# FMapAVL

This module implements maps using AVL trees. It follows the implementation from Ocaml's standard library.

See the comments at the beginning of FSetAVL for more details.

Require Import FMapInterface FMapList ZArith Int.

Notations and helper lemma about pairs

Notation "s #1" := (fst s) (at level 9, format "s '#1'") : pair_scope.
Notation "s #2" := (snd s) (at level 9, format "s '#2'") : pair_scope.

# The Raw functor

Functor of pure functions + separate proofs of invariant preservation

Module Raw (Import I:Int)(X: OrderedType).
Open Local Scope pair_scope.
Open Local Scope lazy_bool_scope.
Open Local Scope Int_scope.

Definition key := X.t.

# Trees

Section Elt.

Variable elt : Type.

# Trees

The fifth field of Node is the height of the tree

Inductive tree :=
| Leaf : tree
| Node : tree -> key -> elt -> tree -> int -> tree.

Notation t := tree.

# Basic functions on trees: height and cardinal

Definition height (m : t) : int :=
match m with
| Leaf => 0
| Node _ _ _ _ h => h
end.

Fixpoint cardinal (m : t) : nat :=
match m with
| Leaf => 0%nat
| Node l _ _ r _ => S (cardinal l + cardinal r)
end.

# Empty Map

Definition empty := Leaf.

# Emptyness test

Definition is_empty m := match m with Leaf => true | _ => false end.

# Appartness

The mem function is deciding appartness. It exploits the bst property to achieve logarithmic complexity.

Fixpoint mem x m : bool :=
match m with
| Leaf => false
| Node l y _ r _ => match X.compare x y with
| LT _ => mem x l
| EQ _ => true
| GT _ => mem x r
end
end.

Fixpoint find x m : option elt :=
match m with
| Leaf => None
| Node l y d r _ => match X.compare x y with
| LT _ => find x l
| EQ _ => Some d
| GT _ => find x r
end
end.

# Helper functions

create l x r creates a node, assuming l and r to be balanced and |height l - height r| <= 2.

Definition create l x e r :=
Node l x e r (max (height l) (height r) + 1).

bal l x e r acts as create, but performs one step of rebalancing if necessary, i.e. assumes |height l - height r| <= 3.

Definition assert_false := create.

Fixpoint bal l x d r :=
let hl := height l in
let hr := height r in
if gt_le_dec hl (hr+2) then
match l with
| Leaf => assert_false l x d r
| Node ll lx ld lr _ =>
if ge_lt_dec (height ll) (height lr) then
create ll lx ld (create lr x d r)
else
match lr with
| Leaf => assert_false l x d r
| Node lrl lrx lrd lrr _ =>
create (create ll lx ld lrl) lrx lrd (create lrr x d r)
end
end
else
if gt_le_dec hr (hl+2) then
match r with
| Leaf => assert_false l x d r
| Node rl rx rd rr _ =>
if ge_lt_dec (height rr) (height rl) then
create (create l x d rl) rx rd rr
else
match rl with
| Leaf => assert_false l x d r
| Node rll rlx rld rlr _ =>
create (create l x d rll) rlx rld (create rlr rx rd rr)
end
end
else
create l x d r.

# Insertion

Fixpoint add x d m :=
match m with
| Leaf => Node Leaf x d Leaf 1
| Node l y d' r h =>
match X.compare x y with
| LT _ => bal (add x d l) y d' r
| EQ _ => Node l y d r h
| GT _ => bal l y d' (add x d r)
end
end.

# Extraction of minimum binding

Morally, remove_min is to be applied to a non-empty tree t = Node l x e r h. Since we can't deal here with assert false for t=Leaf, we pre-unpack t (and forget about h).

Fixpoint remove_min l x d r : t*(key*elt) :=
match l with
| Leaf => (r,(x,d))
| Node ll lx ld lr lh =>
let (l',m) := remove_min ll lx ld lr in
(bal l' x d r, m)
end.

# Merging two trees

merge t1 t2 builds the union of t1 and t2 assuming all elements of t1 to be smaller than all elements of t2, and |height t1 - height t2| <= 2.

Fixpoint merge s1 s2 := match s1,s2 with
| Leaf, _ => s2
| _, Leaf => s1
| _, Node l2 x2 d2 r2 h2 =>
match remove_min l2 x2 d2 r2 with
(s2',(x,d)) => bal s1 x d s2'
end
end.

