pair_NoDup :
  forall l r (Hl: NoDupA (eqk (elt:=oee´)) l)
    (Hl: NoDupA (eqk (elt:=oee´)) r),
    NoDupA (eqk (elt:=oee´)) (fold_right_pair (add (elt:=oee´)) l r).
Hint Resolve fold_right_pair_NoDup.

Lemma combine_NoDup :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´),
  NoDupA (@eqk oee´) (combine m m´).

Definition at_least_left (o:option elt)(o´:option elt´) :=
  match o with
   | None => None
   | _ => Some (o,o´)
  end.

Definition at_least_right (o:option elt)(o´:option elt´) :=
  match o´ with
   | None => None
   | _ => Some (o,o´)
  end.

Lemma combine_l_1 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  find x (combine_l m m´) = at_least_left (find x m) (find x m´).

Lemma combine_r_1 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  find x (combine_r m m´) = at_least_right (find x m) (find x m´).

Definition at_least_one (o:option elt)(o´:option elt´) :=
  match o, o´ with
   | None, None => None
   | _, _ => Some (o,o´)
  end.

Lemma combine_1 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  find x (combine m m´) = at_least_one (find x m) (find x m´).

Variable f : option elt -> option elt´ -> option elt´´.

Definition option_cons (A:Type)(k:key)(o:option A)(l:list (key*A)) :=
  match o with
   | Some e => (k,e)::l
   | None => l
  end.

Definition map2 m m´ :=
  let m0 : t oee´ := combine m m´ in
  let m1 : t (option elt´´) := map (fun p => f (fst p) (snd p)) m0 in
  fold_right_pair (option_cons (A:=elt´´)) m1 nil.

Lemma map2_NoDup :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´),
  NoDupA (@eqk elt´´) (map2 m m´).

Definition at_least_one_then_f (o:option elt)(o´:option elt´) :=
  match o, o´ with
   | None, None => None
   | _, _ => f o o´
  end.

Lemma map2_0 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  find x (map2 m m´) = at_least_one_then_f (find x m) (find x m´).

Lemma map2_1 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  In x m \/ In x m´ ->
  find x (map2 m m´) = f (find x m) (find x m´).

Lemma map2_2 :
  forall m (Hm:NoDupA (@eqk elt) m) m´ (Hm´:NoDupA (@eqk elt´) m´)(x:key),
  In x (map2 m m´) -> In x m \/ In x m´.

End Elt3.
End Raw.

Module Make (X: DecidableType) <: WS with Module E:=X.
 Module Raw := Raw X.

 Module E := X.
 Definition key := E.t.

 Record slist (elt:Type) :=
  {this :> Raw.t elt; NoDup : NoDupA (@Raw.PX.eqk elt) this}.
 Definition t (elt:Type) := slist elt.

Section Elt.
 Variable elt elt´ elt´´:Type.

 Implicit Types m : t elt.
 Implicit Types x y : key.
 Implicit Types e : elt.

 Definition empty : t elt := Build_slist (Raw.empty_NoDup elt).
 Definition is_empty m : bool := Raw.is_empty m.(this).
 Definition add x e m : t elt := Build_slist (Raw.add_NoDup m.(NoDup) x e).
 Definition find x m : option elt := Raw.find x m.(this).
 Definition remove x m : t elt := Build_slist (Raw.remove_NoDup m.(NoDup) x).
 Definition mem x m : bool := Raw.mem x m.(this).
 Definition map f m : t elt´ := Build_slist (Raw.map_NoDup m.(NoDup) f).
 Definition mapi (f:key->elt->elt´) m : t elt´ := Build_slist (Raw.mapi_NoDup m.(NoDup) f).
 Definition map2 f m (m´:t elt´) : t elt´´ :=
   Build_slist (Raw.map2_NoDup f m.(NoDup) m´.(NoDup)).
 Definition elements m : list (key*elt) := @Raw.elements elt m.(this).
 Definition cardinal m := length m.(this).
 Definition fold (A:Type)(f:key->elt->A->A) m (i:A) : A := @Raw.fold elt A f m.(this) i.
 Definition equal cmp m m´ : bool := @Raw.equal elt cmp m.(this) m´.(this).
 Definition MapsTo x e m : Prop := Raw.PX.MapsTo x e m.(this).
 Definition In x m : Prop := Raw.PX.In x m.(this).
 Definition Empty m : Prop := Raw.Empty m.(this).

