# Module Maps

Applicative finite maps are the main data structure used in this project. A finite map associates data to keys. The two main operations are set k d m, which returns a map identical to m except that d is associated to k, and get k m which returns the data associated to key k in map m. In this library, we distinguish two kinds of maps:
• Trees: the get operation returns an option type, either None if no data is associated to the key, or Some d otherwise.
• Maps: the get operation always returns a data. If no data was explicitly associated with the key, a default data provided at map initialization time is returned.
In this library, we provide efficient implementations of trees and maps whose keys range over the type positive of binary positive integers or any type that can be injected into positive. The implementation is based on radix-2 search trees (uncompressed Patricia trees) and guarantees logarithmic-time operations. An inefficient implementation of maps as functions is also provided.

Require Import Equivalence EquivDec.
Require Import Coqlib.

Local Unset Elimination Schemes.
Local Unset Case Analysis Schemes.

Set Implicit Arguments.

# The abstract signatures of trees

Module Type TREE.
Variable elt: Type.
Variable elt_eq: forall (a b: elt), {a = b} + {a <> b}.
Variable t: Type -> Type.
Variable empty: forall (A: Type), t A.
Variable get: forall (A: Type), elt -> t A -> option A.
Variable set: forall (A: Type), elt -> A -> t A -> t A.
Variable remove: forall (A: Type), elt -> t A -> t A.

The ``good variables'' properties for trees, expressing commutations between get, set and remove.
Hypothesis gempty:
forall (A: Type) (i: elt), get i (empty A) = None.
Hypothesis gss:
forall (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = Some x.
Hypothesis gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Hypothesis gsspec:
forall (A: Type) (i j: elt) (x: A) (m: t A),
get i (set j x m) = if elt_eq i j then Some x else get i m.
Hypothesis gsident:
forall (A: Type) (i: elt) (m: t A) (v: A),
get i m = Some v -> set i v m = m.
Hypothesis grs:
forall (A: Type) (i: elt) (m: t A), get i (remove i m) = None.
Hypothesis gro:
forall (A: Type) (i j: elt) (m: t A),
i <> j -> get i (remove j m) = get i m.
Hypothesis grspec:
forall (A: Type) (i j: elt) (m: t A),
get i (remove j m) = if elt_eq i j then None else get i m.

Extensional equality between trees.
Variable beq: forall (A: Type), (A -> A -> bool) -> t A -> t A -> bool.
Hypothesis beq_correct:
forall (A: Type) (eqA: A -> A -> bool) (t1 t2: t A),
beq eqA t1 t2 = true <->
(forall (x: elt),
match get x t1, get x t2 with
| None, None => True
| Some y1, Some y2 => eqA y1 y2 = true
| _, _ => False
end).

Applying a function to all data of a tree.
Variable map:
forall (A B: Type), (elt -> A -> B) -> t A -> t B.
Hypothesis gmap:
forall (A B: Type) (f: elt -> A -> B) (i: elt) (m: t A),
get i (map f m) = option_map (f i) (get i m).

Same as map, but the function does not receive the elt argument.
Variable map1:
forall (A B: Type), (A -> B) -> t A -> t B.
Hypothesis gmap1:
forall (A B: Type) (f: A -> B) (i: elt) (m: t A),
get i (map1 f m) = option_map f (get i m).

Applying a function pairwise to all data of two trees.
Variable combine:
forall (A B C: Type), (option A -> option B -> option C) -> t A -> t B -> t C.
Hypothesis gcombine:
forall (A B C: Type) (f: option A -> option B -> option C),
f None None = None ->
forall (m1: t A) (m2: t B) (i: elt),
get i (combine f m1 m2) = f (get i m1) (get i m2).
Hypothesis combine_commut:
forall (A B: Type) (f g: option A -> option A -> option B),
(forall (i j: option A), f i j = g j i) ->
forall (m1 m2: t A),
combine f m1 m2 = combine g m2 m1.

Enumerating the bindings of a tree.
Variable elements:
forall (A: Type), t A -> list (elt * A).
Hypothesis elements_correct:
forall (A: Type) (m: t A) (i: elt) (v: A),
get i m = Some v -> In (i, v) (elements m).
Hypothesis elements_complete:
forall (A: Type) (m: t A) (i: elt) (v: A),
In (i, v) (elements m) -> get i m = Some v.
Hypothesis elements_keys_norepet:
forall (A: Type) (m: t A),
list_norepet (List.map (@fst elt A) (elements m)).

Folding a function over all bindings of a tree.
Variable fold:
forall (A B: Type), (B -> elt -> A -> B) -> t A -> B -> B.
Hypothesis fold_spec:
forall (A B: Type) (f: B -> elt -> A -> B) (v: B) (m: t A),
fold f m v =
List.fold_left (fun a p => f a (fst p) (snd p)) (elements m) v.
Same as fold, but the function does not receive the elt argument.
Variable fold1:
forall (A B: Type), (B -> A -> B) -> t A -> B -> B.
Hypothesis fold1_spec:
forall (A B: Type) (f: B -> A -> B) (v: B) (m: t A),
fold1 f m v =
List.fold_left (fun a p => f a (snd p)) (elements m) v.
End TREE.

# The abstract signatures of maps

Module Type MAP.
Variable elt: Type.
Variable elt_eq: forall (a b: elt), {a = b} + {a <> b}.
Variable t: Type -> Type.
Variable init: forall (A: Type), A -> t A.
Variable get: forall (A: Type), elt -> t A -> A.
Variable set: forall (A: Type), elt -> A -> t A -> t A.
Hypothesis gi:
forall (A: Type) (i: elt) (x: A), get i (init x) = x.
Hypothesis gss:
forall (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = x.
Hypothesis gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Hypothesis gsspec:
forall (A: Type) (i j: elt) (x: A) (m: t A),
get i (set j x m) = if elt_eq i j then x else get i m.
Hypothesis gsident:
forall (A: Type) (i j: elt) (m: t A), get j (set i (get i m) m) = get j m.
Variable map: forall (A B: Type), (A -> B) -> t A -> t B.
Hypothesis gmap:
forall (A B: Type) (f: A -> B) (i: elt) (m: t A),
get i (map f m) = f(get i m).
End MAP.

# An implementation of trees over type positive

Module PTree <: TREE.
Definition elt := positive.
Definition elt_eq := peq.

Inductive tree (A : Type) : Type :=
| Leaf : tree A
| Node : tree A -> option A -> tree A -> tree A.

Implicit Arguments Leaf [A].
Implicit Arguments Node [A].
Scheme tree_ind := Induction for tree Sort Prop.

Definition t := tree.

Definition empty (A : Type) := (Leaf : t A).

Fixpoint get (A : Type) (i : positive) (m : t A) {struct i} : option A :=
match m with
| Leaf => None
| Node l o r =>
match i with
| xH => o
| xO ii => get ii l
| xI ii => get ii r
end
end.