# Deletion

Fixpoint remove x m := match m with
| Leaf => Leaf
| Node l y d r h =>
match X.compare x y with
| LT _ => bal (remove x l) y d r
| EQ _ => merge l r
| GT _ => bal l y d (remove x r)
end
end.

# join

Same as bal but does not assume anything regarding heights of l and r.

Fixpoint join l : key -> elt -> t -> t :=
match l with
| Node ll lx ld lr lh => fun x d =>
fix join_aux (r:t) : t := match r with
| Leaf => add x d l
| Node rl rx rd rr rh =>
if gt_le_dec lh (rh+2) then bal ll lx ld (join lr x d r)
else if gt_le_dec rh (lh+2) then bal (join_aux rl) rx rd rr
else create l x d r
end
end.

# Splitting

split x m returns a triple (l, o, r) where
• l is the set of elements of m that are < x
• r is the set of elements of m that are > x
• o is the result of find x m.

Record triple := mktriple { t_left:t; t_opt:option elt; t_right:t }.
Notation "<< l , b , r >>" := (mktriple l b r) (at level 9).

Fixpoint split x m : triple := match m with
| Leaf => << Leaf, None, Leaf >>
| Node l y d r h =>
match X.compare x y with
| LT _ => let (ll,o,rl) := split x l in << ll, o, join rl y d r >>
| EQ _ => << l, Some d, r >>
| GT _ => let (rl,o,rr) := split x r in << join l y d rl, o, rr >>
end
end.

# Concatenation

Same as merge but does not assume anything about heights.

Definition concat m1 m2 :=
match m1, m2 with
| Leaf, _ => m2
| _ , Leaf => m1
| _, Node l2 x2 d2 r2 _ =>
let (m2',xd) := remove_min l2 x2 d2 r2 in
join m1 xd#1 xd#2 m2'
end.

# Elements

elements_tree_aux acc t catenates the elements of t in infix order to the list acc

Fixpoint elements_aux (acc : list (key*elt)) m : list (key*elt) :=
match m with
| Leaf => acc
| Node l x d r _ => elements_aux ((x,d) :: elements_aux acc r) l
end.

then elements is an instanciation with an empty acc

Definition elements := elements_aux nil.

# Fold

Fixpoint fold (A : Type) (f : key -> elt -> A -> A) (m : t) : A -> A :=
fun a => match m with
| Leaf => a
| Node l x d r _ => fold f r (f x d (fold f l a))
end.

# Comparison

Variable cmp : elt->elt->bool.

## Enumeration of the elements of a tree

Inductive enumeration :=
| End : enumeration
| More : key -> elt -> t -> enumeration -> enumeration.

cons m e adds the elements of tree m on the head of enumeration e.

Fixpoint cons m e : enumeration :=
match m with
| Leaf => e
| Node l x d r h => cons l (More x d r e)
end.

One step of comparison of elements

Definition equal_more x1 d1 (cont:enumeration->bool) e2 :=
match e2 with
| End => false
| More x2 d2 r2 e2 =>
match X.compare x1 x2 with
| EQ _ => cmp d1 d2 &&& cont (cons r2 e2)
| _ => false
end
end.

Comparison of left tree, middle element, then right tree

Fixpoint equal_cont m1 (cont:enumeration->bool) e2 :=
match m1 with
| Leaf => cont e2
| Node l1 x1 d1 r1 _ =>
equal_cont l1 (equal_more x1 d1 (equal_cont r1 cont)) e2
end.

Initial continuation

Definition equal_end e2 := match e2 with End => true | _ => false end.

The complete comparison

Definition equal m1 m2 := equal_cont m1 equal_end (cons m2 End).

End Elt.
Notation t := tree.
Notation "<< l , b , r >>" := (mktriple l b r) (at level 9).
Notation "t #l" := (t_left t) (at level 9, format "t '#l'").
Notation "t #o" := (t_opt t) (at level 9, format "t '#o'").
Notation "t #r" := (t_right t) (at level 9, format "t '#r'").

# Map

Fixpoint map (elt elt' : Type)(f : elt -> elt')(m : t elt) : t elt' :=
match m with
| Leaf => Leaf _
| Node l x d r h => Node (map f l) x (f d) (map f r) h
end.

Fixpoint mapi (elt elt' : Type)(f : key -> elt -> elt')(m : t elt) : t elt' :=
match m with
| Leaf => Leaf _
| Node l x d r h => Node (mapi f l) x (f x d) (mapi f r) h
end.

