# 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 Coqlib.

Local Unset Elimination Schemes.
Local Unset Case Analysis Schemes.

Set Implicit Arguments.

# The abstract signatures of trees

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

The ``good variables'' properties for trees, expressing commutations between get, set and remove.
Axiom gempty:
forall (A: Type) (i: elt), get i (empty A) = None.
Axiom gss:
forall (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = Some x.
Axiom gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Axiom 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.
Axiom grs:
forall (A: Type) (i: elt) (m: t A), get i (remove i m) = None.
Axiom gro:
forall (A: Type) (i j: elt) (m: t A),
i <> j -> get i (remove j m) = get i m.
Axiom 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.
Parameter beq: forall (A: Type), (A -> A -> bool) -> t A -> t A -> bool.
Axiom 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.
Parameter map:
forall (A B: Type), (elt -> A -> B) -> t A -> t B.
Axiom 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.
Parameter map1:
forall (A B: Type), (A -> B) -> t A -> t B.
Axiom 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.
Parameter combine:
forall (A B C: Type), (option A -> option B -> option C) -> t A -> t B -> t C.
Axiom 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).

Enumerating the bindings of a tree.
Parameter elements:
forall (A: Type), t A -> list (elt * A).
Axiom elements_correct:
forall (A: Type) (m: t A) (i: elt) (v: A),
get i m = Some v -> In (i, v) (elements m).
Axiom elements_complete:
forall (A: Type) (m: t A) (i: elt) (v: A),
In (i, v) (elements m) -> get i m = Some v.
Axiom elements_keys_norepet:
forall (A: Type) (m: t A),
list_norepet (List.map (@fst elt A) (elements m)).
Axiom elements_extensional:
forall (A: Type) (m n: t A),
(forall i, get i m = get i n) ->
elements m = elements n.
Axiom elements_remove:
forall (A: Type) i v (m: t A),
get i m = Some v ->
exists l1 l2, elements m = l1 ++ (i,v) :: l2 /\ elements (remove i m) = l1 ++ l2.

Folding a function over all bindings of a tree.
Parameter fold:
forall (A B: Type), (B -> elt -> A -> B) -> t A -> B -> B.
Axiom 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.
Parameter fold1:
forall (A B: Type), (B -> A -> B) -> t A -> B -> B.
Axiom 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.
Parameter elt: Type.
Parameter elt_eq: forall (a b: elt), {a = b} + {a <> b}.
Parameter t: Type -> Type.
Parameter init: forall (A: Type), A -> t A.
Parameter get: forall (A: Type), elt -> t A -> A.
Parameter set: forall (A: Type), elt -> A -> t A -> t A.
Axiom gi:
forall (A: Type) (i: elt) (x: A), get i (init x) = x.
Axiom gss:
forall (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = x.
Axiom gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Axiom 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.
Axiom gsident:
forall (A: Type) (i j: elt) (m: t A), get j (set i (get i m) m) = get j m.
Parameter map: forall (A B: Type), (A -> B) -> t A -> t B.
Axiom 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.

## Representation of trees

The type tree' of nonempty trees. Each constructor is of the form NodeXYZ, where the bit X says whether there is a left subtree, Y whether there is a value at this node, and Z whether there is a right subtree.

Inductive tree' (A: Type) : Type :=
| Node001: tree' A -> tree' A
| Node010: A -> tree' A
| Node011: A -> tree' A -> tree' A
| Node100: tree' A -> tree' A
| Node101: tree' A -> tree' A ->tree' A
| Node110: tree' A -> A -> tree' A
| Node111: tree' A -> A -> tree' A -> tree' A.

Inductive tree (A: Type) : Type :=
| Empty : tree A
| Nodes: tree' A -> tree A.

Arguments Node001 {A} _.
Arguments Node010 {A} _.
Arguments Node011 {A} _ _.
Arguments Node100 {A} _.
Arguments Node101 {A} _ _.
Arguments Node110 {A} _ _.
Arguments Node111 {A} _ _ _.

Arguments Empty {A}.
Arguments Nodes {A} _.

Scheme tree'_ind := Induction for tree' Sort Prop.

Definition t := tree.

A smart constructor for type tree: given a (possibly empty) left subtree, a (possibly absent) value, and a (possibly empty) right subtree, it builds the corresponding tree.

Definition Node {A} (l: tree A) (o: option A) (r: tree A) : tree A :=
match l,o,r with
| Empty, None, Empty => Empty
| Empty, None, Nodes r' => Nodes (Node001 r')
| Empty, Some x, Empty => Nodes (Node010 x)
| Empty, Some x, Nodes r' => Nodes (Node011 x r')
| Nodes l', None, Empty => Nodes (Node100 l')
| Nodes l', None, Nodes r' => Nodes (Node101 l' r')
| Nodes l', Some x, Empty => Nodes (Node110 l' x)
| Nodes l', Some x, Nodes r' => Nodes (Node111 l' x r')
end.

## Basic operations: empty, get, set, remove

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

Fixpoint get' {A} (p: positive) (m: tree' A) : option A :=
match p, m with
| xH, Node001 _ => None
| xH, Node010 x => Some x
| xH, Node011 x _ => Some x
| xH, Node100 _ => None
| xH, Node101 _ _ => None
| xH, Node110 _ x => Some x
| xH, Node111 _ x _ => Some x
| xO q, Node001 _ => None
| xO q, Node010 _ => None
| xO q, Node011 _ _ => None
| xO q, Node100 m' => get' q m'
| xO q, Node101 m' _ => get' q m'
| xO q, Node110 m' _ => get' q m'
| xO q, Node111 m' _ _ => get' q m'
| xI q, Node001 m' => get' q m'
| xI q, Node010 _ => None
| xI q, Node011 _ m' => get' q m'
| xI q, Node100 m' => None
| xI q, Node101 _ m' => get' q m'
| xI q, Node110 _ _ => None
| xI q, Node111 _ _ m' => get' q m'
end.

Definition get {A} (p: positive) (m: tree A) : option A :=
match m with
| Empty => None
| Nodes m' => get' p m'
end.

set0 p x constructs the singleton tree that maps p to x and has no other bindings.

Fixpoint set0 {A} (p: positive) (x: A) : tree' A :=
match p with
| xH => Node010 x
| xO q => Node100 (set0 q x)
| xI q => Node001 (set0 q x)
end.

