# Library Coq.FSets.FMapFullAVL

# FMapFullAVL

This file contains some complements to FMapAVL.

- Functor AvlProofs proves that trees of FMapAVL are not only

binary search trees, but moreover well-balanced ones. This is done by proving that all operations preserve the balancing.

- We then pack the previous elements in a IntMake functor

similar to the one of FMapAVL, but richer.

- In final IntMake_ord functor, the compare function is

different from the one in FMapAVL: this non-structural version is closer to the original Ocaml code.
Require Import Recdef FMapInterface FMapList ZArith Int FMapAVL ROmega.
Set Implicit Arguments.
Module AvlProofs (Import I:Int)(X: OrderedType).
Module Import Raw := Raw I X.
Module Import II:=MoreInt(I).
Import Raw.Proofs.
Local Open Scope pair_scope.
Local Open Scope Int_scope.
Ltac omega_max := i2z_refl; romega with Z.
Section Elt.
Variable elt : Type.
Implicit Types m r : t elt.

# AVL trees

avl s : s is a properly balanced AVL tree, i.e. for any node the heights of the two children differ by at most 2
Inductive avl : t elt -> Prop :=
| RBLeaf : avl (Leaf _)
| RBNode : forall x e l r h,
avl l ->
avl r ->
-(2) <= height l - height r <= 2 ->
h = max (height l) (height r) + 1 ->
avl (Node l x e r h).

# Automation and dedicated tactics about avl.

Hint Constructors avl.
Lemma height_non_negative : forall (s : t elt), avl s ->
height s >= 0.
Ltac avl_nn_hyp H :=
let nz := fresh "nz" in assert (nz := height_non_negative H).
Ltac avl_nn h :=
let t := type of h in
match type of t with
| Prop => avl_nn_hyp h
| _ => match goal with H : avl h |- _ => avl_nn_hyp H end
end.
Ltac avl_nns :=
match goal with
| H:avl _ |- _ => avl_nn_hyp H; clear H; avl_nns
| _ => idtac
end.

# Basic results about avl, height

Lemma avl_node : forall x e l r, avl l -> avl r ->
-(2) <= height l - height r <= 2 ->
avl (Node l x e r (max (height l) (height r) + 1)).
Hint Resolve avl_node.
Results about height
Lemma height_0 : forall l, avl l -> height l = 0 ->
l = Leaf _.

# Empty map

Lemma empty_avl : avl (empty elt).

# Helper functions

Lemma create_avl :
forall l x e r, avl l -> avl r -> -(2) <= height l - height r <= 2 ->
avl (create l x e r).
Lemma create_height :
forall l x e r, avl l -> avl r -> -(2) <= height l - height r <= 2 ->
height (create l x e r) = max (height l) (height r) + 1.
Lemma bal_avl : forall l x e r, avl l -> avl r ->
-(3) <= height l - height r <= 3 -> avl (bal l x e r).
Lemma bal_height_1 : forall l x e r, avl l -> avl r ->
-(3) <= height l - height r <= 3 ->
0 <= height (bal l x e r) - max (height l) (height r) <= 1.
Lemma bal_height_2 :
forall l x e r, avl l -> avl r -> -(2) <= height l - height r <= 2 ->
height (bal l x e r) == max (height l) (height r) +1.
Ltac omega_bal := match goal with
| H:avl ?l, H':avl ?r |- context [ bal ?l ?x ?e ?r ] =>
generalize (bal_height_1 x e H H') (bal_height_2 x e H H');
omega_max
end.

# Insertion

Lemma add_avl_1 : forall m x e, avl m ->
avl (add x e m) /\ 0 <= height (add x e m) - height m <= 1.
Lemma add_avl : forall m x e, avl m -> avl (add x e m).
Hint Resolve add_avl.

# Extraction of minimum binding

Lemma remove_min_avl_1 : forall l x e r h, avl (Node l x e r h) ->
avl (remove_min l x e r)#1 /\
0 <= height (Node l x e r h) - height (remove_min l x e r)#1 <= 1.
Lemma remove_min_avl : forall l x e r h, avl (Node l x e r h) ->
avl (remove_min l x e r)#1.

# Merging two trees

Lemma merge_avl_1 : forall m1 m2, avl m1 -> avl m2 ->
-(2) <= height m1 - height m2 <= 2 ->
avl (merge m1 m2) /\
0<= height (merge m1 m2) - max (height m1) (height m2) <=1.
Lemma merge_avl : forall m1 m2, avl m1 -> avl m2 ->
-(2) <= height m1 - height m2 <= 2 -> avl (merge m1 m2).

# Deletion

Lemma remove_avl_1 : forall m x, avl m ->
avl (remove x m) /\ 0 <= height m - height (remove x m) <= 1.
Lemma remove_avl : forall m x, avl m -> avl (remove x m).
Hint Resolve remove_avl.