# Map with removal

Fixpoint map_option (elt elt' : Type)(f : key -> elt -> option elt')(m : t elt)
: t elt' :=
match m with
| Leaf => Leaf _
| Node l x d r h =>
match f x d with
| Some d' => join (map_option f l) x d' (map_option f r)
| None => concat (map_option f l) (map_option f r)
end
end.

# Optimized map2

Suggestion by B. Gregoire: a map2 function with specialized arguments allowing to bypass some tree traversal. Instead of one f0 of type key -> option elt -> option elt' -> option elt'', we ask here for:
• f which is a specialisation of f0 when first option isn't None
• mapl treats a tree elt with f0 when second option is None
• mapr treats a tree elt' with f0 when first option is None

The idea is that mapl and mapr can be instantaneous (e.g. the identity or some constant function).

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''.

Fixpoint map2_opt m1 m2 :=
match m1, m2 with
| Leaf, _ => mapr m2
| _, Leaf => mapl m1
| Node l1 x1 d1 r1 h1, _ =>
let (l2',o2,r2') := split x1 m2 in
match f x1 d1 o2 with
| Some e => join (map2_opt l1 l2') x1 e (map2_opt r1 r2')
| None => concat (map2_opt l1 l2') (map2_opt r1 r2')
end
end.

End Map2_opt.

# Map2

The map2 function of the Map interface can be implemented via map2_opt and map_option.

Section Map2.
Variable elt elt' elt'' : Type.
Variable f : option elt -> option elt' -> option elt''.

Definition map2 : t elt -> t elt' -> t elt'' :=
map2_opt
(fun _ d o => f (Some d) o)
(map_option (fun _ d => f (Some d) None))
(map_option (fun _ d' => f None (Some d'))).

End Map2.

# Invariants

Section Invariants.
Variable elt : Type.

## Occurrence in a tree

Inductive MapsTo (x : key)(e : elt) : t elt -> Prop :=
| MapsRoot : forall l r h y,
X.eq x y -> MapsTo x e (Node l y e r h)
| MapsLeft : forall l r h y e',
MapsTo x e l -> MapsTo x e (Node l y e' r h)
| MapsRight : forall l r h y e',
MapsTo x e r -> MapsTo x e (Node l y e' r h).

Inductive In (x : key) : t elt -> Prop :=
| InRoot : forall l r h y e,
X.eq x y -> In x (Node l y e r h)
| InLeft : forall l r h y e',
In x l -> In x (Node l y e' r h)
| InRight : forall l r h y e',
In x r -> In x (Node l y e' r h).

Definition In0 k m := exists e:elt, MapsTo k e m.

## Binary search trees

lt_tree x s: all elements in s are smaller than x (resp. greater for gt_tree)

Definition lt_tree x m := forall y, In y m -> X.lt y x.
Definition gt_tree x m := forall y, In y m -> X.lt x y.

bst t : t is a binary search tree

Inductive bst : t elt -> Prop :=
| BSLeaf : bst (Leaf _)
| BSNode : forall x e l r h, bst l -> bst r ->
lt_tree x l -> gt_tree x r -> bst (Node l x e r h).

End Invariants.

# Correctness proofs, isolated in a sub-module

Module Proofs.
Module MX := OrderedTypeFacts X.
Module PX := KeyOrderedType X.
Module L := FMapList.Raw X.

# Automation and dedicated tactics.

Hint Constructors tree MapsTo In bst.
Hint Unfold lt_tree gt_tree.

Tactic Notation "factornode" ident(l) ident(x) ident(d) ident(r) ident(h)
"as" ident(s) :=
set (s:=Node l x d r h) in *; clearbody s; clear l x d r h.

A tactic for cleaning hypothesis after use of functional induction.

Ltac clearf :=
match goal with
| H : (@Logic.eq (Compare _ _ _ _) _ _) |- _ => clear H; clearf
| H : (@Logic.eq (sumbool _ _) _ _) |- _ => clear H; clearf
| _ => idtac
end.

A tactic to repeat inversion_clear on all hyps of the form (f (Node ...))

Ltac inv f :=
match goal with
| H:f (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ _ _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| _ => idtac
end.

Ltac inv_all f :=
match goal with
| H: f _ |- _ => inversion_clear H; inv f
| H: f _ _ |- _ => inversion_clear H; inv f
| H: f _ _ _ |- _ => inversion_clear H; inv f
| H: f _ _ _ _ |- _ => inversion_clear H; inv f
| _ => idtac
end.

Helper tactic concerning order of elements.