Fixpoint set (A : Type) (i : positive) (v : A) (m : t A) {struct i} : t A :=
match m with
| Leaf =>
match i with
| xH => Node Leaf (Some v) Leaf
| xO ii => Node (set ii v Leaf) None Leaf
| xI ii => Node Leaf None (set ii v Leaf)
end
| Node l o r =>
match i with
| xH => Node l (Some v) r
| xO ii => Node (set ii v l) o r
| xI ii => Node l o (set ii v r)
end
end.

Fixpoint remove (A : Type) (i : positive) (m : t A) {struct i} : t A :=
match i with
| xH =>
match m with
| Leaf => Leaf
| Node Leaf o Leaf => Leaf
| Node l o r => Node l None r
end
| xO ii =>
match m with
| Leaf => Leaf
| Node l None Leaf =>
match remove ii l with
| Leaf => Leaf
| mm => Node mm None Leaf
end
| Node l o r => Node (remove ii l) o r
end
| xI ii =>
match m with
| Leaf => Leaf
| Node Leaf None r =>
match remove ii r with
| Leaf => Leaf
| mm => Node Leaf None mm
end
| Node l o r => Node l o (remove ii r)
end
end.

Theorem gempty:
forall (A: Type) (i: positive), get i (empty A) = None.
Proof.
induction i; simpl; auto.
Qed.

Theorem gss:
forall (A: Type) (i: positive) (x: A) (m: t A), get i (set i x m) = Some x.
Proof.
induction i; destruct m; simpl; auto.
Qed.

Lemma gleaf : forall (A : Type) (i : positive), get i (Leaf : t A) = None.
Proof.
exact gempty. Qed.

Theorem gso:
forall (A: Type) (i j: positive) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Proof.
induction i; intros; destruct j; destruct m; simpl;
try rewrite <- (gleaf A i); auto; try apply IHi; congruence.
Qed.

Theorem gsspec:
forall (A: Type) (i j: positive) (x: A) (m: t A),
get i (set j x m) = if peq i j then Some x else get i m.
Proof.
intros.
destruct (peq i j); [ rewrite e; apply gss | apply gso; auto ].
Qed.

Theorem gsident:
forall (A: Type) (i: positive) (m: t A) (v: A),
get i m = Some v -> set i v m = m.
Proof.
induction i; intros; destruct m; simpl; simpl in H; try congruence.
rewrite (IHi m2 v H); congruence.
rewrite (IHi m1 v H); congruence.
Qed.

Theorem set2:
forall (A: Type) (i: elt) (m: t A) (v1 v2: A),
set i v2 (set i v1 m) = set i v2 m.
Proof.
induction i; intros; destruct m; simpl; try (rewrite IHi); auto.
Qed.

Lemma rleaf : forall (A : Type) (i : positive), remove i (Leaf : t A) = Leaf.
Proof.
destruct i; simpl; auto. Qed.

Theorem grs:
forall (A: Type) (i: positive) (m: t A), get i (remove i m) = None.
Proof.
induction i; destruct m.
simpl; auto.
destruct m1; destruct o; destruct m2 as [ | ll oo rr]; simpl; auto.
rewrite (rleaf A i); auto.
cut (get i (remove i (Node ll oo rr)) = None).
destruct (remove i (Node ll oo rr)); auto; apply IHi.
apply IHi.
simpl; auto.
destruct m1 as [ | ll oo rr]; destruct o; destruct m2; simpl; auto.
rewrite (rleaf A i); auto.
cut (get i (remove i (Node ll oo rr)) = None).
destruct (remove i (Node ll oo rr)); auto; apply IHi.
apply IHi.
simpl; auto.
destruct m1; destruct m2; simpl; auto.
Qed.

Theorem gro:
forall (A: Type) (i j: positive) (m: t A),
i <> j -> get i (remove j m) = get i m.
Proof.
induction i; intros; destruct j; destruct m;
try rewrite (rleaf A (xI j));
try rewrite (rleaf A (xO j));
try rewrite (rleaf A 1); auto;
destruct m1; destruct o; destruct m2;
simpl;
try apply IHi; try congruence;
try rewrite (rleaf A j); auto;
try rewrite (gleaf A i); auto.
cut (get i (remove j (Node m2_1 o m2_2)) = get i (Node m2_1 o m2_2));
[ destruct (remove j (Node m2_1 o m2_2)); try rewrite (gleaf A i); auto
| apply IHi; congruence ].
destruct (remove j (Node m1_1 o0 m1_2)); simpl; try rewrite (gleaf A i);
auto.
destruct (remove j (Node m2_1 o m2_2)); simpl; try rewrite (gleaf A i);
auto.
cut (get i (remove j (Node m1_1 o0 m1_2)) = get i (Node m1_1 o0 m1_2));
[ destruct (remove j (Node m1_1 o0 m1_2)); try rewrite (gleaf A i); auto
| apply IHi; congruence ].
destruct (remove j (Node m2_1 o m2_2)); simpl; try rewrite (gleaf A i);
auto.
destruct (remove j (Node m1_1 o0 m1_2)); simpl; try rewrite (gleaf A i);
auto.
Qed.

Theorem grspec:
forall (A: Type) (i j: elt) (m: t A),
get i (remove j m) = if elt_eq i j then None else get i m.
Proof.
intros. destruct (elt_eq i j). subst j. apply grs. apply gro; auto.
Qed.

Section BOOLEAN_EQUALITY.

Variable A: Type.
Variable beqA: A -> A -> bool.

Fixpoint bempty (m: t A) : bool :=
match m with
| Leaf => true
| Node l None r => bempty l && bempty r
| Node l (Some _) r => false
end.

Fixpoint beq (m1 m2: t A) {struct m1} : bool :=
match m1, m2 with
| Leaf, _ => bempty m2
| _, Leaf => bempty m1
| Node l1 o1 r1, Node l2 o2 r2 =>
match o1, o2 with
| None, None => true
| Some y1, Some y2 => beqA y1 y2
| _, _ => false
end
&& beq l1 l2 && beq r1 r2
end.

Lemma bempty_correct:
forall m, bempty m = true <-> (forall x, get x m = None).
Proof.
induction m; simpl.
split; intros. apply gleaf. auto.
destruct o; split; intros.
congruence.
generalize (H xH); simpl; congruence.
destruct (andb_prop _ _ H). rewrite IHm1 in H0. rewrite IHm2 in H1.
destruct x; simpl; auto.
apply andb_true_intro; split.
apply IHm1. intros; apply (H (xO x)).
apply IHm2. intros; apply (H (xI x)).
Qed.

