Library Coq.MSets.MSetRBT


MSetRBT : Implementation of MSetInterface via Red-Black trees

Initial author: Andrew W. Appel, 2011. Extra modifications by: Pierre Letouzey
The design decisions behind this implementation are described here:
  • Efficient Verified Red-Black Trees, by Andrew W. Appel, September 2011. http://www.cs.princeton.edu/~appel/papers/redblack.pdf
Additional suggested reading:
  • Red-Black Trees in a Functional Setting by Chris Okasaki. Journal of Functional Programming, 9(4):471-477, July 1999. http://www.eecs.usma.edu/webs/people/okasaki/jfp99redblack.pdf
  • Red-black trees with types, by Stefan Kahrs. Journal of Functional Programming, 11(4), 425-432, 2001.
  • Functors for Proofs and Programs, by J.-C. Filliatre and P. Letouzey. ESOP'04: European Symposium on Programming, pp. 370-384, 2004. http://www.lri.fr/~filliatr/ftp/publis/fpp.ps.gz

Require MSetGenTree.
Require Import Bool List BinPos Pnat Setoid SetoidList PeanoNat.
Local Open Scope list_scope.


An extra function not (yet?) in MSetInterface.S

Module Type MSetRemoveMin (Import M:MSetInterface.S).

 Parameter remove_min : t -> option (elt * t).

 Axiom remove_min_spec1 : forall s k s',
  remove_min s = Some (k,s') ->
   min_elt s = Some k /\ remove k s [=] s'.

 Axiom remove_min_spec2 : forall s, remove_min s = None -> Empty s.

End MSetRemoveMin.

The type of color annotation.

Inductive color := Red | Black.

Module Color.
 Definition t := color.
End Color.

Ops : the pure functions


Module Ops (X:Orders.OrderedType) <: MSetInterface.Ops X.

Generic trees instantiated with color

We reuse a generic definition of trees where the information parameter is a color. Functions like mem or fold are also provided by this generic functor.

Include MSetGenTree.Ops X Color.

Definition t := tree.

Basic tree


Definition singleton (k: elt) : tree := Bk Leaf k Leaf.

Changing root color


Definition makeBlack t :=
 match t with
 | Leaf => Leaf
 | Node _ a x b => Bk a x b
 end.

Definition makeRed t :=
 match t with
 | Leaf => Leaf
 | Node _ a x b => Rd a x b
 end.

Balancing

We adapt when one side is not a true red-black tree. Both sides have the same black depth.

Definition lbal l k r :=
 match l with
 | Rd (Rd a x b) y c => Rd (Bk a x b) y (Bk c k r)
 | Rd a x (Rd b y c) => Rd (Bk a x b) y (Bk c k r)
 | _ => Bk l k r
 end.

Definition rbal l k r :=
 match r with
 | Rd (Rd b y c) z d => Rd (Bk l k b) y (Bk c z d)
 | Rd b y (Rd c z d) => Rd (Bk l k b) y (Bk c z d)
 | _ => Bk l k r
 end.

A variant of rbal, with reverse pattern order. Is it really useful ? Should we always use it ?

Definition rbal' l k r :=
 match r with
 | Rd b y (Rd c z d) => Rd (Bk l k b) y (Bk c z d)
 | Rd (Rd b y c) z d => Rd (Bk l k b) y (Bk c z d)
 | _ => Bk l k r
 end.

Balancing with different black depth. One side is almost a red-black tree, while the other is a true red-black tree, but with black depth + 1. Used in deletion.

Definition lbalS l k r :=
 match l with
 | Rd a x b => Rd (Bk a x b) k r
 | _ =>
   match r with
   | Bk a y b => rbal' l k (Rd a y b)
   | Rd (Bk a y b) z c => Rd (Bk l k a) y (rbal' b z (makeRed c))
   | _ => Rd l k r
   end
 end.

Definition rbalS l k r :=
 match r with
 | Rd b y c => Rd l k (Bk b y c)
 | _ =>
   match l with
   | Bk a x b => lbal (Rd a x b) k r
   | Rd a x (Bk b y c) => Rd (lbal (makeRed a) x b) y (Bk c k r)
   | _ => Rd l k r
   end
 end.

Insertion


Fixpoint ins x s :=
 match s with
 | Leaf => Rd Leaf x Leaf
 | Node c l y r =>
   match X.compare x y with
   | Eq => s
   | Lt =>
     match c with
     | Red => Rd (ins x l) y r
     | Black => lbal (ins x l) y r
     end
   | Gt =>
     match c with
     | Red => Rd l y (ins x r)
     | Black => rbal l y (ins x r)
     end
   end
 end.

Definition add x s := makeBlack (ins x s).