Ltac order := match goal with
| U: lt_tree _ ?s, V: In _ ?s |- _ => generalize (U _ V); clear U; order
| U: gt_tree _ ?s, V: In _ ?s |- _ => generalize (U _ V); clear U; order
| _ => MX.order
end.

Ltac intuition_in := repeat progress (intuition; inv In; inv MapsTo).

Ltac join_tac :=
intros l; induction l as [| ll _ lx ld lr Hlr lh];
[ | intros x d r; induction r as [| rl Hrl rx rd rr _ rh]; unfold join;
[ | destruct (gt_le_dec lh (rh+2));
[ match goal with |- context [ bal ?u ?v ?w ?z ] =>
replace (bal u v w z)
with (bal ll lx ld (join lr x d (Node rl rx rd rr rh))); [ | auto]
end
| destruct (gt_le_dec rh (lh+2));
[ match goal with |- context [ bal ?u ?v ?w ?z ] =>
replace (bal u v w z)
with (bal (join (Node ll lx ld lr lh) x d rl) rx rd rr); [ | auto]
end
| ] ] ] ]; intros.

Section Elt.
Variable elt:Type.
Implicit Types m r : t elt.

# Basic results about MapsTo, In, lt_tree, gt_tree, height

Lemma MapsTo_In : forall k e m, MapsTo k e m -> In k m.
Hint Resolve MapsTo_In.

Lemma In_MapsTo : forall k m, In k m -> exists e, MapsTo k e m.

Lemma In_alt : forall k m, In0 k m <-> In k m.

Lemma MapsTo_1 :
forall m x y e, X.eq x y -> MapsTo x e m -> MapsTo y e m.
Hint Immediate MapsTo_1.

Lemma In_1 :
forall m x y, X.eq x y -> In x m -> In y m.

Lemma In_node_iff :
forall l x e r h y,
In y (Node l x e r h) <-> In y l \/ X.eq y x \/ In y r.

Lemma lt_leaf : forall x, lt_tree x (Leaf elt).

Lemma gt_leaf : forall x, gt_tree x (Leaf elt).

Lemma lt_tree_node : forall x y l r e h,
lt_tree x l -> lt_tree x r -> X.lt y x -> lt_tree x (Node l y e r h).

Lemma gt_tree_node : forall x y l r e h,
gt_tree x l -> gt_tree x r -> X.lt x y -> gt_tree x (Node l y e r h).

Hint Resolve lt_leaf gt_leaf lt_tree_node gt_tree_node.

Lemma lt_left : forall x y l r e h,
lt_tree x (Node l y e r h) -> lt_tree x l.

Lemma lt_right : forall x y l r e h,
lt_tree x (Node l y e r h) -> lt_tree x r.

Lemma gt_left : forall x y l r e h,
gt_tree x (Node l y e r h) -> gt_tree x l.

Lemma gt_right : forall x y l r e h,
gt_tree x (Node l y e r h) -> gt_tree x r.

Hint Resolve lt_left lt_right gt_left gt_right.

Lemma lt_tree_not_in :
forall x m, lt_tree x m -> ~ In x m.

Lemma lt_tree_trans :
forall x y, X.lt x y -> forall m, lt_tree x m -> lt_tree y m.

Lemma gt_tree_not_in :
forall x m, gt_tree x m -> ~ In x m.

Lemma gt_tree_trans :
forall x y, X.lt y x -> forall m, gt_tree x m -> gt_tree y m.

Hint Resolve lt_tree_not_in lt_tree_trans gt_tree_not_in gt_tree_trans.

# Empty map

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

Lemma empty_bst : bst (empty elt).

Lemma empty_1 : Empty (empty elt).

# Emptyness test

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

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

# Appartness

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

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

Lemma find_1 : forall m x e, bst m -> 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 find_iff : forall m x e, bst m ->
(find x m = Some e <-> MapsTo x e m).

Lemma find_in : forall m x, find x m <> None -> In x m.

Lemma in_find : forall m x, bst m -> In x m -> find x m <> None.

Lemma find_in_iff : forall m x, bst m ->
(find x m <> None <-> In x m).

Lemma not_find_iff : forall m x, bst m ->
(find x m = None <-> ~In x m).

Lemma find_find : forall m m' x,
find x m = find x m' <->
(forall d, find x m = Some d <-> find x m' = Some d).

Lemma find_mapsto_equiv : forall m m' x, bst m -> bst m' ->
(find x m = find x m' <->
(forall d, MapsTo x d m <-> MapsTo x d m')).

Lemma find_in_equiv : forall m m' x, bst m -> bst m' ->
find x m = find x m' ->
(In x m <-> In x m').