Lemma beq_correct:
forall m1 m2,
beq m1 m2 = true <->
(forall (x: elt),
match get x m1, get x m2 with
| None, None => True
| Some y1, Some y2 => beqA y1 y2 = true
| _, _ => False
end).
Proof.
induction m1; intros.
- simpl. rewrite bempty_correct. split; intros.
rewrite gleaf. rewrite H. auto.
generalize (H x). rewrite gleaf. destruct (get x m2); tauto.
- destruct m2.
+ unfold beq. rewrite bempty_correct. split; intros.
rewrite H. rewrite gleaf. auto.
generalize (H x). rewrite gleaf. destruct (get x (Node m1_1 o m1_2)); tauto.
+ simpl. split; intros.
* destruct (andb_prop _ _ H). destruct (andb_prop _ _ H0).
rewrite IHm1_1 in H3. rewrite IHm1_2 in H1.
destruct x; simpl. apply H1. apply H3.
destruct o; destruct o0; auto || congruence.
* apply andb_true_intro. split. apply andb_true_intro. split.
generalize (H xH); simpl. destruct o; destruct o0; tauto.
apply IHm1_1. intros; apply (H (xO x)).
apply IHm1_2. intros; apply (H (xI x)).
Qed.

End BOOLEAN_EQUALITY.

Fixpoint prev_append (i j: positive) {struct i} : positive :=
match i with
| xH => j
| xI i' => prev_append i' (xI j)
| xO i' => prev_append i' (xO j)
end.

Definition prev (i: positive) : positive :=
prev_append i xH.

Lemma prev_append_prev i j:
prev (prev_append i j) = prev_append j i.
Proof.
revert j. unfold prev.
induction i as [i IH|i IH|]. 3: reflexivity.
intros j. simpl. rewrite IH. reflexivity.
intros j. simpl. rewrite IH. reflexivity.
Qed.

Lemma prev_involutive i :
prev (prev i) = i.
Proof (prev_append_prev i xH).

Lemma prev_append_inj i j j' :
prev_append i j = prev_append i j' -> j = j'.
Proof.
revert j j'.
induction i as [i Hi|i Hi|]; intros j j' H; auto;
specialize (Hi _ _ H); congruence.
Qed.

Fixpoint xmap (A B : Type) (f : positive -> A -> B) (m : t A) (i : positive)
{struct m} : t B :=
match m with
| Leaf => Leaf
| Node l o r => Node (xmap f l (xO i))
(match o with None => None | Some x => Some (f (prev i) x) end)
(xmap f r (xI i))
end.

Definition map (A B : Type) (f : positive -> A -> B) m := xmap f m xH.

Lemma xgmap:
forall (A B: Type) (f: positive -> A -> B) (i j : positive) (m: t A),
get i (xmap f m j) = option_map (f (prev (prev_append i j))) (get i m).
Proof.
induction i; intros; destruct m; simpl; auto.
Qed.

Theorem gmap:
forall (A B: Type) (f: positive -> A -> B) (i: positive) (m: t A),
get i (map f m) = option_map (f i) (get i m).
Proof.
intros A B f i m.
unfold map.
rewrite xgmap. repeat f_equal. exact (prev_involutive i).
Qed.

Fixpoint map1 (A B: Type) (f: A -> B) (m: t A) {struct m} : t B :=
match m with
| Leaf => Leaf
| Node l o r => Node (map1 f l) (option_map f o) (map1 f r)
end.

Theorem gmap1:
forall (A B: Type) (f: A -> B) (i: elt) (m: t A),
get i (map1 f m) = option_map f (get i m).
Proof.
induction i; intros; destruct m; simpl; auto.
Qed.

Definition Node' (A: Type) (l: t A) (x: option A) (r: t A): t A :=
match l, x, r with
| Leaf, None, Leaf => Leaf
| _, _, _ => Node l x r
end.

Lemma gnode':
forall (A: Type) (l r: t A) (x: option A) (i: positive),
get i (Node' l x r) = get i (Node l x r).
Proof.
intros. unfold Node'.
destruct l; destruct x; destruct r; auto.
destruct i; simpl; auto; rewrite gleaf; auto.
Qed.

Fixpoint filter1 (A: Type) (pred: A -> bool) (m: t A) {struct m} : t A :=
match m with
| Leaf => Leaf
| Node l o r =>
let o' := match o with None => None | Some x => if pred x then o else None end in
Node' (filter1 pred l) o' (filter1 pred r)
end.

Theorem gfilter1:
forall (A: Type) (pred: A -> bool) (i: elt) (m: t A),
get i (filter1 pred m) =
match get i m with None => None | Some x => if pred x then Some x else None end.
Proof.
intros until m. revert m i. induction m; simpl; intros.
rewrite gleaf; auto.
rewrite gnode'. destruct i; simpl; auto. destruct o; auto.
Qed.

Section COMBINE.

Variables A B C: Type.
Variable f: option A -> option B -> option C.
Hypothesis f_none_none: f None None = None.

Fixpoint xcombine_l (m : t A) {struct m} : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node' (xcombine_l l) (f o None) (xcombine_l r)
end.

Lemma xgcombine_l :
forall (m: t A) (i : positive),
get i (xcombine_l m) = f (get i m) None.
Proof.
induction m; intros; simpl.
repeat rewrite gleaf. auto.
rewrite gnode'. destruct i; simpl; auto.
Qed.

Fixpoint xcombine_r (m : t B) {struct m} : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node' (xcombine_r l) (f None o) (xcombine_r r)
end.

Lemma xgcombine_r :
forall (m: t B) (i : positive),
get i (xcombine_r m) = f None (get i m).
Proof.
induction m; intros; simpl.
repeat rewrite gleaf. auto.
rewrite gnode'. destruct i; simpl; auto.
Qed.

Fixpoint combine (m1: t A) (m2: t B) {struct m1} : t C :=
match m1 with
| Leaf => xcombine_r m2
| Node l1 o1 r1 =>
match m2 with
| Leaf => xcombine_l m1
| Node l2 o2 r2 => Node' (combine l1 l2) (f o1 o2) (combine r1 r2)
end
end.

Theorem gcombine:
forall (m1: t A) (m2: t B) (i: positive),
get i (combine m1 m2) = f (get i m1) (get i m2).
Proof.
induction m1; intros; simpl.
rewrite gleaf. apply xgcombine_r.
destruct m2; simpl.
rewrite gleaf. rewrite <- xgcombine_l. auto.
repeat rewrite gnode'. destruct i; simpl; auto.
Qed.

End COMBINE.

Lemma xcombine_lr :
forall (A B: Type) (f g : option A -> option A -> option B) (m : t A),
(forall (i j : option A), f i j = g j i) ->
xcombine_l f m = xcombine_r g m.
Proof.
induction m; intros; simpl; auto.
rewrite IHm1; auto.
rewrite IHm2; auto.
rewrite H; auto.
Qed.

Theorem combine_commut:
forall (A B: Type) (f g: option A -> option A -> option B),
(forall (i j: option A), f i j = g j i) ->
forall (m1 m2: t A),
combine f m1 m2 = combine g m2 m1.
Proof.
intros A B f g EQ1.
assert (EQ2: forall (i j: option A), g i j = f j i).
intros; auto.
induction m1; intros; destruct m2; simpl;
try rewrite EQ1;
repeat rewrite (xcombine_lr f g);
repeat rewrite (xcombine_lr g f);
auto.
rewrite IHm1_1.
rewrite IHm1_2.
auto.
Qed.

