Library CompOptCert.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.
  Parameter elt: Type.
  Parameter elt_eq: ∀ (a b: elt), {a = b } + {a ≠ b }.
  Parameter t: Type → Type.
  Parameter empty: ∀ (A: Type), t A.
  Parameter get: ∀ (A: Type), elt → t A → option A.
  Parameter set: ∀ (A: Type), elt → A → t A → t A.
  Parameter remove: ∀ (A: Type), elt → t A → t A.

The ``good variables'' properties for trees, expressing commutations between get, set and remove.

  Axiom gempty:
    ∀ (A: Type) (i: elt), get i (empty A) = None.
  Axiom gss:
    ∀ (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = Some x.
  Axiom gso:
    ∀ (A: Type) (i j: elt) (x: A) (m: t A),
    i ≠ j → get i (set j x m) = get i m.
  Axiom gsspec:
    ∀ (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:
    ∀ (A: Type) (i: elt) (m: t A), get i (remove i m) = None.
  Axiom gro:
    ∀ (A: Type) (i j: elt) (m: t A),
    i ≠ j → get i (remove j m) = get i m.
  Axiom grspec:
    ∀ (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: ∀ (A: Type), (A → A → bool ) → t A → t A → bool.
  Axiom beq_correct:
    ∀ (A: Type) (eqA: A → A → bool) (t1 t2: t A),
    beq eqA t1 t2 = true ↔
    (∀ (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:
    ∀ (A B: Type), (elt → A → B ) → t A → t B.
  Axiom gmap:
    ∀ (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:
    ∀ (A B: Type), (A → B ) → t A → t B.
  Axiom gmap1:
    ∀ (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:
    ∀ (A B C: Type), (option A → option B → option C ) → t A → t B → t C.
  Axiom gcombine:
    ∀ (A B C: Type) (f: option A → option B → option C),
    f None None = None →
    ∀ (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:
    ∀ (A: Type), t A → list (elt × A).
  Axiom elements_correct:
    ∀ (A: Type) (m: t A) (i: elt) (v: A),
    get i m = Some v → In (i , v ) (elements m).
  Axiom elements_complete:
    ∀ (A: Type) (m: t A) (i: elt) (v: A),
    In (i , v ) (elements m) → get i m = Some v.
  Axiom elements_keys_norepet:
    ∀ (A: Type) (m: t A),
    list_norepet (List.map (@fst elt A) (elements m)).
  Axiom elements_extensional:
    ∀ (A: Type) (m n: t A),
    (∀ i, get i m = get i n ) →
    elements m = elements n.
  Axiom elements_remove:
    ∀ (A: Type) i v (m: t A),
    get i m = Some v →
    ∃ 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:
    ∀ (A B: Type), (B → elt → A → B ) → t A → B → B.
  Axiom fold_spec:
    ∀ (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:
    ∀ (A B: Type), (B → A → B ) → t A → B → B.
  Axiom fold1_spec:
    ∀ (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: ∀ (a b: elt), {a = b } + {a ≠ b }.
  Parameter t: Type → Type.
  Parameter init: ∀ (A: Type), A → t A.
  Parameter get: ∀ (A: Type), elt → t A → A.
  Parameter set: ∀ (A: Type), elt → A → t A → t A.
  Axiom gi:
    ∀ (A: Type) (i: elt) (x: A), get i (init x) = x.
  Axiom gss:
    ∀ (A: Type) (i: elt) (x: A) (m: t A), get i (set i x m) = x.
  Axiom gso:
    ∀ (A: Type) (i j: elt) (x: A) (m: t A),
    i ≠ j → get i (set j x m) = get i m.
  Axiom gsspec:
    ∀ (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:
    ∀ (A: Type) (i j: elt) (m: t A), get j (set i (get i m) m) = get j m.
  Parameter map: ∀ (A B: Type), (A → B ) → t A → t B.
  Axiom gmap:
    ∀ (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.