# Helper functions

Lemma create_bst :
forall l x e r, bst l -> bst r -> lt_tree x l -> gt_tree x r ->
bst (create l x e r).
Hint Resolve create_bst.

Lemma create_in :
forall l x e r y,
In y (create l x e r) <-> X.eq y x \/ In y l \/ In y r.

Lemma bal_bst : forall l x e r, bst l -> bst r ->
lt_tree x l -> gt_tree x r -> bst (bal l x e r).
Hint Resolve bal_bst.

Lemma bal_in : forall l x e r y,
In y (bal l x e r) <-> X.eq y x \/ In y l \/ In y r.

Lemma bal_mapsto : forall l x e r y e',
MapsTo y e' (bal l x e r) <-> MapsTo y e' (create l x e r).

Lemma bal_find : forall l x e r y,
bst l -> bst r -> lt_tree x l -> gt_tree x r ->
find y (bal l x e r) = find y (create l x e r).

# Insertion

Lemma add_in : forall m x y e,
In y (add x e m) <-> X.eq y x \/ In y m.

Lemma add_bst : forall m x e, bst m -> bst (add x e m).

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_find : forall m x y e, bst m ->
find y (add x e m) =
match X.compare y x with EQ _ => Some e | _ => find y m end.

# Extraction of minimum binding

Lemma remove_min_in : forall l x e r h y,
In y (Node l x e r h) <->
X.eq y (remove_min l x e r)#2#1 \/ In y (remove_min l x e r)#1.

Lemma remove_min_mapsto : forall l x e r h y e',
MapsTo y e' (Node l x e r h) <->
((X.eq y (remove_min l x e r)#2#1) /\ e' = (remove_min l x e r)#2#2)
\/ MapsTo y e' (remove_min l x e r)#1.

Lemma remove_min_bst : forall l x e r h,
bst (Node l x e r h) -> bst (remove_min l x e r)#1.
Hint Resolve remove_min_bst.

Lemma remove_min_gt_tree : forall l x e r h,
bst (Node l x e r h) ->
gt_tree (remove_min l x e r)#2#1 (remove_min l x e r)#1.
Hint Resolve remove_min_gt_tree.

Lemma remove_min_find : forall l x e r h y,
bst (Node l x e r h) ->
find y (Node l x e r h) =
match X.compare y (remove_min l x e r)#2#1 with
| LT _ => None
| EQ _ => Some (remove_min l x e r)#2#2
| GT _ => find y (remove_min l x e r)#1
end.

# Merging two trees

Lemma merge_in : forall m1 m2 y, bst m1 -> bst m2 ->
(In y (merge m1 m2) <-> In y m1 \/ In y m2).

Lemma merge_mapsto : forall m1 m2 y e, bst m1 -> bst m2 ->
(MapsTo y e (merge m1 m2) <-> MapsTo y e m1 \/ MapsTo y e m2).

Lemma merge_bst : forall m1 m2, bst m1 -> bst m2 ->
(forall y1 y2 : key, In y1 m1 -> In y2 m2 -> X.lt y1 y2) ->
bst (merge m1 m2).

# Deletion

Lemma remove_in : forall m x y, bst m ->
(In y (remove x m) <-> ~ X.eq y x /\ In y m).

Lemma remove_bst : forall m x, bst m -> bst (remove x m).

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

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

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

# join

Lemma join_in : forall l x d r y,
In y (join l x d r) <-> X.eq y x \/ In y l \/ In y r.

Lemma join_bst : forall l x d r, bst l -> bst r ->
lt_tree x l -> gt_tree x r -> bst (join l x d r).
Hint Resolve join_bst.

Lemma join_find : forall l x d r y,
bst l -> bst r -> lt_tree x l -> gt_tree x r ->
find y (join l x d r) = find y (create l x d r).

# split

Lemma split_in_1 : forall m x, bst m -> forall y,
(In y (split x m)#l <-> In y m /\ X.lt y x).

Lemma split_in_2 : forall m x, bst m -> forall y,
(In y (split x m)#r <-> In y m /\ X.lt x y).

Lemma split_in_3 : forall m x, bst m ->
(split x m)#o = find x m.

Lemma split_bst : forall m x, bst m ->
bst (split x m)#l /\ bst (split x m)#r.

Lemma split_lt_tree : forall m x, bst m -> lt_tree x (split x m)#l.

Lemma split_gt_tree : forall m x, bst m -> gt_tree x (split x m)#r.