Fixpoint xelements (A : Type) (m : t A) (i : positive)
(k: list (positive * A)) {struct m}
: list (positive * A) :=
match m with
| Leaf => k
| Node l None r =>
xelements l (xO i) (xelements r (xI i) k)
| Node l (Some x) r =>
xelements l (xO i)
((prev i, x) :: xelements r (xI i) k)
end.

Definition elements (A: Type) (m : t A) := xelements m xH nil.

Lemma xelements_incl:
forall (A: Type) (m: t A) (i : positive) k x,
In x k -> In x (xelements m i k).
Proof.
induction m; intros; simpl.
auto.
destruct o.
apply IHm1. simpl; right; auto.
auto.
Qed.

Lemma xelements_correct:
forall (A: Type) (m: t A) (i j : positive) (v: A) k,
get i m = Some v -> In (prev (prev_append i j), v) (xelements m j k).
Proof.
induction m; intros.
rewrite (gleaf A i) in H; congruence.
destruct o; destruct i; simpl; simpl in H.
apply xelements_incl. right. auto.
auto.
inv H. apply xelements_incl. left. reflexivity.
apply xelements_incl. auto.
auto.
inv H.
Qed.

Theorem elements_correct:
forall (A: Type) (m: t A) (i: positive) (v: A),
get i m = Some v -> In (i, v) (elements m).
Proof.
intros A m i v H.
generalize (xelements_correct m i xH nil H). rewrite prev_append_prev. exact id.
Qed.

Lemma in_xelements:
forall (A: Type) (m: t A) (i k: positive) (v: A) l,
In (k, v) (xelements m i l) ->
(exists j, k = prev (prev_append j i) /\ get j m = Some v) \/ In (k, v) l.
Proof.
induction m as [|l IHl o r IHr]; intros i j v k H.
right; exact H.
destruct o as [o|].
specialize (IHl _ _ _ _ H). destruct IHl as [(j' & -> & Hj')|[IHl|IHl]].
left. exists (xO j'). split. reflexivity. exact Hj'.
inv IHl. left. exists xH. split. reflexivity. reflexivity.
destruct (IHr _ _ _ _ IHl) as [(j' & -> & Hj') | IH]; clear IHr.
left. exists (xI j'). split. reflexivity. exact Hj'.
right; assumption.
specialize (IHl _ _ _ _ H). destruct IHl as [(j' & -> & Hj' )|IHl].
left. exists (xO j'). split. reflexivity. exact Hj'.
destruct (IHr _ _ _ _ IHl) as [(j' & -> & Hj') | IH]; clear IHr.
left. exists (xI j'). split. reflexivity. exact Hj'.
right; assumption.
Qed.

Lemma xelements_complete:
forall (A: Type) (m: t A) (i j : positive) (v: A) k,
In (prev_append j i, v) (xelements m j k) -> get i m = Some v \/ In (prev_append j i, v) k.
Proof.
intros A m i j v k H.
apply in_xelements in H. destruct H as [(j' & EQ & Hj')|H].
rewrite prev_append_prev in EQ. apply prev_append_inj in EQ. left; congruence.
right; assumption.
Qed.

Theorem elements_complete:
forall (A: Type) (m: t A) (i: positive) (v: A),
In (i, v) (elements m) -> get i m = Some v.
Proof.
unfold elements. intros A m i v H.
change i with (prev_append 1 i) in H.
destruct (xelements_complete _ _ _ _ _ H) as [K|()]. exact K.
Qed.

Definition xkeys (A: Type) (m: t A) (i: positive) (l: list (positive * A)) :=
List.map (@fst positive A) (xelements m i l).

Lemma in_xkeys:
forall (A: Type) (m: t A) (i k: positive) l,
In k (xkeys m i l) ->
(exists j, k = prev (prev_append j i)) \/ In k (List.map fst l).
Proof.
unfold xkeys; intros.
apply (list_in_map_inv) in H. destruct H as ((j, v) & -> & H).
change j with (prev_append 1 j) in H. apply in_xelements in H.
destruct H as [(j' & Hj' & _) | H]. simpl in Hj'; subst j.
left. simpl. exists j'. reflexivity.
right. apply in_map. exact H.
Qed.

Lemma xelements_keys_norepet:
forall (A: Type) (m: t A) (i: positive) l,
(forall k v, get k m = Some v -> ~In (prev (prev_append k i)) (List.map fst l)) ->
list_norepet (List.map fst l) ->
list_norepet (xkeys m i l).
Proof.
unfold xkeys.
intros A m.
induction m as [|l Hl o r Hr]; intros i elem H NR.
auto.
destruct o as [o|].
- apply Hl.
intros k v Hkv IN.
destruct IN as [IN | IN].
rewrite prev_append_prev in IN. simpl in IN. apply prev_append_inj in IN. discriminate.
apply in_xkeys in IN. destruct IN as [(j & EQ) | IN].
rewrite !prev_append_prev in EQ. simpl in EQ. apply prev_append_inj in EQ. discriminate.
elim (H (xO k) v); auto; fail.
simpl. constructor.
intros IN.
apply in_xkeys in IN. destruct IN as [(j & EQ) | IN].
rewrite !prev_append_prev in EQ. simpl in EQ. apply prev_append_inj in EQ. discriminate.
elim (H xH o); auto; fail.
apply Hr; auto.
intros k.
exact (H (xI k)).
- apply Hl.
intros k v Hkv IN.
apply in_xkeys in IN. destruct IN as [(j & EQ) | IN].
rewrite !prev_append_prev in EQ. simpl in EQ. apply prev_append_inj in EQ. discriminate.
exact (H (xO k) _ Hkv IN).
apply Hr; auto.
intros k.
exact (H (xI k)).
Qed.

Theorem elements_keys_norepet:
forall (A: Type) (m: t A),
list_norepet (List.map (@fst elt A) (elements m)).
Proof.
intros. change (list_norepet (xkeys m 1 nil)). apply xelements_keys_norepet.
intros; red; intros. elim H0. constructor.
Qed.

Remark xelements_empty:
forall (A: Type) (m: t A) i l, (forall i, get i m = None) -> xelements m i l = l.
Proof.
induction m; simpl; intros.
auto.
destruct o. generalize (H xH); simpl; congruence.
rewrite IHm1. apply IHm2.
intros. apply (H (xI i0)).
intros. apply (H (xO i0)).
Qed.