  Arguments Leaf {A}.
  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:
    ∀ (A: Type) (i: positive), get i (empty A) = None.
  Proof.
    induction i; simpl; auto.
  Qed.

  Theorem gss:
    ∀ (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 : ∀ (A : Type) (i : positive), get i (Leaf : t A) = None.
    Proof. exact gempty. Qed.

  Theorem gso:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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 : ∀ (A : Type) (i : positive), remove i (Leaf : t A) = Leaf.
  Proof. destruct i; simpl; auto. Qed.

  Theorem grs:
    ∀ (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:
    ∀ (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:
    ∀ (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:
      ∀ m, bempty m = true ↔ (∀ 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:
      ∀ m1 m2,
      beq m1 m2 = true ↔
      (∀ (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:
      ∀ (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:
    ∀ (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:
    ∀ (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':
    ∀ (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:
    ∀ (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 :
          ∀ (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 :
          ∀ (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:
      ∀ (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 :
    ∀ (A B: Type) (f g : option A → option A → option B) (m : t A),
    (∀ (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:
    ∀ (A B: Type) (f g: option A → option A → option B),
    (∀ (i j: option A), f i j = g j i ) →
    ∀ (m1 m2: t A),
    combine f m1 m2 = combine g m2 m1.
  Proof.
    intros A B f g EQ1.
    assert (EQ2: ∀ (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.

  Remark xelements_append:
    ∀ A (m: t A) i k1 k2,
    xelements m i (k1 ++ k2) = xelements m i k1 ++ k2.
  Proof.
    induction m; intros; simpl.
  - auto.
  - destruct o; rewrite IHm2; rewrite <- IHm1; auto.
  Qed.

  Remark xelements_leaf:
    ∀ A i, xelements (@Leaf A) i nil = nil.
  Proof.
    intros; reflexivity.
  Qed.

  Remark xelements_node:
    ∀ A (m1: t A) o (m2: t A) i,
    xelements (Node m1 o m2) i nil =
       xelements m1 (xO i) nil
    ++ match o with None ⇒ nil | Some v ⇒ (prev i , v) :: nil end
    ++ xelements m2 (xI i) nil.
  Proof.
    intros. simpl. destruct o; simpl; rewrite <- xelements_append; auto.
  Qed.

    Lemma xelements_incl:
      ∀ (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:
      ∀ (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:
    ∀ (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:
    ∀ (A: Type) (m: t A) (i k: positive) (v: A) ,
    In (k , v ) (xelements m i nil) →
    ∃ j , k = prev (prev_append j i) ∧ get j m = Some v.
  Proof.
    induction m; intros.
  - rewrite xelements_leaf in H. contradiction.
  - rewrite xelements_node in H. rewrite ! in_app_iff in H. destruct H as [P | [P | P]].
    + exploit IHm1; eauto. intros (j & Q & R). ∃ (xO j); auto.
    + destruct o; simpl in P; intuition auto. inv H. ∃ xH; auto.
    + exploit IHm2; eauto. intros (j & Q & R). ∃ (xI j); auto.
  Qed.

  Theorem elements_complete:
    ∀ (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. 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: Type) (m: t A) (i: positive) :=
    List.map (@fst positive A) (xelements m i nil).

  Remark xkeys_leaf:
    ∀ A i, xkeys (@Leaf A) i = nil.
  Proof.
    intros; reflexivity.
  Qed.

  Remark xkeys_node:
    ∀ 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. rewrite ! map_app. destruct o; auto.
  Qed.

  Lemma in_xkeys:
    ∀ (A: Type) (m: t A) (i k: positive),
    In k (xkeys m i) →
    (∃ 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). ∃ k; auto.
  Qed.