Lemma split_find : forall m x y, bst m ->
find y m = match X.compare y x with
| LT _ => find y (split x m)#l
| EQ _ => (split x m)#o
| GT _ => find y (split x m)#r
end.

# Concatenation

Lemma concat_in : forall m1 m2 y,
In y (concat m1 m2) <-> In y m1 \/ In y m2.

Lemma concat_bst : forall m1 m2, bst m1 -> bst m2 ->
(forall y1 y2, In y1 m1 -> In y2 m2 -> X.lt y1 y2) ->
bst (concat m1 m2).
Hint Resolve concat_bst.

Lemma concat_find : forall m1 m2 y, bst m1 -> bst m2 ->
(forall y1 y2, In y1 m1 -> In y2 m2 -> X.lt y1 y2) ->
find y (concat m1 m2) =
match find y m2 with Some d => Some d | None => find y m1 end.

# Elements

Notation eqk := (PX.eqk (elt:= elt)).
Notation eqke := (PX.eqke (elt:= elt)).
Notation ltk := (PX.ltk (elt:= elt)).

Lemma elements_aux_mapsto : forall (s:t elt) acc x e,
InA eqke (x,e) (elements_aux acc s) <-> MapsTo x e s \/ InA eqke (x,e) acc.

Lemma elements_mapsto : forall (s:t elt) x e, InA eqke (x,e) (elements s) <-> MapsTo x e s.

Lemma elements_in : forall (s:t elt) x, L.PX.In x (elements s) <-> In x s.

Lemma elements_aux_sort : forall (s:t elt) acc, bst s -> sort ltk acc ->
(forall x e y, InA eqke (x,e) acc -> In y s -> X.lt y x) ->
sort ltk (elements_aux acc s).

Lemma elements_sort : forall s : t elt, bst s -> sort ltk (elements s).
Hint Resolve elements_sort.

Lemma elements_nodup : forall s : t elt, bst s -> NoDupA eqk (elements s).

Lemma elements_aux_cardinal :
forall (m:t elt) acc, (length acc + cardinal m)%nat = length (elements_aux acc m).

Lemma elements_cardinal : forall (m:t elt), cardinal m = length (elements m).

Lemma elements_app :
forall (s:t elt) acc, elements_aux acc s = elements s ++ acc.

Lemma elements_node :
forall (t1 t2:t elt) x e z l,
elements t1 ++ (x,e) :: elements t2 ++ l =
elements (Node t1 x e t2 z) ++ l.

# Fold

Definition fold' (A : Type) (f : key -> elt -> A -> A)(s : t elt) :=
L.fold f (elements s).

Lemma fold_equiv_aux :
forall (A : Type) (s : t elt) (f : key -> elt -> A -> A) (a : A) acc,
L.fold f (elements_aux acc s) a = L.fold f acc (fold f s a).

Lemma fold_equiv :
forall (A : Type) (s : t elt) (f : key -> elt -> A -> A) (a : A),
fold f s a = fold' f s a.

Lemma fold_1 :
forall (s:t elt)(Hs:bst s)(A : Type)(i:A)(f : key -> elt -> A -> A),
fold f s i = fold_left (fun a p => f p#1 p#2 a) (elements s) i.

# Comparison

flatten_e e returns the list of elements of the enumeration e i.e. the list of elements actually compared

Fixpoint flatten_e (e : enumeration elt) : list (key*elt) := match e with
| End => nil
| More x e t r => (x,e) :: elements t ++ flatten_e r
end.

Lemma flatten_e_elements :
forall (l:t elt) r x d z e,
elements l ++ flatten_e (More x d r e) =
elements (Node l x d r z) ++ flatten_e e.

Lemma cons_1 : forall (s:t elt) e,
flatten_e (cons s e) = elements s ++ flatten_e e.

Proof of correction for the comparison

Variable cmp : elt->elt->bool.

Definition IfEq b l1 l2 := L.equal cmp l1 l2 = b.

Lemma cons_IfEq : forall b x1 x2 d1 d2 l1 l2,
X.eq x1 x2 -> cmp d1 d2 = true ->
IfEq b l1 l2 ->
IfEq b ((x1,d1)::l1) ((x2,d2)::l2).

Lemma equal_end_IfEq : forall e2,
IfEq (equal_end e2) nil (flatten_e e2).

Lemma equal_more_IfEq :
forall x1 d1 (cont:enumeration elt -> bool) x2 d2 r2 e2 l,
IfEq (cont (cons r2 e2)) l (elements r2 ++ flatten_e e2) ->
IfEq (equal_more cmp x1 d1 cont (More x2 d2 r2 e2)) ((x1,d1)::l)
(flatten_e (More x2 d2 r2 e2)).