# Join

Lemma join_avl_1 : forall l x d r, avl l -> avl r ->
avl (join l x d r) /\
0<= height (join l x d r) - max (height l) (height r) <= 1.
Lemma join_avl : forall l x d r, avl l -> avl r -> avl (join l x d r).
Hint Resolve join_avl.
concat
Lemma concat_avl : forall m1 m2, avl m1 -> avl m2 -> avl (concat m1 m2).
Hint Resolve concat_avl.
split
Lemma split_avl : forall m x, avl m ->
avl (split x m)#l /\ avl (split x m)#r.
End Elt.
Hint Constructors avl.
Section Map.
Variable elt elt' : Type.
Variable f : elt -> elt'.
Lemma map_height : forall m, height (map f m) = height m.
Lemma map_avl : forall m, avl m -> avl (map f m).
End Map.
Section Mapi.
Variable elt elt' : Type.
Variable f : key -> elt -> elt'.
Lemma mapi_height : forall m, height (mapi f m) = height m.
Lemma mapi_avl : forall m, avl m -> avl (mapi f m).
End Mapi.
Section Map_option.
Variable elt elt' : Type.
Variable f : key -> elt -> option elt'.
Lemma map_option_avl : forall m, avl m -> avl (map_option f m).
End Map_option.
Section Map2_opt.
Variable elt elt' elt'' : Type.
Variable f : key -> elt -> option elt' -> option elt''.
Variable mapl : t elt -> t elt''.
Variable mapr : t elt' -> t elt''.
Hypothesis mapl_avl : forall m, avl m -> avl (mapl m).
Hypothesis mapr_avl : forall m', avl m' -> avl (mapr m').
Notation map2_opt := (map2_opt f mapl mapr).
Lemma map2_opt_avl : forall m1 m2, avl m1 -> avl m2 ->
avl (map2_opt m1 m2).
End Map2_opt.
Section Map2.
Variable elt elt' elt'' : Type.
Variable f : option elt -> option elt' -> option elt''.
Lemma map2_avl : forall m1 m2, avl m1 -> avl m2 -> avl (map2 f m1 m2).
End Map2.
End AvlProofs.

# Encapsulation

We can implement S with balanced binary search trees. When compared to FMapAVL, we maintain here two invariants (bst and avl) instead of only bst, which is enough for fulfilling the FMap interface.
Module IntMake (I:Int)(X: OrderedType) <: S with Module E := X.
Module E := X.
Module Import AvlProofs := AvlProofs I X.
Import Raw.
Import Raw.Proofs.
Record bbst (elt:Type) :=
Bbst {this :> tree elt; is_bst : bst this; is_avl: avl this}.
Definition t := bbst.
Definition key := E.t.
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 := Bbst (empty_bst elt) (empty_avl elt).
Definition is_empty m : bool := is_empty m.(this).
Definition add x e m : t elt :=
Bbst (add_bst x e m.(is_bst)) (add_avl x e m.(is_avl)).
Definition remove x m : t elt :=
Bbst (remove_bst x m.(is_bst)) (remove_avl x m.(is_avl)).
Definition mem x m : bool := mem x m.(this).
Definition find x m : option elt := find x m.(this).
Definition map f m : t elt' :=
Bbst (map_bst f m.(is_bst)) (map_avl f m.(is_avl)).
Definition mapi (f:key->elt->elt') m : t elt' :=
Bbst (mapi_bst f m.(is_bst)) (mapi_avl f m.(is_avl)).
Definition map2 f m (m':t elt') : t elt'' :=
Bbst (map2_bst f m.(is_bst) m'.(is_bst)) (map2_avl f m.(is_avl) m'.(is_avl)).
Definition elements m : list (key*elt) := elements m.(this).
Definition cardinal m := cardinal m.(this).
Definition fold (A:Type) (f:key->elt->A->A) m i := fold (A:=A) f m.(this) i.
Definition equal cmp m m' : bool := equal cmp m.(this) m'.(this).
Definition MapsTo x e m : Prop := MapsTo x e m.(this).
Definition In x m : Prop := In0 x m.(this).
Definition Empty m : Prop := Empty m.(this).
Definition eq_key : (key*elt) -> (key*elt) -> Prop := @PX.eqk elt.
Definition eq_key_elt : (key*elt) -> (key*elt) -> Prop := @PX.eqke elt.
Definition lt_key : (key*elt) -> (key*elt) -> Prop := @PX.ltk 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 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 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_3 : forall m, sort lt_key (elements m).
Lemma elements_3w : forall m, NoDupA eq_key (elements m).
Lemma cardinal_1 : forall m, cardinal m = length (elements m).
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 := Equiv (Cmp cmp).
Lemma Equivb_Equivb : forall cmp m m',
Equivb cmp m m' <-> Raw.Proofs.Equivb cmp m m'.
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 IntMake.
Module IntMake_ord (I:Int)(X: OrderedType)(D : OrderedType) <:
Sord with Module Data := D
with Module MapS.E := X.
Module Data := D.
Module Import MapS := IntMake(I)(X).
Import AvlProofs.
Import Raw.Proofs.
Module Import MD := OrderedTypeFacts(D).
Module LO := FMapList.Make_ord(X)(D).
Definition t := MapS.t D.t.
Definition cmp e e' :=
match D.compare e e' with EQ _ => true | _ => false end.
Definition elements (m:t) :=
LO.MapS.Build_slist (Raw.Proofs.elements_sort m.(is_bst)).