Fixpoint set' {A} (p: positive) (x: A) (m: tree' A) : tree' A :=
match p, m with
| xH, Node001 r => Node011 x r
| xH, Node010 _ => Node010 x
| xH, Node011 _ r => Node011 x r
| xH, Node100 l => Node110 l x
| xH, Node101 l r => Node111 l x r
| xH, Node110 l _ => Node110 l x
| xH, Node111 l _ r => Node111 l x r
| xO q, Node001 r => Node101 (set0 q x) r
| xO q, Node010 y => Node110 (set0 q x) y
| xO q, Node011 y r => Node111 (set0 q x) y r
| xO q, Node100 l => Node100 (set' q x l)
| xO q, Node101 l r => Node101 (set' q x l) r
| xO q, Node110 l y => Node110 (set' q x l) y
| xO q, Node111 l y r => Node111 (set' q x l) y r
| xI q, Node001 r => Node001 (set' q x r)
| xI q, Node010 y => Node011 y (set0 q x)
| xI q, Node011 y r => Node011 y (set' q x r)
| xI q, Node100 l => Node101 l (set0 q x)
| xI q, Node101 l r => Node101 l (set' q x r)
| xI q, Node110 l y => Node111 l y (set0 q x)
| xI q, Node111 l y r => Node111 l y (set' q x r)
end.

Definition set {A} (p: positive) (x: A) (m: tree A) : tree A :=
match m with
| Empty => Nodes (set0 p x)
| Nodes m' => Nodes (set' p x m')
end.

Removal in a nonempty tree produces a possibly empty tree. To simplify the code, we use the Node smart constructor in the cases where the result can be empty or nonempty, depending on the results of the recursive calls.

Fixpoint rem' {A} (p: positive) (m: tree' A) : tree A :=
match p, m with
| xH, Node001 r => Nodes m
| xH, Node010 _ => Empty
| xH, Node011 _ r => Nodes (Node001 r)
| xH, Node100 l => Nodes m
| xH, Node101 l r => Nodes m
| xH, Node110 l _ => Nodes (Node100 l)
| xH, Node111 l _ r => Nodes (Node101 l r)
| xO q, Node001 r => Nodes m
| xO q, Node010 y => Nodes m
| xO q, Node011 y r => Nodes m
| xO q, Node100 l => Node (rem' q l) None Empty
| xO q, Node101 l r => Node (rem' q l) None (Nodes r)
| xO q, Node110 l y => Node (rem' q l) (Some y) Empty
| xO q, Node111 l y r => Node (rem' q l) (Some y) (Nodes r)
| xI q, Node001 r => Node Empty None (rem' q r)
| xI q, Node010 y => Nodes m
| xI q, Node011 y r => Node Empty (Some y) (rem' q r)
| xI q, Node100 l => Nodes m
| xI q, Node101 l r => Node (Nodes l) None (rem' q r)
| xI q, Node110 l y => Nodes m
| xI q, Node111 l y r => Node (Nodes l) (Some y) (rem' q r)
end.

This use of Node causes some run-time overhead, which we eliminate by partial evaluation within Coq.

Definition remove' := Eval cbv [rem' Node] in @rem'.

Definition remove {A} (p: positive) (m: tree A) : tree A :=
match m with
| Empty => Empty
| Nodes m' => remove' p m'
end.

## Good variable properties for the basic operations

Theorem gempty:
forall (A: Type) (i: positive), get i (empty A) = None.
Proof.
reflexivity. Qed.

Lemma gEmpty:
forall (A: Type) (i: positive), get i (@Empty A) = None.
Proof.
reflexivity. Qed.

Lemma gss0: forall {A} p (x: A), get' p (set0 p x) = Some x.
Proof.
induction p; simpl; auto. Qed.

Lemma gso0: forall {A} p q (x: A), p<>q -> get' p (set0 q x) = None.
Proof.
induction p; destruct q; simpl; intros; auto; try apply IHp; congruence.
Qed.

Theorem gss:
forall (A: Type) (i: positive) (x: A) (m: tree A), get i (set i x m) = Some x.
Proof.
intros. destruct m as [|m]; simpl.
- apply gss0.
- revert m; induction i; destruct m; simpl; intros; auto using gss0.
Qed.

Theorem gso:
forall (A: Type) (i j: positive) (x: A) (m: tree A),
i <> j -> get i (set j x m) = get i m.
Proof.
intros. destruct m as [|m]; simpl.
- apply gso0; auto.
- revert m j H; induction i; destruct j,m; simpl; intros; auto;
solve [apply IHi; congruence | apply gso0; congruence | 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.

Lemma gNode:
forall {A} (i: positive) (l: tree A) (x: option A) (r: tree A),
get i (Node l x r) = match i with xH => x | xO j => get j l | xI j => get j r end.
Proof.
intros. destruct l, x, r; simpl; auto; destruct i; auto.
Qed.

Theorem grs:
forall (A: Type) (i: positive) (m: tree A), get i (remove i m) = None.
Proof.
Local Opaque Node.
destruct m as [ |m]; simpl. auto.
change (remove' i m) with (rem' i m).
revert m. induction i; destruct m; simpl; auto; rewrite gNode; auto.
Qed.

Theorem gro:
forall (A: Type) (i j: positive) (m: tree A),
i <> j -> get i (remove j m) = get i m.
Proof.
Local Opaque Node.
destruct m as [ |m]; simpl. auto.
change (remove' j m) with (rem' j m).
revert j m. induction i; destruct j, m; simpl; intros; auto;
solve [ congruence
| rewrite gNode; auto; apply IHi; congruence ].
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.

## Custom case analysis principles and induction principles

We can view trees as being of one of two (non-exclusive) cases: either Empty for an empty tree, or Node l o r for a nonempty tree, with l and r the left and right subtrees and o an optional value. The Empty constructor and the Node smart function defined above provide one half of the view: the one that lets us construct values of type tree A. We now define the other half of the view: the one that lets us inspect and recurse over values of type tree A. This is achieved by defining appropriate principles for case analysis and induction.

Definition not_trivially_empty {A} (l: tree A) (o: option A) (r: tree A) :=
match l, o, r with
| Empty, None, Empty => False
| _, _, _ => True
end.

A case analysis principle

Section TREE_CASE.

Context {A B: Type}
(empty: B)
(node: tree A -> option A -> tree A -> B).

Definition tree_case (m: tree A) : B :=
match m with
| Empty => empty
| Nodes (Node001 r) => node Empty None (Nodes r)
| Nodes (Node010 x) => node Empty (Some x) Empty
| Nodes (Node011 x r) => node Empty (Some x) (Nodes r)
| Nodes (Node100 l) => node (Nodes l) None Empty
| Nodes (Node101 l r) => node (Nodes l) None (Nodes r)
| Nodes (Node110 l x) => node (Nodes l) (Some x) Empty
| Nodes (Node111 l x r) => node (Nodes l) (Some x) (Nodes r)
end.