Theorem elements_canonical_order:
forall (A B: Type) (R: A -> B -> Prop) (m: t A) (n: t B),
(forall i x, get i m = Some x -> exists y, get i n = Some y /\ R x y) ->
(forall i y, get i n = Some y -> exists x, get i m = Some x /\ R x y) ->
list_forall2
(fun i_x i_y => fst i_x = fst i_y /\ R (snd i_x) (snd i_y))
(elements m) (elements n).
Proof.
intros until R.
assert (forall m n j l1 l2,
(forall i x, get i m = Some x -> exists y, get i n = Some y /\ R x y) ->
(forall i y, get i n = Some y -> exists x, get i m = Some x /\ R x y) ->
list_forall2
(fun i_x i_y => fst i_x = fst i_y /\ R (snd i_x) (snd i_y))
l1 l2 ->
list_forall2
(fun i_x i_y => fst i_x = fst i_y /\ R (snd i_x) (snd i_y))
(xelements m j l1) (xelements n j l2)).
{
induction m; simpl; intros.
rewrite xelements_empty. auto.
intros. destruct (get i n) eqn:E; auto. exploit H0; eauto.
intros [x [P Q]]. rewrite gleaf in P; congruence.
destruct o.
destruct n. exploit (H xH a); auto. simpl. intros [y [P Q]]; congruence.
exploit (H xH a); auto. intros [y [P Q]]. simpl in P. subst o.
simpl. apply IHm1.
intros i x. exact (H (xO i) x).
intros i x. exact (H0 (xO i) x).
constructor. simpl; auto.
apply IHm2.
intros i x. exact (H (xI i) x).
intros i x. exact (H0 (xI i) x).
auto.
destruct n. simpl.
rewrite ! xelements_empty. auto.
intros. destruct (get i m2) eqn:E; auto. exploit (H (xI i)); eauto.
rewrite gleaf. intros [y [P Q]]; congruence.
intros. destruct (get i m1) eqn:E; auto. exploit (H (xO i)); eauto.
rewrite gleaf. intros [y [P Q]]; congruence.
destruct o.
exploit (H0 xH); simpl; eauto. intros [y [P Q]]; congruence.
simpl. apply IHm1.
intros i x. exact (H (xO i) x).
intros i x. exact (H0 (xO i) x).
apply IHm2.
intros i x. exact (H (xI i) x).
intros i x. exact (H0 (xI i) x).
auto.
}
intros. apply H. auto. auto. constructor.
Qed.

Theorem elements_extensional:
forall (A: Type) (m n: t A),
(forall i, get i m = get i n) ->
elements m = elements n.
Proof.
intros.
exploit (elements_canonical_order (fun (x y: A) => x = y) m n).
intros. rewrite H in H0. exists x; auto.
intros. rewrite <- H in H0. exists y; auto.
induction 1. auto. destruct a1 as [a2 a3]; destruct b1 as [b2 b3]; simpl in *.
destruct H0. congruence.
Qed.

Fixpoint xfold (A B: Type) (f: B -> positive -> A -> B)
(i: positive) (m: t A) (v: B) {struct m} : B :=
match m with
| Leaf => v
| Node l None r =>
let v1 := xfold f (xO i) l v in
xfold f (xI i) r v1
| Node l (Some x) r =>
let v1 := xfold f (xO i) l v in
let v2 := f v1 (prev i) x in
xfold f (xI i) r v2
end.

Definition fold (A B : Type) (f: B -> positive -> A -> B) (m: t A) (v: B) :=
xfold f xH m v.

Lemma xfold_xelements:
forall (A B: Type) (f: B -> positive -> A -> B) m i v l,
List.fold_left (fun a p => f a (fst p) (snd p)) l (xfold f i m v) =
List.fold_left (fun a p => f a (fst p) (snd p)) (xelements m i l) v.
Proof.
induction m; intros.
simpl. auto.
destruct o; simpl.
rewrite <- IHm1. simpl. rewrite <- IHm2. auto.
rewrite <- IHm1. rewrite <- IHm2. auto.
Qed.

Theorem fold_spec:
forall (A B: Type) (f: B -> positive -> A -> B) (v: B) (m: t A),
fold f m v =
List.fold_left (fun a p => f a (fst p) (snd p)) (elements m) v.
Proof.
intros. unfold fold, elements. rewrite <- xfold_xelements. auto.
Qed.

Fixpoint fold1 (A B: Type) (f: B -> A -> B) (m: t A) (v: B) {struct m} : B :=
match m with
| Leaf => v
| Node l None r =>
let v1 := fold1 f l v in
fold1 f r v1
| Node l (Some x) r =>
let v1 := fold1 f l v in
let v2 := f v1 x in
fold1 f r v2
end.

Lemma fold1_xelements:
forall (A B: Type) (f: B -> A -> B) m i v l,
List.fold_left (fun a p => f a (snd p)) l (fold1 f m v) =
List.fold_left (fun a p => f a (snd p)) (xelements m i l) v.
Proof.
induction m; intros.
simpl. auto.
destruct o; simpl.
rewrite <- IHm1. simpl. rewrite <- IHm2. auto.
rewrite <- IHm1. rewrite <- IHm2. auto.
Qed.

Theorem fold1_spec:
forall (A B: Type) (f: B -> A -> B) (v: B) (m: t A),
fold1 f m v =
List.fold_left (fun a p => f a (snd p)) (elements m) v.
Proof.
intros. apply fold1_xelements with (l := @nil (positive * A)).
Qed.

End PTree.

# An implementation of maps over type positive

Module PMap <: MAP.
Definition elt := positive.
Definition elt_eq := peq.

Definition t (A : Type) : Type := (A * PTree.t A)%type.

Definition init (A : Type) (x : A) :=
(x, PTree.empty A).

Definition get (A : Type) (i : positive) (m : t A) :=
match PTree.get i (snd m) with
| Some x => x
| None => fst m
end.

Definition set (A : Type) (i : positive) (x : A) (m : t A) :=
(fst m, PTree.set i x (snd m)).

Theorem gi:
forall (A: Type) (i: positive) (x: A), get i (init x) = x.
Proof.
intros. unfold init. unfold get. simpl. rewrite PTree.gempty. auto.
Qed.

Theorem gss:
forall (A: Type) (i: positive) (x: A) (m: t A), get i (set i x m) = x.
Proof.
intros. unfold get. unfold set. simpl. rewrite PTree.gss. auto.
Qed.

Theorem gso:
forall (A: Type) (i j: positive) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Proof.
intros. unfold get. unfold set. simpl. rewrite PTree.gso; auto.
Qed.

Theorem gsspec:
forall (A: Type) (i j: positive) (x: A) (m: t A),
get i (set j x m) = if peq i j then x else get i m.
Proof.
intros. destruct (peq i j).
rewrite e. apply gss. auto.
apply gso. auto.
Qed.

Theorem gsident:
forall (A: Type) (i j: positive) (m: t A),
get j (set i (get i m) m) = get j m.
Proof.
intros. destruct (peq i j).
rewrite e. rewrite gss. auto.
rewrite gso; auto.
Qed.

Definition map (A B : Type) (f : A -> B) (m : t A) : t B :=
(f (fst m), PTree.map1 f (snd m)).

Theorem gmap:
forall (A B: Type) (f: A -> B) (i: positive) (m: t A),
get i (map f m) = f(get i m).
Proof.
intros. unfold map. unfold get. simpl. rewrite PTree.gmap1.
unfold option_map. destruct (PTree.get i (snd m)); auto.
Qed.