Lemma equal_cont_IfEq : forall m1 cont e2 l,
(forall e, IfEq (cont e) l (flatten_e e)) ->
IfEq (equal_cont cmp m1 cont e2) (elements m1 ++ l) (flatten_e e2).

Lemma equal_IfEq : forall (m1 m2:t elt),
IfEq (equal cmp m1 m2) (elements m1) (elements m2).

Definition Equivb 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 Equivb_elements : forall s s',
Equivb s s' <-> L.Equivb cmp (elements s) (elements s').

Lemma equal_Equivb : forall (s s': t elt), bst s -> bst s' ->
(equal cmp s s' = true <-> Equivb s s').

End Elt.

Section Map.
Variable elt elt' : Type.
Variable f : elt -> elt'.

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

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

Lemma map_bst : forall m, bst m -> bst (map f m).

End Map.
Section Mapi.
Variable elt elt' : Type.
Variable f : key -> elt -> elt'.

Lemma mapi_1 : forall (m: tree elt)(x:key)(e: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),
In x (mapi f m) -> In x m.

Lemma mapi_bst : forall m, bst m -> bst (mapi f m).

End Mapi.

Section Map_option.
Variable elt elt' : Type.
Variable f : key -> elt -> option elt'.
Hypothesis f_compat : forall x x' d, X.eq x x' -> f x d = f x' d.

Lemma map_option_2 : forall (m:t elt)(x:key),
In x (map_option f m) -> exists d, MapsTo x d m /\ f x d <> None.

Lemma map_option_bst : forall m, bst m -> bst (map_option f m).
Hint Resolve map_option_bst.

Ltac nonify e :=
replace e with (@None elt) by
(symmetry; rewrite not_find_iff; auto; intro; order).

Lemma map_option_find : forall (m:t elt)(x:key),
bst m ->
find x (map_option f m) =
match (find x m) with Some d => f x d | None => None end.

End Map_option.

Section Map2_opt.
Variable elt elt' elt'' : Type.
Variable f0 : key -> option elt -> option elt' -> option elt''.
Variable f : key -> elt -> option elt' -> option elt''.
Variable mapl : t elt -> t elt''.
Variable mapr : t elt' -> t elt''.
Hypothesis f0_f : forall x d o, f x d o = f0 x (Some d) o.
Hypothesis mapl_bst : forall m, bst m -> bst (mapl m).
Hypothesis mapr_bst : forall m', bst m' -> bst (mapr m').
Hypothesis mapl_f0 : forall x m, bst m ->
find x (mapl m) =
match find x m with Some d => f0 x (Some d) None | None => None end.
Hypothesis mapr_f0 : forall x m', bst m' ->
find x (mapr m') =
match find x m' with Some d' => f0 x None (Some d') | None => None end.
Hypothesis f0_compat : forall x x' o o', X.eq x x' -> f0 x o o' = f0 x' o o'.

Notation map2_opt := (map2_opt f mapl mapr).

Lemma map2_opt_2 : forall m m' y, bst m -> bst m' ->
In y (map2_opt m m') -> In y m \/ In y m'.

Lemma map2_opt_bst : forall m m', bst m -> bst m' ->
bst (map2_opt m m').
Hint Resolve map2_opt_bst.

Ltac map2_aux :=
match goal with
| H : In ?x _ \/ In ?x ?m,
H' : find ?x ?m = find ?x ?m', B:bst ?m, B':bst ?m' |- _ =>
destruct H; [ intuition_in; order |
rewrite <-(find_in_equiv B B' H'); auto ]
end.

Ltac nonify t :=
match t with (find ?y (map2_opt ?m ?m')) =>
replace t with (@None elt'');
[ | symmetry; rewrite not_find_iff; auto; intro;
destruct (@map2_opt_2 m m' y); auto; order ]
end.

Lemma map2_opt_1 : forall m m' y, bst m -> bst m' ->
In y m \/ In y m' ->
find y (map2_opt m m') = f0 y (find y m) (find y m').

End Map2_opt.

Section Map2.
Variable elt elt' elt'' : Type.
Variable f : option elt -> option elt' -> option elt''.

Lemma map2_bst : forall m m', bst m -> bst m' -> bst (map2 f m m').

Lemma map2_1 : forall m m' y, bst m -> bst m' ->
In y m \/ In y m' -> find y (map2 f m m') = f (find y m) (find y m').