Lemma unroll_tree_case: forall l o r,
not_trivially_empty l o r ->
tree_case (Node l o r) = node l o r.
Proof.
destruct l, o, r; simpl; intros; auto. contradiction.
Qed.

End TREE_CASE.

A recursion principle

Section TREE_REC.

Context {A B: Type}
(empty: B)
(node: tree A -> B -> option A -> tree A -> B -> B).

Fixpoint tree_rec' (m: tree' A) : B :=
match m with
| Node001 r => node Empty empty None (Nodes r) (tree_rec' r)
| Node010 x => node Empty empty (Some x) Empty empty
| Node011 x r => node Empty empty (Some x) (Nodes r) (tree_rec' r)
| Node100 l => node (Nodes l) (tree_rec' l) None Empty empty
| Node101 l r => node (Nodes l) (tree_rec' l) None (Nodes r) (tree_rec' r)
| Node110 l x => node (Nodes l) (tree_rec' l) (Some x) Empty empty
| Node111 l x r => node (Nodes l) (tree_rec' l) (Some x) (Nodes r) (tree_rec' r)
end.

Definition tree_rec (m: tree A) : B :=
match m with
| Empty => empty
| Nodes m' => tree_rec' m'
end.

Lemma unroll_tree_rec: forall l o r,
not_trivially_empty l o r ->
tree_rec (Node l o r) = node l (tree_rec l) o r (tree_rec r).
Proof.
destruct l, o, r; simpl; intros; auto. contradiction.
Qed.

End TREE_REC.

A double recursion principle

Section TREE_REC2.

Context {A B C: Type}
(base: C)
(base1: tree B -> C)
(base2: tree A -> C)
(nodes: forall (l1: tree A) (o1: option A) (r1: tree A)
(l2: tree B) (o2: option B) (r2: tree B)
(lrec: C) (rrec: C), C).

Fixpoint tree_rec2' (m1: tree' A) (m2: tree' B) : C.
Proof.
destruct m1 as [ r1 | x1 | x1 r1 | l1 | l1 r1 | l1 x1 | l1 x1 r1 ];
destruct m2 as [ r2 | x2 | x2 r2 | l2 | l2 r2 | l2 x2 | l2 x2 r2 ];
(apply nodes;
[ solve [ exact (Nodes l1) | exact Empty ]
| solve [ exact (Some x1) | exact None ]
| solve [ exact (Nodes r1) | exact Empty ]
| solve [ exact (Nodes l2) | exact Empty ]
| solve [ exact (Some x2) | exact None ]
| solve [ exact (Nodes r2) | exact Empty ]
| solve [ exact (tree_rec2' l1 l2) | exact (base2 (Nodes l1)) | exact (base1 (Nodes l2)) | exact base ]
| solve [ exact (tree_rec2' r1 r2) | exact (base2 (Nodes r1)) | exact (base1 (Nodes r2)) | exact base ]
]).
Defined.

Definition tree_rec2 (a: tree A) (b: tree B) : C :=
match a, b with
| Empty, Empty => base
| Empty, _ => base1 b
| _, Empty => base2 a
| Nodes a', Nodes b' => tree_rec2' a' b'
end.

Lemma unroll_tree_rec2_NE:
forall l1 o1 r1,
not_trivially_empty l1 o1 r1 ->
tree_rec2 (Node l1 o1 r1) Empty = base2 (Node l1 o1 r1).
Proof.
intros. destruct l1, o1, r1; try contradiction; reflexivity.
Qed.

Lemma unroll_tree_rec2_EN:
forall l2 o2 r2,
not_trivially_empty l2 o2 r2 ->
tree_rec2 Empty (Node l2 o2 r2) = base1 (Node l2 o2 r2).
Proof.
intros. destruct l2, o2, r2; try contradiction; reflexivity.
Qed.

Lemma unroll_tree_rec2_NN:
forall l1 o1 r1 l2 o2 r2,
not_trivially_empty l1 o1 r1 -> not_trivially_empty l2 o2 r2 ->
tree_rec2 (Node l1 o1 r1) (Node l2 o2 r2) =
nodes l1 o1 r1 l2 o2 r2 (tree_rec2 l1 l2) (tree_rec2 r1 r2).
Proof.
intros.
destruct l1, o1, r1; try contradiction; destruct l2, o2, r2; try contradiction; reflexivity.
Qed.

End TREE_REC2.

An induction principle

Section TREE_IND.

Context {A: Type} (P: tree A -> Type)
(empty: P Empty)
(node: forall l, P l -> forall o r, P r -> not_trivially_empty l o r ->
P (Node l o r)).

Program Fixpoint tree_ind' (m: tree' A) : P (Nodes m) :=
match m with
| Node001 r => @node Empty empty None (Nodes r) (tree_ind' r) _
| Node010 x => @node Empty empty (Some x) Empty empty _
| Node011 x r => @node Empty empty (Some x) (Nodes r) (tree_ind' r) _
| Node100 l => @node (Nodes l) (tree_ind' l) None Empty empty _
| Node101 l r => @node (Nodes l) (tree_ind' l) None (Nodes r) (tree_ind' r) _
| Node110 l x => @node (Nodes l) (tree_ind' l) (Some x) Empty empty _
| Node111 l x r => @node (Nodes l) (tree_ind' l) (Some x) (Nodes r) (tree_ind' r) _
end.

Definition tree_ind (m: tree A) : P m :=
match m with
| Empty => empty
| Nodes m' => tree_ind' m'
end.

End TREE_IND.

## Extensionality property

Lemma tree'_not_empty:
forall {A} (m: tree' A), exists i, get' i m <> None.
Proof.
induction m; simpl; try destruct IHm as [p H].
- exists (xI p); auto.
- exists xH; simpl; congruence.
- exists xH; simpl; congruence.
- exists (xO p); auto.
- destruct IHm1 as [p H]; exists (xO p); auto.
- exists xH; simpl; congruence.
- exists xH; simpl; congruence.
Qed.

Corollary extensionality_empty:
forall {A} (m: tree A),
(forall i, get i m = None) -> m = Empty.
Proof.
intros. destruct m as [ | m]; auto. destruct (tree'_not_empty m) as [i GET].
elim GET. apply H.
Qed.