 Definition Equal m m´ := forall y, find y m = find y m´.
 Definition Equiv (eq_elt:elt->elt->Prop) m m´ :=
         (forall k, In k m <-> In k m´) /\
         (forall k e e´, MapsTo k e m -> MapsTo k e´ m´ -> eq_elt e e´).
 Definition Equivb cmp m m´ : Prop := @Raw.Equivb elt cmp m.(this) m´.(this).

 Definition eq_key : (key*elt) -> (key*elt) -> Prop := @Raw.PX.eqk elt.
 Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop:= @Raw.PX.eqke elt.

 Lemma MapsTo_1 : forall m x y e, E.eq x y -> MapsTo x e m -> MapsTo y e m.

 Lemma mem_1 : forall m x, In x m -> mem x m = true.
 Lemma mem_2 : forall m x, mem x m = true -> In x m.

 Lemma empty_1 : Empty empty.

 Lemma is_empty_1 : forall m, Empty m -> is_empty m = true.
 Lemma is_empty_2 : forall m, is_empty m = true -> Empty m.

 Lemma add_1 : forall m x y e, E.eq x y -> MapsTo y e (add x e m).
 Lemma add_2 : forall m x y e e´, ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e´ m).
 Lemma add_3 : forall m x y e e´, ~ E.eq x y -> MapsTo y e (add x e´ m) -> MapsTo y e m.

 Lemma remove_1 : forall m x y, E.eq x y -> ~ In y (remove x m).
 Lemma remove_2 : forall m x y e, ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
 Lemma remove_3 : forall m x y e, MapsTo y e (remove x m) -> MapsTo y e m.

 Lemma find_1 : forall m x e, MapsTo x e m -> find x m = Some e.
 Lemma find_2 : forall m x e, find x m = Some e -> MapsTo x e m.

 Lemma elements_1 : forall m x e, MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
 Lemma elements_2 : forall m x e, InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
 Lemma elements_3w : forall m, NoDupA eq_key (elements m).

 Lemma cardinal_1 : forall m, cardinal m = length (elements m).

 Lemma fold_1 : forall m (A : Type) (i : A) (f : key -> elt -> A -> A),
        fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.

 Lemma equal_1 : forall m m´ cmp, Equivb cmp m m´ -> equal cmp m m´ = true.
 Lemma equal_2 : forall m m´ cmp, equal cmp m m´ = true -> Equivb cmp m m´.

 End Elt.

 Lemma map_1 : forall (elt elt´:Type)(m: t elt)(x:key)(e:elt)(f:elt->elt´),
        MapsTo x e m -> MapsTo x (f e) (map f m).
 Lemma map_2 : forall (elt elt´:Type)(m: t elt)(x:key)(f:elt->elt´),
        In x (map f m) -> In x m.

 Lemma mapi_1 : forall (elt elt´:Type)(m: t elt)(x:key)(e:elt)
        (f:key->elt->elt´), MapsTo x e m ->
        exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m).
 Lemma mapi_2 : forall (elt elt´:Type)(m: t elt)(x:key)
        (f:key->elt->elt´), In x (mapi f m) -> In x m.

 Lemma map2_1 : forall (elt elt´ elt´´:Type)(m: t elt)(m´: t elt´)
        (x:key)(f:option elt->option elt´->option elt´´),
        In x m \/ In x m´ ->
        find x (map2 f m m´) = f (find x m) (find x m´).
 Lemma map2_2 : forall (elt elt´ elt´´:Type)(m: t elt)(m´: t elt´)
        (x:key)(f:option elt->option elt´->option elt´´),
        In x (map2 f m m´) -> In x m \/ In x m´.

End Make.