Theorem set2:
forall (A: Type) (i: elt) (x y: A) (m: t A),
set i y (set i x m) = set i y m.
Proof.
intros. unfold set. simpl. decEq. apply PTree.set2.
Qed.

End PMap.

# An implementation of maps over any type that injects into type positive

Module Type INDEXED_TYPE.
Variable t: Type.
Variable index: t -> positive.
Hypothesis index_inj: forall (x y: t), index x = index y -> x = y.
Variable eq: forall (x y: t), {x = y} + {x <> y}.
End INDEXED_TYPE.

Module IMap(X: INDEXED_TYPE).

Definition elt := X.t.
Definition elt_eq := X.eq.
Definition t : Type -> Type := PMap.t.
Definition init (A: Type) (x: A) := PMap.init x.
Definition get (A: Type) (i: X.t) (m: t A) := PMap.get (X.index i) m.
Definition set (A: Type) (i: X.t) (v: A) (m: t A) := PMap.set (X.index i) v m.
Definition map (A B: Type) (f: A -> B) (m: t A) : t B := PMap.map f m.

Lemma gi:
forall (A: Type) (x: A) (i: X.t), get i (init x) = x.
Proof.
intros. unfold get, init. apply PMap.gi.
Qed.

Lemma gss:
forall (A: Type) (i: X.t) (x: A) (m: t A), get i (set i x m) = x.
Proof.
intros. unfold get, set. apply PMap.gss.
Qed.

Lemma gso:
forall (A: Type) (i j: X.t) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Proof.
intros. unfold get, set. apply PMap.gso.
red. intro. apply H. apply X.index_inj; auto.
Qed.

Lemma gsspec:
forall (A: Type) (i j: X.t) (x: A) (m: t A),
get i (set j x m) = if X.eq i j then x else get i m.
Proof.
intros. unfold get, set.
rewrite PMap.gsspec.
case (X.eq i j); intro.
subst j. rewrite peq_true. reflexivity.
rewrite peq_false. reflexivity.
red; intro. elim n. apply X.index_inj; auto.
Qed.

Lemma gmap:
forall (A B: Type) (f: A -> B) (i: X.t) (m: t A),
get i (map f m) = f(get i m).
Proof.
intros. unfold map, get. apply PMap.gmap.
Qed.

Lemma set2:
forall (A: Type) (i: elt) (x y: A) (m: t A),
set i y (set i x m) = set i y m.
Proof.
intros. unfold set. apply PMap.set2.
Qed.

End IMap.

Module ZIndexed.
Definition t := Z.
Definition index (z: Z): positive :=
match z with
| Z0 => xH
| Zpos p => xO p
| Zneg p => xI p
end.
Lemma index_inj: forall (x y: Z), index x = index y -> x = y.
Proof.
unfold index; destruct x; destruct y; intros;
try discriminate; try reflexivity.
congruence.
congruence.
Qed.
Definition eq := zeq.
End ZIndexed.

Module ZMap := IMap(ZIndexed).

Module NIndexed.
Definition t := N.
Definition index (n: N): positive :=
match n with
| N0 => xH
| Npos p => xO p
end.
Lemma index_inj: forall (x y: N), index x = index y -> x = y.
Proof.
unfold index; destruct x; destruct y; intros;
try discriminate; try reflexivity.
congruence.
Qed.
Lemma eq: forall (x y: N), {x = y} + {x <> y}.
Proof.
decide equality. apply peq.
Qed.
End NIndexed.

Module NMap := IMap(NIndexed).

# An implementation of maps over any type with decidable equality

Module Type EQUALITY_TYPE.
Variable t: Type.
Variable eq: forall (x y: t), {x = y} + {x <> y}.
End EQUALITY_TYPE.

Module EMap(X: EQUALITY_TYPE) <: MAP.

Definition elt := X.t.
Definition elt_eq := X.eq.
Definition t (A: Type) := X.t -> A.
Definition init (A: Type) (v: A) := fun (_: X.t) => v.
Definition get (A: Type) (x: X.t) (m: t A) := m x.
Definition set (A: Type) (x: X.t) (v: A) (m: t A) :=
fun (y: X.t) => if X.eq y x then v else m y.
Lemma gi:
forall (A: Type) (i: elt) (x: A), init x i = x.
Proof.
intros. reflexivity.
Qed.
Lemma gss:
forall (A: Type) (i: elt) (x: A) (m: t A), (set i x m) i = x.
Proof.
intros. unfold set. case (X.eq i i); intro.
reflexivity. tauto.
Qed.
Lemma gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> (set j x m) i = m i.
Proof.
intros. unfold set. case (X.eq i j); intro.
congruence. reflexivity.
Qed.
Lemma gsspec:
forall (A: Type) (i j: elt) (x: A) (m: t A),
get i (set j x m) = if elt_eq i j then x else get i m.
Proof.
intros. unfold get, set, elt_eq. reflexivity.
Qed.
Lemma gsident:
forall (A: Type) (i j: elt) (m: t A), get j (set i (get i m) m) = get j m.
Proof.
intros. unfold get, set. case (X.eq j i); intro.
congruence. reflexivity.
Qed.
Definition map (A B: Type) (f: A -> B) (m: t A) :=
fun (x: X.t) => f(m x).
Lemma gmap:
forall (A B: Type) (f: A -> B) (i: elt) (m: t A),
get i (map f m) = f(get i m).
Proof.
intros. unfold get, map. reflexivity.
Qed.
End EMap.

# Additional properties over trees

Module Tree_Properties(T: TREE).

An induction principle over fold.

Section TREE_FOLD_IND.

Variables V A: Type.
Variable f: A -> T.elt -> V -> A.
Variable P: T.t V -> A -> Prop.
Variable init: A.
Variable m_final: T.t V.

Hypothesis P_compat:
forall m m' a,
(forall x, T.get x m = T.get x m') ->
P m a -> P m' a.

Hypothesis H_base:
P (T.empty _) init.

Hypothesis H_rec:
forall m a k v,
T.get k m = None -> T.get k m_final = Some v -> P m a -> P (T.set k v m) (f a k v).

Let f' (a: A) (p : T.elt * V) := f a (fst p) (snd p).

Let P' (l: list (T.elt * V)) (a: A) : Prop :=
forall m, list_equiv l (T.elements m) -> P m a.

Remark H_base':
P' nil init.
Proof.
red; intros. apply P_compat with (T.empty _); auto.
intros. rewrite T.gempty. symmetry. case_eq (T.get x m); intros; auto.
assert (In (x, v) nil). rewrite (H (x, v)). apply T.elements_correct. auto.
contradiction.
Qed.