Theorem extensionality:
forall (A: Type) (m1 m2: tree A),
(forall i, get i m1 = get i m2) -> m1 = m2.
Proof.
intros A. induction m1 using tree_ind; induction m2 using tree_ind; intros.
- auto.
- symmetry. apply extensionality_empty. intros; symmetry; apply H0.
- apply extensionality_empty. apply H0.
- f_equal.
+ apply IHm1_1. intros. specialize (H1 (xO i)); rewrite ! gNode in H1. auto.
+ specialize (H1 xH); rewrite ! gNode in H1. auto.
+ apply IHm1_2. intros. specialize (H1 (xI i)); rewrite ! gNode in H1. auto.
Qed.

Some consequences of extensionality

Theorem gsident:
forall {A} (i: positive) (m: t A) (v: A),
get i m = Some v -> set i v m = m.
Proof.
intros; apply extensionality; intros j.
rewrite gsspec. destruct (peq j i); congruence.
Qed.

Theorem set2:
forall {A} (i: elt) (m: t A) (v1 v2: A),
set i v2 (set i v1 m) = set i v2 m.
Proof.
intros; apply extensionality; intros j.
rewrite ! gsspec. destruct (peq j i); auto.
Qed.

## Boolean equality

Section BOOLEAN_EQUALITY.

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

Fixpoint beq' (m1 m2: tree' A) {struct m1} : bool :=
match m1, m2 with
| Node001 r1, Node001 r2 => beq' r1 r2
| Node010 x1, Node010 x2 => beqA x1 x2
| Node011 x1 r1, Node011 x2 r2 => beqA x1 x2 && beq' r1 r2
| Node100 l1, Node100 l2 => beq' l1 l2
| Node101 l1 r1, Node101 l2 r2 => beq' l1 l2 && beq' r1 r2
| Node110 l1 x1, Node110 l2 x2 => beqA x1 x2 && beq' l1 l2
| Node111 l1 x1 r1, Node111 l2 x2 r2 => beqA x1 x2 && beq' l1 l2 && beq' r1 r2
| _, _ => false
end.

Definition beq (m1 m2: t A) : bool :=
match m1, m2 with
| Empty, Empty => true
| Nodes m1', Nodes m2' => beq' m1' m2'
| _, _ => false
end.

Let beq_optA (o1 o2: option A) : bool :=
match o1, o2 with
| None, None => true
| Some a1, Some a2 => beqA a1 a2
| _, _ => false
end.

Lemma beq_correct_bool:
forall m1 m2,
beq m1 m2 = true <-> (forall x, beq_optA (get x m1) (get x m2) = true).
Proof.
Local Transparent Node.
assert (beq_NN: forall l1 o1 r1 l2 o2 r2,
not_trivially_empty l1 o1 r1 ->
not_trivially_empty l2 o2 r2 ->
beq (Node l1 o1 r1) (Node l2 o2 r2) =
beq l1 l2 && beq_optA o1 o2 && beq r1 r2).
{ intros.
destruct l1, o1, r1; try contradiction; destruct l2, o2, r2; try contradiction;
simpl; rewrite ? andb_true_r, ? andb_false_r; auto.
rewrite andb_comm; auto.
f_equal; rewrite andb_comm; auto. }
induction m1 using tree_ind; [|induction m2 using tree_ind].
- intros. simpl; split; intros.
+ destruct m2; try discriminate. simpl; auto.
+ replace m2 with (@Empty A); auto. apply extensionality; intros x.
specialize (H x). destruct (get x m2); simpl; congruence.
- split; intros.
+ destruct (Node l o r); try discriminate. simpl; auto.
+ replace (Node l o r) with (@Empty A); auto. apply extensionality; intros x.
specialize (H0 x). destruct (get x (Node l o r)); simpl in *; congruence.
- rewrite beq_NN by auto. split; intros.
+ InvBooleans. rewrite ! gNode. destruct x.
* apply IHm0; auto.
* apply IHm1; auto.
* auto.
+ apply andb_true_intro; split; [apply andb_true_intro; split|].
* apply IHm1. intros. specialize (H1 (xO x)); rewrite ! gNode in H1; auto.
* specialize (H1 xH); rewrite ! gNode in H1; auto.
* apply IHm0. intros. specialize (H1 (xI x)); rewrite ! gNode in H1; auto.
Qed.

Theorem 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.
intros. rewrite beq_correct_bool. unfold beq_optA. split; intros.
- specialize (H x). destruct (get x m1), (get x m2); intuition congruence.
- specialize (H x). destruct (get x m1), (get x m2); intuition auto.
Qed.

End BOOLEAN_EQUALITY.

## Collective operations

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 map' {A B} (f: positive -> A -> B) (m: tree' A) (i: positive)
{struct m} : tree' B :=
match m with
| Node001 r => Node001 (map' f r (xI i))
| Node010 x => Node010 (f (prev i) x)
| Node011 x r => Node011 (f (prev i) x) (map' f r (xI i))
| Node100 l => Node100 (map' f l (xO i))
| Node101 l r => Node101 (map' f l (xO i)) (map' f r (xI i))
| Node110 l x => Node110 (map' f l (xO i)) (f (prev i) x)
| Node111 l x r => Node111 (map' f l (xO i)) (f (prev i) x) (map' f r (xI i))
end.

Definition map {A B} (f: positive -> A -> B) (m: tree A) :=
match m with
| Empty => Empty
| Nodes m => Nodes (map' f m xH)
end.

Lemma gmap':
forall {A B} (f: positive -> A -> B) (i j : positive) (m: tree' A),
get' i (map' 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} (f: positive -> A -> B) (i: positive) (m: t A),
get i (map f m) = option_map (f i) (get i m).
Proof.
intros; destruct m as [|m]; simpl. auto. rewrite gmap'. repeat f_equal. exact (prev_involutive i).
Qed.

Fixpoint map1' {A B} (f: A -> B) (m: tree' A) {struct m} : tree' B :=
match m with
| Node001 r => Node001 (map1' f r)
| Node010 x => Node010 (f x)
| Node011 x r => Node011 (f x) (map1' f r)
| Node100 l => Node100 (map1' f l)
| Node101 l r => Node101 (map1' f l) (map1' f r)
| Node110 l x => Node110 (map1' f l) (f x)
| Node111 l x r => Node111 (map1' f l) (f x) (map1' f r)
end.

Definition map1 {A B} (f: A -> B) (m: t A) : t B :=
match m with
| Empty => Empty
| Nodes m => Nodes (map1' f m)
end.

Theorem gmap1:
forall {A B} (f: A -> B) (i: elt) (m: t A),
get i (map1 f m) = option_map f (get i m).
Proof.
intros. destruct m as [|m]; simpl. auto.
revert i; induction m; destruct i; simpl; auto.
Qed.