# As comparison function, we propose here a non-structural

version faithful to the code of Ocaml's Map library, instead of the structural version of FMapAVL
Fixpoint cardinal_e (e:Raw.enumeration D.t) :=
match e with
| Raw.End => 0%nat
| Raw.More _ _ r e => S (Raw.cardinal r + cardinal_e e)
end.
Lemma cons_cardinal_e : forall m e,
cardinal_e (Raw.cons m e) = (Raw.cardinal m + cardinal_e e)%nat.
Definition cardinal_e_2 ee :=
(cardinal_e (fst ee) + cardinal_e (snd ee))%nat.
Function compare_aux (ee:Raw.enumeration D.t * Raw.enumeration D.t)
{ measure cardinal_e_2 ee } : comparison :=
match ee with
| (Raw.End, Raw.End) => Eq
| (Raw.End, Raw.More _ _ _ _) => Lt
| (Raw.More _ _ _ _, Raw.End) => Gt
| (Raw.More x1 d1 r1 e1, Raw.More x2 d2 r2 e2) =>
match X.compare x1 x2 with
| EQ _ => match D.compare d1 d2 with
| EQ _ => compare_aux (Raw.cons r1 e1, Raw.cons r2 e2)
| LT _ => Lt
| GT _ => Gt
end
| LT _ => Lt
| GT _ => Gt
end
end.
Definition Cmp c :=
match c with
| Eq => LO.eq_list
| Lt => LO.lt_list
| Gt => (fun l1 l2 => LO.lt_list l2 l1)
end.
Lemma cons_Cmp : forall c x1 x2 d1 d2 l1 l2,
X.eq x1 x2 -> D.eq d1 d2 ->
Cmp c l1 l2 -> Cmp c ((x1,d1)::l1) ((x2,d2)::l2).
Hint Resolve cons_Cmp.
Lemma compare_aux_Cmp : forall e,
Cmp (compare_aux e) (flatten_e (fst e)) (flatten_e (snd e)).
Lemma compare_Cmp : forall m1 m2,
Cmp (compare_aux (Raw.cons m1 (Raw.End _), Raw.cons m2 (Raw.End _)))
(Raw.elements m1) (Raw.elements m2).
Definition eq (m1 m2 : t) := LO.eq_list (Raw.elements m1) (Raw.elements m2).
Definition lt (m1 m2 : t) := LO.lt_list (Raw.elements m1) (Raw.elements m2).
Definition compare (s s':t) : Compare lt eq s s'.
Definition selements (m1 : t) :=
LO.MapS.Build_slist (elements_sort m1.(is_bst)).
Definition seq (m1 m2 : t) := LO.eq (selements m1) (selements m2).
Definition slt (m1 m2 : t) := LO.lt (selements m1) (selements m2).
Lemma eq_seq : forall m1 m2, eq m1 m2 <-> seq m1 m2.
Lemma lt_slt : forall m1 m2, lt m1 m2 <-> slt m1 m2.
Lemma eq_1 : forall (m m' : t), MapS.Equivb cmp m m' -> eq m m'.
Lemma eq_2 : forall m m', eq m m' -> MapS.Equivb cmp m m'.
Lemma eq_refl : forall m : t, eq m m.
Lemma eq_sym : forall m1 m2 : t, eq m1 m2 -> eq m2 m1.
Lemma eq_trans : forall m1 m2 m3 : t, eq m1 m2 -> eq m2 m3 -> eq m1 m3.
Lemma lt_trans : forall m1 m2 m3 : t, lt m1 m2 -> lt m2 m3 -> lt m1 m3.
Lemma lt_not_eq : forall m1 m2 : t, lt m1 m2 -> ~ eq m1 m2.
End IntMake_ord.
Module Make (X: OrderedType) <: S with Module E := X
:=IntMake(Z_as_Int)(X).
Module Make_ord (X: OrderedType)(D: OrderedType)
<: Sord with Module Data := D
with Module MapS.E := X
:=IntMake_ord(Z_as_Int)(X)(D).