Remark H_rec':
forall k v l a,
~In k (List.map (@fst T.elt V) l) ->
In (k, v) (T.elements m_final) ->
P' l a ->
P' (l ++ (k, v) :: nil) (f a k v).
Proof.
unfold P'; intros.
set (m0 := T.remove k m).
apply P_compat with (T.set k v m0).
intros. unfold m0. rewrite T.gsspec. destruct (T.elt_eq x k).
symmetry. apply T.elements_complete. rewrite <- (H2 (x, v)).
apply in_or_app. simpl. intuition congruence.
apply T.gro. auto.
apply H_rec. unfold m0. apply T.grs. apply T.elements_complete. auto.
apply H1. red. intros [k' v'].
split; intros.
apply T.elements_correct. unfold m0. rewrite T.gro. apply T.elements_complete.
rewrite <- (H2 (k', v')). apply in_or_app. auto.
red; intro; subst k'. elim H. change k with (fst (k, v')). apply in_map. auto.
assert (T.get k' m0 = Some v'). apply T.elements_complete. auto.
unfold m0 in H4. rewrite T.grspec in H4. destruct (T.elt_eq k' k). congruence.
assert (In (k', v') (T.elements m)). apply T.elements_correct; auto.
rewrite <- (H2 (k', v')) in H5. destruct (in_app_or _ _ _ H5). auto.
simpl in H6. intuition congruence.
Qed.

Lemma fold_rec_aux:
forall l1 l2 a,
list_equiv (l2 ++ l1) (T.elements m_final) ->
list_disjoint (List.map (@fst T.elt V) l1) (List.map (@fst T.elt V) l2) ->
list_norepet (List.map (@fst T.elt V) l1) ->
P' l2 a -> P' (l2 ++ l1) (List.fold_left f' l1 a).
Proof.
induction l1; intros; simpl.
rewrite <- List.app_nil_end. auto.
destruct a as [k v]; simpl in *. inv H1.
change ((k, v) :: l1) with (((k, v) :: nil) ++ l1). rewrite <- List.app_ass. apply IHl1.
rewrite app_ass. auto.
red; intros. rewrite map_app in H3. destruct (in_app_or _ _ _ H3). apply H0; auto with coqlib.
simpl in H4. intuition congruence.
auto.
unfold f'. simpl. apply H_rec'; auto. eapply list_disjoint_notin; eauto with coqlib.
rewrite <- (H (k, v)). apply in_or_app. simpl. auto.
Qed.

Theorem fold_rec:
P m_final (T.fold f m_final init).
Proof.
intros. rewrite T.fold_spec. fold f'.
assert (P' (nil ++ T.elements m_final) (List.fold_left f' (T.elements m_final) init)).
apply fold_rec_aux.
simpl. red; intros; tauto.
simpl. red; intros. elim H0.
apply T.elements_keys_norepet.
apply H_base'.
simpl in H. red in H. apply H. red; intros. tauto.
Qed.

End TREE_FOLD_IND.

A nonnegative measure over trees

Section MEASURE.

Variable V: Type.

Definition cardinal (x: T.t V) : nat := List.length (T.elements x).

Remark list_incl_length:
forall (A: Type) (l1: list A), list_norepet l1 ->
forall (l2: list A), List.incl l1 l2 -> (List.length l1 <= List.length l2)%nat.
Proof.
induction 1; simpl; intros.
omega.
exploit (List.in_split hd l2). auto with coqlib. intros [l3 [l4 EQ]]. subst l2.
assert (length tl <= length (l3 ++ l4))%nat.
apply IHlist_norepet. red; intros.
exploit (H1 a); auto with coqlib.
repeat rewrite in_app_iff. simpl. intuition. subst. contradiction.
repeat rewrite app_length in *. simpl. omega.
Qed.

Remark list_length_incl:
forall (A: Type) (l1: list A), list_norepet l1 ->
forall l2, List.incl l1 l2 -> List.length l1 = List.length l2 -> List.incl l2 l1.
Proof.
induction 1; simpl; intros.
destruct l2; simpl in *. auto with coqlib. discriminate.
exploit (List.in_split hd l2). auto with coqlib. intros [l3 [l4 EQ]]. subst l2.
assert (incl (l3 ++ l4) tl).
apply IHlist_norepet. red; intros.
exploit (H1 a); auto with coqlib.
repeat rewrite in_app_iff. simpl. intuition. subst. contradiction.
repeat rewrite app_length in *. simpl in H2. omega.
red; simpl; intros. rewrite in_app_iff in H4; simpl in H4. intuition.
Qed.

Remark list_strict_incl_length:
forall (A: Type) (l1 l2: list A) (x: A),
list_norepet l1 -> List.incl l1 l2 -> ~In x l1 -> In x l2 ->
(List.length l1 < List.length l2)%nat.
Proof.
intros. exploit list_incl_length; eauto. intros.
assert (length l1 = length l2 \/ length l1 < length l2)%nat by omega.
destruct H4; auto. elim H1. eapply list_length_incl; eauto.
Qed.

Remark list_norepet_map:
forall (A B: Type) (f: A -> B) (l: list A),
list_norepet (List.map f l) -> list_norepet l.
Proof.
induction l; simpl; intros.
constructor.
inv H. constructor; auto. red; intros; elim H2. apply List.in_map; auto.
Qed.

Theorem cardinal_remove:
forall x m y, T.get x m = Some y -> (cardinal (T.remove x m) < cardinal m)%nat.
Proof.
unfold cardinal; intros. apply list_strict_incl_length with (x := (x, y)).
apply list_norepet_map with (f := @fst T.elt V). apply T.elements_keys_norepet.
red; intros. destruct a as [x' y']. exploit T.elements_complete; eauto.
rewrite T.grspec. destruct (T.elt_eq x' x); intros; try discriminate.
apply T.elements_correct; auto.
red; intros. exploit T.elements_complete; eauto. rewrite T.grspec. rewrite dec_eq_true. congruence.
apply T.elements_correct; auto.
Qed.

End MEASURE.

Forall and exists

Section FORALL_EXISTS.

Variable A: Type.

Definition for_all (m: T.t A) (f: T.elt -> A -> bool) : bool :=
T.fold (fun b x a => b && f x a) m true.

Lemma for_all_correct:
forall m f,
for_all m f = true <-> (forall x a, T.get x m = Some a -> f x a = true).
Proof.
intros m0 f.
unfold for_all. apply fold_rec; intros.
-
rewrite H0. split; intros. rewrite <- H in H2; auto. rewrite H in H2; auto.
-
split; intros. rewrite T.gempty in H0; congruence. auto.
-
split; intros.
destruct (andb_prop _ _ H2). rewrite T.gsspec in H3. destruct (T.elt_eq x k).
inv H3. auto.
apply H1; auto.
apply andb_true_intro. split.
rewrite H1. intros. apply H2. rewrite T.gso; auto. congruence.
apply H2. apply T.gss.
Qed.

Definition exists_ (m: T.t A) (f: T.elt -> A -> bool) : bool :=
T.fold (fun b x a => b || f x a) m false.