Definition map_filter1_nonopt {A B} (f: A -> option B) (m: tree A) : tree B :=
tree_rec
Empty
(fun l lrec o r rrec => Node lrec (match o with None => None | Some a => f a end) rrec)
m.

Local Transparent Node.

Definition map_filter1 :=
Eval cbv [map_filter1_nonopt tree_rec tree_rec' Node] in @map_filter1_nonopt.

Lemma gmap_filter1:
forall {A B} (f: A -> option B) (m: tree A) (i: positive),
get i (map_filter1 f m) = match get i m with None => None | Some a => f a end.
Proof.
change @map_filter1 with @map_filter1_nonopt. unfold map_filter1_nonopt.
intros until f. induction m using tree_ind; intros.
- auto.
- rewrite unroll_tree_rec by auto. rewrite ! gNode; destruct i; auto.
Qed.

Definition filter1 {A} (pred: A -> bool) (m: t A) : t A :=
map_filter1 (fun a => if pred a then Some a else None) m.

Theorem gfilter1:
forall {A} (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. apply gmap_filter1.
Qed.

Section COMBINE.

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

Let combine_l := map_filter1 (fun a => f (Some a) None).
Let combine_r := map_filter1 (fun b => f None (Some b)).

Let combine_nonopt (m1: tree A) (m2: tree B) : tree C :=
tree_rec2
Empty
combine_r
combine_l
(fun l1 o1 r1 l2 o2 r2 lrec rrec =>
Node lrec
(match o1, o2 with None, None => None | _, _ => f o1 o2 end)
rrec)
m1 m2.

Definition combine :=
Eval cbv [combine_nonopt tree_rec2 tree_rec2'] in combine_nonopt.

Lemma gcombine_l: forall m i, get i (combine_l m) = f (get i m) None.
Proof.
intros; unfold combine_l; rewrite gmap_filter1. destruct (get i m); auto.
Qed.

Lemma gcombine_r: forall m i, get i (combine_r m) = f None (get i m).
Proof.
intros; unfold combine_r; rewrite gmap_filter1. destruct (get i m); auto.
Qed.

Theorem gcombine:
forall (m1: t A) (m2: t B) (i: positive),
get i (combine m1 m2) = f (get i m1) (get i m2).
Proof.
change combine with combine_nonopt.
induction m1 using tree_ind; induction m2 using tree_ind; intros.
- auto.
- unfold combine_nonopt; rewrite unroll_tree_rec2_EN by auto. apply gcombine_r.
- unfold combine_nonopt; rewrite unroll_tree_rec2_NE by auto. apply gcombine_l.
- unfold combine_nonopt; rewrite unroll_tree_rec2_NN by auto.
rewrite ! gNode; destruct i; auto. destruct o, o0; auto.
Qed.

End COMBINE.

Theorem combine_commut:
forall (A B: Type) (f g: option A -> option A -> option B),
f None None = None -> g None None = None ->
(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. apply extensionality; intros i. rewrite ! gcombine by auto. auto.
Qed.

## List of all bindings

Fixpoint xelements' {A} (m : tree' A) (i: positive) (k: list (positive * A))
{struct m} : list (positive * A) :=
match m with
| Node001 r => xelements' r (xI i) k
| Node010 x => (prev i, x) :: k
| Node011 x r => (prev i, x) :: xelements' r (xI i) k
| Node100 l => xelements' l (xO i) k
| Node101 l r => xelements' l (xO i) (xelements' r (xI i) k)
| Node110 l x => xelements' l (xO i) ((prev i, x)::k)
| Node111 l x r => xelements' l (xO i) ((prev i, x):: (xelements' r (xI i) k))
end.

Definition elements {A} (m : t A) : list (positive * A) :=
match m with Empty => nil | Nodes m' => xelements' m' xH nil end.

Definition xelements {A} (m : t A) (i: positive) : list (positive * A) :=
match m with Empty => nil | Nodes m' => xelements' m' i nil end.

Lemma xelements'_append:
forall A (m: tree' A) i k1 k2,
xelements' m i (k1 ++ k2) = xelements' m i k1 ++ k2.
Proof.
induction m; intros; simpl; auto.
- f_equal; auto.
- rewrite IHm2, IHm1. auto.
- rewrite <- IHm. auto.
- rewrite IHm2, <- IHm1. auto.
Qed.

Lemma xelements_Node:
forall A (l: tree A) (o: option A) (r: tree A) i,
xelements (Node l o r) i =
xelements l (xO i)
++ match o with None => nil | Some v => (prev i, v) :: nil end
++ xelements r (xI i).
Proof.
Local Transparent Node.
intros; destruct l, o, r; simpl; rewrite <- ? xelements'_append; auto.
Qed.

Lemma xelements_correct:
forall A (m: tree A) i j v,
get i m = Some v -> In (prev (prev_append i j), v) (xelements m j).
Proof.
intros A. induction m using tree_ind; intros.
- discriminate.
- rewrite xelements_Node, ! in_app. rewrite gNode in H0. destruct i.
+ right; right. apply (IHm2 i (xI j)); auto.
+ left. apply (IHm1 i (xO j)); auto.
+ right; left. subst o. rewrite prev_append_prev. auto with coqlib.
Qed.

Theorem elements_correct:
forall A (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 H). rewrite prev_append_prev. auto.
Qed.

Lemma in_xelements:
forall A (m: tree A) (i k: positive) (v: A) ,
In (k, v) (xelements m i) ->
exists j, k = prev (prev_append j i) /\ get j m = Some v.
Proof.
intros A. induction m using tree_ind; intros.
- elim H.
- rewrite xelements_Node, ! in_app in H0. destruct H0 as [P | [P | P]].
+ exploit IHm1; eauto. intros (j & Q & R). exists (xO j); rewrite gNode; auto.
+ destruct o; simpl in P; intuition auto. inv H0. exists xH; rewrite gNode; auto.
+ exploit IHm2; eauto. intros (j & Q & R). exists (xI j); rewrite gNode; auto.
Qed.

Theorem elements_complete:
forall A (m: t A) (i: positive) (v: A),
In (i, v) (elements m) -> get i m = Some v.
Proof.
intros A m i v H. exploit in_xelements; eauto. intros (j & P & Q).
rewrite prev_append_prev in P. change i with (prev_append 1 i) in P.
exploit prev_append_inj; eauto. intros; congruence.
Qed.

Definition xkeys {A} (m: t A) (i: positive) := List.map fst (xelements m i).