Deletion


Fixpoint append (l:tree) : tree -> tree :=
 match l with
 | Leaf => fun r => r
 | Node lc ll lx lr =>
   fix append_l (r:tree) : tree :=
   match r with
   | Leaf => l
   | Node rc rl rx rr =>
     match lc, rc with
     | Red, Red =>
       let lrl := append lr rl in
       match lrl with
       | Rd lr' x rl' => Rd (Rd ll lx lr') x (Rd rl' rx rr)
       | _ => Rd ll lx (Rd lrl rx rr)
       end
     | Black, Black =>
       let lrl := append lr rl in
       match lrl with
       | Rd lr' x rl' => Rd (Bk ll lx lr') x (Bk rl' rx rr)
       | _ => lbalS ll lx (Bk lrl rx rr)
       end
     | Black, Red => Rd (append_l rl) rx rr
     | Red, Black => Rd ll lx (append lr r)
     end
   end
 end.

Fixpoint del x t :=
 match t with
 | Leaf => Leaf
 | Node _ a y b =>
   match X.compare x y with
   | Eq => append a b
   | Lt =>
     match a with
     | Bk _ _ _ => lbalS (del x a) y b
     | _ => Rd (del x a) y b
     end
   | Gt =>
     match b with
     | Bk _ _ _ => rbalS a y (del x b)
     | _ => Rd a y (del x b)
     end
   end
 end.

Definition remove x t := makeBlack (del x t).

Removing minimal element


Fixpoint delmin l x r : (elt * tree) :=
 match l with
 | Leaf => (x,r)
 | Node lc ll lx lr =>
   let (k,l') := delmin ll lx lr in
   match lc with
   | Black => (k, lbalS l' x r)
   | Red => (k, Rd l' x r)
   end
 end.

Definition remove_min t : option (elt * tree) :=
 match t with
 | Leaf => None
 | Node _ l x r =>
   let (k,t) := delmin l x r in
   Some (k, makeBlack t)
 end.

Tree-ification

We rebuild a tree of size if pred then n-1 else n as soon as the list l has enough elements

Definition bogus : tree * list elt := (Leaf, nil).

Notation treeify_t := (list elt -> tree * list elt).

Definition treeify_zero : treeify_t :=
 fun acc => (Leaf,acc).

Definition treeify_one : treeify_t :=
 fun acc => match acc with
 | x::acc => (Rd Leaf x Leaf, acc)
 | _ => bogus
 end.

Definition treeify_cont (f g : treeify_t) : treeify_t :=
 fun acc =>
 match f acc with
 | (l, x::acc) =>
   match g acc with
   | (r, acc) => (Bk l x r, acc)
   end
 | _ => bogus
 end.

Fixpoint treeify_aux (pred:bool)(n: positive) : treeify_t :=
 match n with
 | xH => if pred then treeify_zero else treeify_one
 | xO n => treeify_cont (treeify_aux pred n) (treeify_aux true n)
 | xI n => treeify_cont (treeify_aux false n) (treeify_aux pred n)
 end.

Fixpoint plength_aux (l:list elt)(p:positive) := match l with
 | nil => p
 | _::l => plength_aux l (Pos.succ p)
end.

Definition plength l := plength_aux l 1.

Definition treeify (l:list elt) :=
 fst (treeify_aux true (plength l) l).

Filtering


Fixpoint filter_aux (f: elt -> bool) s acc :=
 match s with
 | Leaf => acc
 | Node _ l k r =>
   let acc := filter_aux f r acc in
   if f k then filter_aux f l (k::acc)
   else filter_aux f l acc
 end.

Definition filter (f: elt -> bool) (s: t) : t :=
 treeify (filter_aux f s nil).

Fixpoint partition_aux (f: elt -> bool) s acc1 acc2 :=
 match s with
 | Leaf => (acc1,acc2)
 | Node _ sl k sr =>
   let (acc1, acc2) := partition_aux f sr acc1 acc2 in
   if f k then partition_aux f sl (k::acc1) acc2
   else partition_aux f sl acc1 (k::acc2)
 end.

Definition partition (f: elt -> bool) (s:t) : t*t :=
  let (ok,ko) := partition_aux f s nil nil in
  (treeify ok, treeify ko).

Union, intersection, difference

union of the elements of l1 and l2 into a third acc list.

Fixpoint union_list l1 : list elt -> list elt -> list elt :=
 match l1 with
 | nil => @rev_append _
 | x::l1' =>
    fix union_l1 l2 acc :=
    match l2 with
    | nil => rev_append l1 acc
    | y::l2' =>
       match X.compare x y with
       | Eq => union_list l1' l2' (x::acc)
       | Lt => union_l1 l2' (y::acc)
       | Gt => union_list l1' l2 (x::acc)
       end
    end
 end.

Definition linear_union s1 s2 :=
  treeify (union_list (rev_elements s1) (rev_elements s2) nil).

Fixpoint inter_list l1 : list elt -> list elt -> list elt :=
 match l1 with
 | nil => fun _ acc => acc
 | x::l1' =>
    fix inter_l1 l2 acc :=
    match l2 with
    | nil => acc
    | y::l2' =>
       match X.compare x y with
       | Eq => inter_list l1' l2' (x::acc)
       | Lt => inter_l1 l2' acc
       | Gt => inter_list l1' l2 acc
       end
    end
 end.

Definition linear_inter s1 s2 :=
  treeify (inter_list (rev_elements s1) (rev_elements s2) nil).