Lemma exists_correct:
forall m f,
exists_ m f = true <-> (exists x a, T.get x m = Some a /\ f x a = true).
Proof.
intros m0 f.
unfold exists_. apply fold_rec; intros.
-
rewrite H0. split; intros (x0 & a0 & P & Q); exists x0; exists a0; split; auto; congruence.
-
split; intros. congruence. destruct H as (x & a & P & Q). rewrite T.gempty in P; congruence.
-
split; intros.
destruct (orb_true_elim _ _ H2).
rewrite H1 in e. destruct e as (x1 & a1 & P & Q).
exists x1; exists a1; split; auto. rewrite T.gso; auto. congruence.
exists k; exists v; split; auto. apply T.gss.
destruct H2 as (x1 & a1 & P & Q). apply orb_true_intro.
rewrite T.gsspec in P. destruct (T.elt_eq x1 k).
inv P. right; auto.
left. apply H1. exists x1; exists a1; auto.
Qed.

Remark exists_for_all:
forall m f,
exists_ m f = negb (for_all m (fun x a => negb (f x a))).
Proof.
intros. unfold exists_, for_all. rewrite ! T.fold_spec.
change false with (negb true). generalize (T.elements m) true.
induction l; simpl; intros.
auto.
rewrite <- IHl. f_equal.
destruct b; destruct (f (fst a) (snd a)); reflexivity.
Qed.

Remark for_all_exists:
forall m f,
for_all m f = negb (exists_ m (fun x a => negb (f x a))).
Proof.
intros. unfold exists_, for_all. rewrite ! T.fold_spec.
change true with (negb false). generalize (T.elements m) false.
induction l; simpl; intros.
auto.
rewrite <- IHl. f_equal.
destruct b; destruct (f (fst a) (snd a)); reflexivity.
Qed.

Lemma for_all_false:
forall m f,
for_all m f = false <-> (exists x a, T.get x m = Some a /\ f x a = false).
Proof.
intros. rewrite for_all_exists.
rewrite negb_false_iff. rewrite exists_correct.
split; intros (x & a & P & Q); exists x; exists a; split; auto.
rewrite negb_true_iff in Q. auto.
rewrite Q; auto.
Qed.

Lemma exists_false:
forall m f,
exists_ m f = false <-> (forall x a, T.get x m = Some a -> f x a = false).
Proof.
intros. rewrite exists_for_all.
rewrite negb_false_iff. rewrite for_all_correct.
split; intros. apply H in H0. rewrite negb_true_iff in H0. auto. rewrite H; auto.
Qed.

End FORALL_EXISTS.

More about beq

Section BOOLEAN_EQUALITY.

Variable A: Type.
Variable beqA: A -> A -> bool.

Theorem beq_false:
forall m1 m2,
T.beq beqA m1 m2 = false <->
exists x, match T.get x m1, T.get x m2 with
| None, None => False
| Some a1, Some a2 => beqA a1 a2 = false
| _, _ => True
end.
Proof.
intros; split; intros.
-
set (p1 := fun x a1 => match T.get x m2 with None => false | Some a2 => beqA a1 a2 end).
set (p2 := fun x a2 => match T.get x m1 with None => false | Some a1 => beqA a1 a2 end).
destruct (for_all m1 p1) eqn:F1; [destruct (for_all m2 p2) eqn:F2 | idtac].
+ cut (T.beq beqA m1 m2 = true). congruence.
rewrite for_all_correct in *. rewrite T.beq_correct; intros.
destruct (T.get x m1) as [a1|] eqn:X1.
generalize (F1 _ _ X1). unfold p1. destruct (T.get x m2); congruence.
destruct (T.get x m2) as [a2|] eqn:X2; auto.
generalize (F2 _ _ X2). unfold p2. rewrite X1. congruence.
+ rewrite for_all_false in F2. destruct F2 as (x & a & P & Q).
exists x. rewrite P. unfold p2 in Q. destruct (T.get x m1); auto.
+ rewrite for_all_false in F1. destruct F1 as (x & a & P & Q).
exists x. rewrite P. unfold p1 in Q. destruct (T.get x m2); auto.
-
destruct H as [x P].
destruct (T.beq beqA m1 m2) eqn:E; auto.
rewrite T.beq_correct in E.
generalize (E x). destruct (T.get x m1); destruct (T.get x m2); tauto || congruence.
Qed.

End BOOLEAN_EQUALITY.

Extensional equality between trees

Section EXTENSIONAL_EQUALITY.

Variable A: Type.
Variable eqA: A -> A -> Prop.
Hypothesis eqAeq: Equivalence eqA.

Definition Equal (m1 m2: T.t A) : Prop :=
forall x, match T.get x m1, T.get x m2 with
| None, None => True
| Some a1, Some a2 => a1 === a2
| _, _ => False
end.

Lemma Equal_refl: forall m, Equal m m.
Proof.
intros; red; intros. destruct (T.get x m); auto. reflexivity.
Qed.

Lemma Equal_sym: forall m1 m2, Equal m1 m2 -> Equal m2 m1.
Proof.
intros; red; intros. generalize (H x). destruct (T.get x m1); destruct (T.get x m2); auto. intros; symmetry; auto.
Qed.

Lemma Equal_trans: forall m1 m2 m3, Equal m1 m2 -> Equal m2 m3 -> Equal m1 m3.
Proof.
intros; red; intros. generalize (H x) (H0 x).
destruct (T.get x m1); destruct (T.get x m2); try tauto;
destruct (T.get x m3); try tauto.
intros. transitivity a0; auto.
Qed.

Instance Equal_Equivalence : Equivalence Equal := {
Equivalence_Reflexive := Equal_refl;
Equivalence_Symmetric := Equal_sym;
Equivalence_Transitive := Equal_trans
}.

Hypothesis eqAdec: EqDec A eqA.

Program Definition Equal_dec (m1 m2: T.t A) : { m1 === m2 } + { m1 =/= m2 } :=
match T.beq (fun a1 a2 => proj_sumbool (a1 == a2)) m1 m2 with
| true => left _
| false => right _
end.
Next Obligation.
rename Heq_anonymous into B.
symmetry in B. rewrite T.beq_correct in B.
red; intros. generalize (B x).
destruct (T.get x m1); destruct (T.get x m2); auto.
intros. eapply proj_sumbool_true; eauto.
Qed.
Next Obligation.
assert (T.beq (fun a1 a2 => proj_sumbool (a1 == a2)) m1 m2 = true).
apply T.beq_correct; intros.
generalize (H x).
destruct (T.get x m1); destruct (T.get x m2); try tauto.
intros. apply proj_sumbool_is_true; auto.
unfold equiv, complement in H0. congruence.
Qed.

Instance Equal_EqDec : EqDec (T.t A) Equal := Equal_dec.

End EXTENSIONAL_EQUALITY.

End Tree_Properties.

Module PTree_Properties := Tree_Properties(PTree).

# Useful notations

Notation "a ! b" := (PTree.get b a) (at level 1).
Notation "a !! b" := (PMap.get b a) (at level 1).