Lemma xkeys_Node:
forall A (m1: t A) o (m2: t A) i,
xkeys (Node m1 o m2) i =
xkeys m1 (xO i)
++ match o with None => nil | Some v => prev i :: nil end
++ xkeys m2 (xI i).
Proof.
intros. unfold xkeys. rewrite xelements_Node, ! map_app. destruct o; auto.
Qed.

Lemma in_xkeys:
forall (A: Type) (m: t A) (i k: positive),
In k (xkeys m i) ->
(exists j, k = prev (prev_append j i)).
Proof.
unfold xkeys; intros.
apply (list_in_map_inv) in H. destruct H as ((j, v) & -> & H).
exploit in_xelements; eauto. intros (k & P & Q). exists k; auto.
Qed.

Lemma xelements_keys_norepet:
forall (A: Type) (m: t A) (i: positive),
list_norepet (xkeys m i).
Proof.
intros A; induction m using tree_ind; intros.
- constructor.
- assert (NOTIN1: ~ In (prev i) (xkeys l (xO i))).
{ red; intros. exploit in_xkeys; eauto. intros (j & EQ).
rewrite prev_append_prev in EQ. simpl in EQ. apply prev_append_inj in EQ. discriminate. }
assert (NOTIN2: ~ In (prev i) (xkeys r (xI i))).
{ red; intros. exploit in_xkeys; eauto. intros (j & EQ).
rewrite prev_append_prev in EQ. simpl in EQ. apply prev_append_inj in EQ. discriminate. }
assert (DISJ: forall x, In x (xkeys l (xO i)) -> In x (xkeys r (xI i)) -> False).
{ intros. exploit in_xkeys. eexact H0. intros (j1 & EQ1).
exploit in_xkeys. eexact H1. intros (j2 & EQ2).
rewrite prev_append_prev in *. simpl in *. rewrite EQ2 in EQ1. apply prev_append_inj in EQ1. discriminate. }
rewrite xkeys_Node. apply list_norepet_append. auto.
destruct o; simpl; auto. constructor; auto.
red; intros. red; intros; subst y. destruct o; simpl in H1.
destruct H1. subst x. tauto. eauto. eauto.
Qed.

Theorem elements_keys_norepet:
forall A (m: t A),
list_norepet (List.map (@fst elt A) (elements m)).
Proof.
intros. apply (xelements_keys_norepet m xH).
Qed.

Remark xelements_empty:
forall A (m: t A) i, (forall i, get i m = None) -> xelements m i = nil.
Proof.
intros. replace m with (@Empty A). auto.
apply extensionality; intros. symmetry; auto.
Qed.

Theorem elements_canonical_order':
forall (A B: Type) (R: A -> B -> Prop) (m: t A) (n: t B),
(forall i, option_rel R (get i m) (get i n)) ->
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 n.
change (elements m) with (xelements m xH). change (elements n) with (xelements n xH).
generalize 1%positive. revert m n.
induction m using tree_ind; [ | induction n using tree_ind]; intros until p; intros REL.
- replace n with (@Empty B). constructor.
apply extensionality; intros. specialize (REL i). simpl in *. inv REL; auto.
- replace (Node l o r) with (@Empty A). constructor.
apply extensionality; intros. specialize (REL i). simpl in *. inv REL; auto.
- rewrite ! xelements_Node. repeat apply list_forall2_app.
+ apply IHm. intros. specialize (REL (xO i)). rewrite ! gNode in REL; auto.
+ specialize (REL xH). rewrite ! gNode in REL. inv REL; constructor; auto using list_forall2_nil.
+ apply IHm0. intros. specialize (REL (xI i)). rewrite ! gNode in REL; auto.
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. apply elements_canonical_order'.
intros. destruct (get i m) as [x|] eqn:GM.
exploit H; eauto. intros (y & P & Q). rewrite P; constructor; auto.
destruct (get i n) as [y|] eqn:GN.
exploit H0; eauto. intros (x & P & Q). congruence.
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. replace n with m; auto. apply extensionality; auto.
Qed.

Lemma xelements_remove:
forall A v (m: t A) i j,
get i m = Some v ->
exists l1 l2,
xelements m j = l1 ++ (prev (prev_append i j), v) :: l2
/\ xelements (remove i m) j = l1 ++ l2.
Proof.
intros A; induction m using tree_ind; intros.
- discriminate.
- assert (REMOVE: remove i (Node l o r) =
match i with
| xH => Node l None r
| xO ii => Node (remove ii l) o r
| xI ii => Node l o (remove ii r)
end).
{ destruct l, o, r, i; reflexivity. }
rewrite gNode in H0. rewrite REMOVE. destruct i; rewrite ! xelements_Node.
+ destruct (IHm0 i (xI j) H0) as (l1 & l2 & EQ & EQ').
exists (xelements l (xO j) ++
match o with None => nil | Some x => (prev j, x) :: nil end ++
l1);
exists l2; split.
rewrite EQ, ! app_ass. auto.
rewrite EQ', ! app_ass. auto.
+ destruct (IHm i (xO j) H0) as (l1 & l2 & EQ & EQ').
exists l1;
exists (l2 ++
match o with None => nil | Some x => (prev j, x) :: nil end ++
xelements r (xI j));
split.
rewrite EQ, ! app_ass. auto.
rewrite EQ', ! app_ass. auto.
+ subst o. exists (xelements l (xO j)); exists (xelements r (xI j)); split.
rewrite prev_append_prev. auto.
auto.
Qed.

Theorem elements_remove:
forall A i v (m: t A),
get i m = Some v ->
exists l1 l2, elements m = l1 ++ (i,v) :: l2 /\ elements (remove i m) = l1 ++ l2.
Proof.
intros. exploit xelements_remove. eauto. instantiate (1 := xH).
rewrite prev_append_prev. auto.
Qed.

## Folding over a tree

Fixpoint fold' {A B} (f: B -> positive -> A -> B)
(i: positive) (m: tree' A) (v: B) {struct m} : B :=
match m with
| Node001 r => fold' f (xI i) r v
| Node010 x => f v (prev i) x
| Node011 x r => fold' f (xI i) r (f v (prev i) x)
| Node100 l => fold' f (xO i) l v
| Node101 l r => fold' f (xI i) r (fold' f (xO i) l v)
| Node110 l x => f (fold' f (xO i) l v) (prev i) x
| Node111 l x r => fold' f (xI i) r (f (fold' f (xO i) l v) (prev i) x)
end.

Definition fold {A B} (f: B -> positive -> A -> B) (m: t A) (v: B) :=
match m with Empty => v | Nodes m' => fold' f xH m' v end.