Lemma map2_2 : forall m m' y, bst m -> bst m' ->
In y (map2 f m m') -> In y m \/ In y m'.

End Map2.
End Proofs.
End Raw.

# Encapsulation

Now, in order to really provide a functor implementing S, we need to encapsulate everything into a type of balanced binary search trees.

Module IntMake (I:Int)(X: OrderedType) <: S with Module E := X.

Module E := X.
Module Raw := Raw I X.
Import Raw.Proofs.

Record bst (elt:Type) :=
Bst {this :> Raw.tree elt; is_bst : Raw.bst this}.

Definition t := bst.
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 := Bst (empty_bst elt).
Definition is_empty m : bool := Raw.is_empty m.(this).
Definition add x e m : t elt := Bst (add_bst x e m.(is_bst)).
Definition remove x m : t elt := Bst (remove_bst x m.(is_bst)).
Definition mem x m : bool := Raw.mem x m.(this).
Definition find x m : option elt := Raw.find x m.(this).
Definition map f m : t elt' := Bst (map_bst f m.(is_bst)).
Definition mapi (f:key->elt->elt') m : t elt' :=
Bst (mapi_bst f m.(is_bst)).
Definition map2 f m (m':t elt') : t elt'' :=
Bst (map2_bst f m.(is_bst) m'.(is_bst)).
Definition elements m : list (key*elt) := Raw.elements m.(this).
Definition cardinal m := Raw.cardinal m.(this).
Definition fold (A:Type) (f:key->elt->A->A) m i := Raw.fold (A:=A) f m.(this) i.
Definition equal cmp m m' : bool := Raw.equal cmp m.(this) m'.(this).

Definition MapsTo x e m : Prop := Raw.MapsTo x e m.(this).
Definition In x m : Prop := Raw.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).
Module LO := FMapList.Make_ord(X)(D).
Module R := Raw.
Module P := Raw.Proofs.

Definition t := MapS.t D.t.

Definition cmp e e' :=
match D.compare e e' with EQ _ => true | _ => false end.

One step of comparison of elements

Definition compare_more x1 d1 (cont:R.enumeration D.t -> comparison) e2 :=
match e2 with
| R.End => Gt
| R.More x2 d2 r2 e2 =>
match X.compare x1 x2 with
| EQ _ => match D.compare d1 d2 with
| EQ _ => cont (R.cons r2 e2)
| LT _ => Lt
| GT _ => Gt
end
| LT _ => Lt
| GT _ => Gt
end
end.

Comparison of left tree, middle element, then right tree

Fixpoint compare_cont s1 (cont:R.enumeration D.t -> comparison) e2 :=
match s1 with
| R.Leaf => cont e2
| R.Node l1 x1 d1 r1 _ =>
compare_cont l1 (compare_more x1 d1 (compare_cont r1 cont)) e2
end.

Initial continuation

Definition compare_end (e2:R.enumeration D.t) :=
match e2 with R.End => Eq | _ => Lt end.

The complete comparison

Definition compare_pure s1 s2 :=
compare_cont s1 compare_end (R.cons s2 (Raw.End _)).

Correctness of this comparison

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_end_Cmp :
forall e2, Cmp (compare_end e2) nil (P.flatten_e e2).

Lemma compare_more_Cmp : forall x1 d1 cont x2 d2 r2 e2 l,
Cmp (cont (R.cons r2 e2)) l (R.elements r2 ++ P.flatten_e e2) ->
Cmp (compare_more x1 d1 cont (R.More x2 d2 r2 e2)) ((x1,d1)::l)
(P.flatten_e (R.More x2 d2 r2 e2)).

Lemma compare_cont_Cmp : forall s1 cont e2 l,
(forall e, Cmp (cont e) l (P.flatten_e e)) ->
Cmp (compare_cont s1 cont e2) (R.elements s1 ++ l) (P.flatten_e e2).

Lemma compare_Cmp : forall s1 s2,
Cmp (compare_pure s1 s2) (R.elements s1) (R.elements s2).

The dependent-style compare

Definition eq (m1 m2 : t) := LO.eq_list (elements m1) (elements m2).
Definition lt (m1 m2 : t) := LO.lt_list (elements m1) (elements m2).

Definition compare (s s':t) : Compare lt eq s s'.

Definition selements (m1 : t) :=
LO.MapS.Build_slist (P.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), Equivb cmp m m' -> eq m m'.

Lemma eq_2 : forall m m', eq m m' -> 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).