  Lemma xelements_keys_norepet:
    ∀ (A: Type) (m: t A) (i: positive),
    list_norepet (xkeys m i).
  Proof.
    induction m; intros.
  - rewrite xkeys_leaf; constructor.
  - assert (NOTIN1: ¬ In (prev i) (xkeys m1 (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 m2 (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: ∀ x, In x (xkeys m1 (xO i)) → In x (xkeys m2 (xI i)) → False).
    { intros. exploit in_xkeys. eexact H. intros (j1 & EQ1).
      exploit in_xkeys. eexact H0. 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 H0.
    destruct H0. subst x. tauto. eauto. eauto.
  Qed.

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

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

  Theorem elements_canonical_order':
    ∀ (A B: Type) (R: A → B → Prop) (m: t A) (n: t B),
    (∀ 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. unfold elements. generalize 1%positive. revert m n.
    induction m; intros.
  - simpl. rewrite xelements_empty. constructor.
    intros. specialize (H i). rewrite gempty in H. inv H; auto.
  - destruct n as [ | n1 o' n2 ].
  + rewrite (xelements_empty (Node m1 o m2)). simpl; constructor.
    intros. specialize (H i). rewrite gempty in H. inv H; auto.
  + rewrite ! xelements_node. repeat apply list_forall2_app.
    apply IHm1. intros. apply (H (xO i)).
    generalize (H xH); simpl; intros OR; inv OR.
    constructor.
    constructor. auto. constructor.
    apply IHm2. intros. apply (H (xI i)).
  Qed.

  Theorem elements_canonical_order:
    ∀ (A B: Type) (R: A → B → Prop) (m: t A) (n: t B),
    (∀ i x, get i m = Some x → ∃ y , get i n = Some y ∧ R x y ) →
    (∀ i y, get i n = Some y → ∃ 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:
    ∀ (A: Type) (m n: t A),
    (∀ 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. destruct (get i n); constructor; auto.
    induction 1. auto. destruct a1 as [a2 a3]; destruct b1 as [b2 b3]; simpl in ×.
    destruct H0. congruence.
  Qed.

  Lemma xelements_remove:
    ∀ (A: Type) v (m: t A) i j,
    get i m = Some v →
    ∃ l1 l2 ,
    xelements m j nil = l1 ++ (prev (prev_append i j), v ) :: l2
    ∧ xelements (remove i m) j nil = l1 ++ l2.
  Proof.
    induction m; intros.
  - rewrite gleaf in H; discriminate.
  - assert (REMOVE: xelements (remove i (Node m1 o m2)) j nil =
                    xelements (match i with
                               | xH ⇒ Node m1 None m2
                               | xO ii ⇒ Node (remove ii m1) o m2
                               | xI ii ⇒ Node m1 o (remove ii m2) end)
                              j nil).
    {
      destruct i; simpl remove.
      destruct m1; auto. destruct o; auto. destruct (remove i m2); auto.
      destruct o; auto. destruct m2; auto. destruct (remove i m1); auto.
      destruct m1; auto. destruct m2; auto.
    }
    rewrite REMOVE. destruct i; simpl in H.
    + destruct (IHm2 i (xI j) H) as (l1 & l2 & EQ & EQ').
      ∃ (xelements m1 (xO j) nil ++
              match o with None ⇒ nil | Some x ⇒ (prev j, x) :: nil end ++
              l1);
      ∃ l2; split.
      rewrite xelements_node, EQ, ! app_ass. auto.
      rewrite xelements_node, EQ', ! app_ass. auto.
    + destruct (IHm1 i (xO j) H) as (l1 & l2 & EQ & EQ').
      ∃ l1;
      ∃ (l2 ++
              match o with None ⇒ nil | Some x ⇒ (prev j, x) :: nil end ++
              xelements m2 (xI j) nil);
      split.
      rewrite xelements_node, EQ, ! app_ass. auto.
      rewrite xelements_node, EQ', ! app_ass. auto.
    + subst o. ∃ (xelements m1 (xO j) nil); ∃ (xelements m2 (xI j) nil); split.
      rewrite xelements_node. rewrite prev_append_prev. auto.
      rewrite xelements_node; auto.
  Qed.