Lemma fold'_xelements':
forall A B (f: B -> positive -> A -> B) m i v l,
List.fold_left (fun a p => f a (fst p) (snd p)) l (fold' 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.
rewrite <- IHm1, <- IHm2; auto.
rewrite <- IHm; auto.
rewrite <- IHm1. simpl. rewrite <- IHm2; auto.
Qed.

Theorem fold_spec:
forall A B (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. destruct m; auto. rewrite <- fold'_xelements'. auto.
Qed.

Fixpoint fold1' {A B} (f: B -> A -> B) (m: tree' A) (v: B) {struct m} : B :=
match m with
| Node001 r => fold1' f r v
| Node010 x => f v x
| Node011 x r => fold1' f r (f v x)
| Node100 l => fold1' f l v
| Node101 l r => fold1' f r (fold1' f l v)
| Node110 l x => f (fold1' f l v) x
| Node111 l x r => fold1' f r (f (fold1' f l v) x)
end.

Definition fold1 {A B} (f: B -> A -> B) (m: t A) (v: B) : B :=
match m with Empty => v | Nodes m' => fold1' f m' v end.

Lemma fold1'_xelements':
forall A B (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; simpl; intros; auto.
rewrite <- IHm1. rewrite <- IHm2. auto.
rewrite <- IHm. auto.
rewrite <- IHm1. simpl. rewrite <- IHm2. auto.
Qed.

Theorem fold1_spec:
forall A B (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. destruct m as [|m]. reflexivity.
apply fold1'_xelements' with (l := @nil (positive * A)).
Qed.

Arguments empty A : simpl never.
Arguments get {A} p m : simpl never.
Arguments set {A} p x m : simpl never.
Arguments remove {A} p m : simpl never.

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. reflexivity.
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.
Parameter t: Type.
Parameter index: t -> positive.
Axiom index_inj: forall (x y: t), index x = index y -> x = y.
Parameter 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.
Parameter t: Type.
Parameter 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.

# A partial implementation of trees over any type that injects into type positive

Module ITree(X: INDEXED_TYPE).

Definition elt := X.t.
Definition elt_eq := X.eq.
Definition t : Type -> Type := PTree.t.

Definition empty (A: Type): t A := PTree.empty A.
Definition get (A: Type) (k: elt) (m: t A): option A := PTree.get (X.index k) m.
Definition set (A: Type) (k: elt) (v: A) (m: t A): t A := PTree.set (X.index k) v m.
Definition remove (A: Type) (k: elt) (m: t A): t A := PTree.remove (X.index k) m.

Theorem gempty:
forall (A: Type) (i: elt), get i (empty A) = None.
Proof.
intros. apply PTree.gempty.
Qed.
Theorem gss:
forall (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = Some x.
Proof.
intros. apply PTree.gss.
Qed.
Theorem gso:
forall (A: Type) (i j: elt) (x: A) (m: t A),
i <> j -> get i (set j x m) = get i m.
Proof.
intros. apply PTree.gso. red; intros; elim H; apply X.index_inj; auto.
Qed.
Theorem 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.
Proof.
intros. destruct (elt_eq i j). subst j; apply gss. apply gso; auto.
Qed.
Theorem grs:
forall (A: Type) (i: elt) (m: t A), get i (remove i m) = None.
Proof.
intros. apply PTree.grs.
Qed.
Theorem gro:
forall (A: Type) (i j: elt) (m: t A),
i <> j -> get i (remove j m) = get i m.
Proof.
intros. apply PTree.gro. red; intros; elim H; apply X.index_inj; 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.

Definition beq: forall (A: Type), (A -> A -> bool) -> t A -> t A -> bool := PTree.beq.
Theorem beq_sound:
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.
Proof.
unfold beq, get. intros. rewrite PTree.beq_correct in H. apply H.
Qed.

Definition combine: forall (A B C: Type), (option A -> option B -> option C) -> t A -> t B -> t C := PTree.combine.
Theorem 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).
Proof.
intros. apply PTree.gcombine. auto.
Qed.
End ITree.

Module ZTree := ITree(ZIndexed).

# Additional properties over trees

Require Import Equivalence EquivDec.

Module Tree_Properties(T: TREE).

Two induction principles over fold.

Section TREE_FOLD_IND.

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

Hypothesis H_base:
forall m,
(forall k, T.get k m = None) ->
P m init.

Hypothesis H_rec:
forall m a k v,
T.get k m = Some v -> T.get k m_final = Some v ->
P (T.remove k m) a -> P m (f a k v).

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

Let P' (l: list (T.elt * V)) (a: A) : Type :=
forall m, (forall k v, In (k, v) l <-> T.get k m = Some v) -> P m a.

Let H_base':
P' nil init.
Proof.
intros m EQV. apply H_base.
intros. destruct (T.get k m) as [v|] eqn:G; auto.
apply EQV in G. contradiction.
Defined.

Let H_rec':
forall k v l a,
~In k (List.map fst l) ->
T.get k m_final = Some v ->
P' l a ->
P' ((k, v) :: l) (f a k v).
Proof.
unfold P'; intros k v l a NOTIN FINAL HR m EQV.
set (m0 := T.remove k m).
apply H_rec.
- apply EQV. simpl; auto.
- auto.
- apply HR. intros k' v'. rewrite T.grspec. split; intros; destruct (T.elt_eq k' k).
+ subst k'. elim NOTIN. change k with (fst (k, v')). apply List.in_map; auto.
+ apply EQV. simpl; auto.
+ congruence.
+ apply EQV in H. simpl in H. intuition congruence.
Defined.

Lemma fold_ind_aux:
forall l,
(forall k v, In (k, v) l -> T.get k m_final = Some v) ->
list_norepet (List.map fst l) ->
P' l (List.fold_right f' init l).
Proof.
induction l as [ | [k v] l ]; simpl; intros FINAL NOREPET.
- apply H_base'.
- apply H_rec'.
+ inv NOREPET. auto.
+ apply FINAL. auto.
+ apply IHl. auto. inv NOREPET; auto.
Defined.

Theorem fold_ind:
P m_final (T.fold f m_final init).
Proof.
intros.
set (l' := List.rev (T.elements m_final)).
assert (P' l' (List.fold_right f' init l')).
{ apply fold_ind_aux.
intros. apply T.elements_complete. apply List.in_rev. auto.
unfold l'; rewrite List.map_rev. apply list_norepet_rev. apply T.elements_keys_norepet. }
unfold l', f' in X; rewrite fold_left_rev_right in X.
rewrite T.fold_spec. apply X.
intros; simpl. rewrite <- List.in_rev.
split. apply T.elements_complete. apply T.elements_correct.
Defined.

End TREE_FOLD_IND.

Section TREE_FOLD_REC.

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).

Theorem fold_rec:
P m_final (T.fold f m_final init).
Proof.
apply fold_ind.
- intros. apply P_compat with (T.empty V); auto.
+ intros. rewrite T.gempty. auto.
- intros. apply P_compat with (T.set k v (T.remove k m)).
+ intros. rewrite T.gsspec, T.grspec. destruct (T.elt_eq x k); auto. congruence.
+ apply H_rec; auto. apply T.grs.
Qed.

End TREE_FOLD_REC.

A nonnegative measure over trees

Section MEASURE.

Variable V: Type.

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

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.
exploit T.elements_remove; eauto. intros (l1 & l2 & P & Q).
rewrite P, Q. rewrite ! app_length. simpl. lia.
Qed.

Theorem cardinal_set:
forall x m y, T.get x m = None -> (cardinal m < cardinal (T.set x y m))%nat.
Proof.
intros. set (m' := T.set x y m).
replace (cardinal m) with (cardinal (T.remove x m')).
apply cardinal_remove with y. unfold m'; apply T.gss.
unfold cardinal. f_equal. apply T.elements_extensional.
intros. unfold m'. rewrite T.grspec, T.gsspec.
destruct (T.elt_eq i x); auto. congruence.
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.
- (* Extensionality *)
rewrite H0. split; intros. rewrite <- H in H2; auto. rewrite H in H2; auto.
- (* Base case *)
split; intros. rewrite T.gempty in H0; congruence. auto.
- (* Inductive case *)
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.
- (* Extensionality *)
rewrite H0. split; intros (x0 & a0 & P & Q); exists x0; exists a0; split; auto; congruence.
- (* Base case *)
split; intros. congruence. destruct H as (x & a & P & Q). rewrite T.gempty in P; congruence.
- (* Inductive case *)
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.
- (* beq = false -> existence *)
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.
- (* existence -> beq = false *)
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.

Global 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.

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

End EXTENSIONAL_EQUALITY.

Creating a tree from a list of (key, value) pairs.

Section OF_LIST.

Variable A: Type.

Let f := fun (m: T.t A) (k_v: T.elt * A) => T.set (fst k_v) (snd k_v) m.

Definition of_list (l: list (T.elt * A)) : T.t A :=
List.fold_left f l (T.empty _).

Lemma in_of_list:
forall l k v, T.get k (of_list l) = Some v -> In (k, v) l.
Proof.
assert (REC: forall k v l m,
T.get k (fold_left f l m) = Some v -> In (k, v) l \/ T.get k m = Some v).
{ induction l as [ | [k1 v1] l]; simpl; intros.
- tauto.
- apply IHl in H. unfold f in H. simpl in H. rewrite T.gsspec in H.
destruct H; auto.
destruct (T.elt_eq k k1). inv H. auto. auto.
}
intros. apply REC in H. rewrite T.gempty in H. intuition congruence.
Qed.

Lemma of_list_dom:
forall l k, In k (map fst l) -> exists v, T.get k (of_list l) = Some v.
Proof.
assert (REC: forall k l m,
In k (map fst l) \/ (exists v, T.get k m = Some v) ->
exists v, T.get k (fold_left f l m) = Some v).
{ induction l as [ | [k1 v1] l]; simpl; intros.
- tauto.
- apply IHl. unfold f; rewrite T.gsspec. simpl. destruct (T.elt_eq k k1).
right; econstructor; eauto.
intuition congruence.
}
intros. apply REC. auto.
Qed.

Remark of_list_unchanged:
forall k l m, ~In k (map fst l) -> T.get k (List.fold_left f l m) = T.get k m.
Proof.
induction l as [ | [k1 v1] l]; simpl; intros.
- auto.
- rewrite IHl by tauto. unfold f; apply T.gso; intuition auto.
Qed.

Lemma of_list_unique:
forall k v l1 l2,
~In k (map fst l2) -> T.get k (of_list (l1 ++ (k, v) :: l2)) = Some v.
Proof.
intros. unfold of_list. rewrite fold_left_app. simpl.
rewrite of_list_unchanged by auto. unfold f; apply T.gss.
Qed.

Lemma of_list_norepet:
forall l k v, list_norepet (map fst l) -> In (k, v) l -> T.get k (of_list l) = Some v.
Proof.
assert (REC: forall k v l m,
list_norepet (map fst l) ->
In (k, v) l ->
T.get k (fold_left f l m) = Some v).
{ induction l as [ | [k1 v1] l]; simpl; intros.
contradiction.
inv H. destruct H0.
inv H. rewrite of_list_unchanged by auto. apply T.gss.
apply IHl; auto.
}
intros; apply REC; auto.
Qed.

Lemma of_list_elements:
forall m k, T.get k (of_list (T.elements m)) = T.get k m.
Proof.
intros. destruct (T.get k m) as [v|] eqn:M.
- apply of_list_norepet. apply T.elements_keys_norepet. apply T.elements_correct; auto.
- destruct (T.get k (of_list (T.elements m))) as [v|] eqn:M'; auto.
apply in_of_list in M'. apply T.elements_complete in M'. congruence.
Qed.

End OF_LIST.

Lemma of_list_related:
forall (A B: Type) (R: A -> B -> Prop) k l1 l2,
list_forall2 (fun ka kb => fst ka = fst kb /\ R (snd ka) (snd kb)) l1 l2 ->
option_rel R (T.get k (of_list l1)) (T.get k (of_list l2)).
Proof.
intros until k. unfold of_list.
set (R' := fun ka kb => fst ka = fst kb /\ R (snd ka) (snd kb)).
set (fa := fun (m : T.t A) (k_v : T.elt * A) => T.set (fst k_v) (snd k_v) m).
set (fb := fun (m : T.t B) (k_v : T.elt * B) => T.set (fst k_v) (snd k_v) m).
assert (REC: forall l1 l2, list_forall2 R' l1 l2 ->
forall m1 m2, option_rel R (T.get k m1) (T.get k m2) ->
option_rel R (T.get k (fold_left fa l1 m1)) (T.get k (fold_left fb l2 m2))).
{ induction 1; intros; simpl.
- auto.
- apply IHlist_forall2. unfold fa, fb. rewrite ! T.gsspec.
destruct H as [E F]. rewrite E. destruct (T.elt_eq k (fst b1)).
constructor; auto.
auto. }
intros. apply REC; auto. rewrite ! T.gempty. constructor.
Qed.

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).