  Theorem elements_remove:
    ∀ (A: Type) i v (m: t A),
    get i m = Some v →
    ∃ 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.

  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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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: ∀ (x y: t), index x = index y → x = y.
  Parameter eq: ∀ (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:
    ∀ (A: Type) (x: A) (i: X.t), get i (init x) = x.
  Proof.
    intros. unfold get, init. apply PMap.gi.
  Qed.

  Lemma gss:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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: ∀ (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: ∀ (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: ∀ (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: ∀ (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:
    ∀ (A: Type) (i: elt) (x: A), init x i = x.
  Proof.
    intros. reflexivity.
  Qed.
  Lemma gss:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (A: Type) (i: elt), get i (empty A) = None.
  Proof.
    intros. apply PTree.gempty.
  Qed.
  Theorem gss:
    ∀ (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:
    ∀ (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:
    ∀ (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:
    ∀ (A: Type) (i: elt) (m: t A), get i (remove i m) = None.
  Proof.
    intros. apply PTree.grs.
  Qed.
  Theorem gro:
    ∀ (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:
    ∀ (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: ∀ (A: Type), (A → A → bool ) → t A → t A → bool := PTree.beq.
  Theorem beq_sound:
    ∀ (A: Type) (eqA: A → A → bool) (t1 t2: t A),
    beq eqA t1 t2 = true →
    ∀ (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: ∀ (A B C: Type), (option A → option B → option C ) → t A → t B → t C := PTree.combine.
  Theorem gcombine:
    ∀ (A B C: Type) (f: option A → option B → option C),
    f None None = None →
    ∀ (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

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:
  ∀ m,
  (∀ k, T.get k m = None ) →
  P m init.

Hypothesis H_rec:
  ∀ 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 :=
  ∀ m, (∀ 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.
Qed.

Let H_rec':
  ∀ 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.
Qed.

Lemma fold_ind_aux:
  ∀ l,
  (∀ 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:
  ∀ m m' a,
  (∀ 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:
  ∀ 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:
  ∀ 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:
  ∀ 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:
  ∀ m f,
  for_all m f = true ↔ (∀ 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:
  ∀ m f,
  exists_ m f = true ↔ (∃ 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); ∃ x0; ∃ 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).
  ∃ x1; ∃ a1; split; auto. rewrite T.gso; auto. congruence.
  ∃ k; ∃ 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. ∃ x1; ∃ a1; auto.
Qed.

Remark exists_for_all:
  ∀ 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:
  ∀ 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:
  ∀ m f,
  for_all m f = false ↔ (∃ 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); ∃ x; ∃ a; split; auto.
  rewrite negb_true_iff in Q. auto.
  rewrite Q; auto.
Qed.

Lemma exists_false:
  ∀ m f,
  exists_ m f = false ↔ (∀ 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:
  ∀ m1 m2,
  T.beq beqA m1 m2 = false ↔
  ∃ 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).
    ∃ 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).
    ∃ 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 :=
  ∀ 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: ∀ m, Equal m m.
Proof.
  intros; red; intros. destruct (T.get x m); auto. reflexivity.
Qed.

Lemma Equal_sym: ∀ 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: ∀ 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:
  ∀ l k v, T.get k (of_list l) = Some v → In (k , v ) l.
Proof.
  assert (REC: ∀ 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:
  ∀ l k, In k (map fst l) → ∃ v , T.get k (of_list l) = Some v.
Proof.
  assert (REC: ∀ k l m,
            In k (map fst l) ∨ (∃ v , T.get k m = Some v ) →
            ∃ 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:
  ∀ 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:
  ∀ 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:
  ∀ l k v, list_norepet (map fst l) → In (k , v ) l → T.get k (of_list l) = Some v.
Proof.
  assert (REC: ∀ 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:
  ∀ 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:
  ∀ (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: ∀ l1 l2, list_forall2 R' l1 l2 →
               ∀ 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).