Fixpoint diff_list l1 : list elt -> list elt -> list elt :=
 match l1 with
 | nil => fun _ acc => acc
 | x::l1' =>
    fix diff_l1 l2 acc :=
    match l2 with
    | nil => rev_append l1 acc
    | y::l2' =>
       match X.compare x y with
       | Eq => diff_list l1' l2' acc
       | Lt => diff_l1 l2' acc
       | Gt => diff_list l1' l2 (x::acc)
       end
    end
 end.

Definition linear_diff s1 s2 :=
  treeify (diff_list (rev_elements s1) (rev_elements s2) nil).

compare_height returns:
  • Lt if height s2 is at least twice height s1;
  • Gt if height s1 is at least twice height s2;
  • Eq if heights are approximately equal.
Warning: this is not an equivalence relation! but who cares....

Definition skip_red t :=
 match t with
 | Rd t' _ _ => t'
 | _ => t
 end.

Definition skip_black t :=
 match skip_red t with
 | Bk t' _ _ => t'
 | t' => t'
 end.

Fixpoint compare_height (s1x s1 s2 s2x: tree) : comparison :=
 match skip_red s1x, skip_red s1, skip_red s2, skip_red s2x with
 | Node _ s1x' _ _, Node _ s1' _ _, Node _ s2' _ _, Node _ s2x' _ _ =>
   compare_height (skip_black s1x') s1' s2' (skip_black s2x')
 | _, Leaf, _, Node _ _ _ _ => Lt
 | Node _ _ _ _, _, Leaf, _ => Gt
 | Node _ s1x' _ _, Node _ s1' _ _, Node _ s2' _ _, Leaf =>
   compare_height (skip_black s1x') s1' s2' Leaf
 | Leaf, Node _ s1' _ _, Node _ s2' _ _, Node _ s2x' _ _ =>
   compare_height Leaf s1' s2' (skip_black s2x')
 | _, _, _, _ => Eq
 end.

When one tree is quite smaller than the other, we simply adds repeatively all its elements in the big one. For trees of comparable height, we rather use linear_union.

Definition union (t1 t2: t) : t :=
 match compare_height t1 t1 t2 t2 with
 | Lt => fold add t1 t2
 | Gt => fold add t2 t1
 | Eq => linear_union t1 t2
 end.

Definition diff (t1 t2: t) : t :=
 match compare_height t1 t1 t2 t2 with
 | Lt => filter (fun k => negb (mem k t2)) t1
 | Gt => fold remove t2 t1
 | Eq => linear_diff t1 t2
 end.

Definition inter (t1 t2: t) : t :=
 match compare_height t1 t1 t2 t2 with
 | Lt => filter (fun k => mem k t2) t1
 | Gt => filter (fun k => mem k t1) t2
 | Eq => linear_inter t1 t2
 end.

End Ops.

MakeRaw : the pure functions and their specifications


Module Type MakeRaw (X:Orders.OrderedType) <: MSetInterface.RawSets X.
Include Ops X.

Generic definition of binary-search-trees and proofs of specifications for generic functions such as mem or fold.

Include MSetGenTree.Props X Color.


Local Hint Immediate MX.eq_sym.
Local Hint Unfold In lt_tree gt_tree Ok.
Local Hint Constructors InT bst.
Local Hint Resolve MX.eq_refl MX.eq_trans MX.lt_trans ok.
Local Hint Resolve lt_leaf gt_leaf lt_tree_node gt_tree_node.
Local Hint Resolve lt_tree_not_in lt_tree_trans gt_tree_not_in gt_tree_trans.
Local Hint Resolve elements_spec2.

Singleton set


Lemma singleton_spec x y : InT y (singleton x) <-> X.eq y x.

Instance singleton_ok x : Ok (singleton x).

makeBlack, MakeRed


Lemma makeBlack_spec s x : InT x (makeBlack s) <-> InT x s.

Lemma makeRed_spec s x : InT x (makeRed s) <-> InT x s.

Instance makeBlack_ok s `{Ok s} : Ok (makeBlack s).

Instance makeRed_ok s `{Ok s} : Ok (makeRed s).

Generic handling for red-matching and red-red-matching


Definition isblack t :=
 match t with Bk _ _ _ => True | _ => False end.

Definition notblack t :=
 match t with Bk _ _ _ => False | _ => True end.

Definition notred t :=
 match t with Rd _ _ _ => False | _ => True end.

Definition rcase {A} f g t : A :=
 match t with
 | Rd a x b => f a x b
 | _ => g t
 end.

Inductive rspec {A} f g : tree -> A -> Prop :=
 | rred a x b : rspec f g (Rd a x b) (f a x b)
 | relse t : notred t -> rspec f g t (g t).

Fact rmatch {A} f g t : rspec (A:=A) f g t (rcase f g t).

Definition rrcase {A} f g t : A :=
 match t with
 | Rd (Rd a x b) y c => f a x b y c
 | Rd a x (Rd b y c) => f a x b y c
 | _ => g t
 end.

Notation notredred := (rrcase (fun _ _ _ _ _ => False) (fun _ => True)).

Inductive rrspec {A} f g : tree -> A -> Prop :=
 | rrleft a x b y c : rrspec f g (Rd (Rd a x b) y c) (f a x b y c)
 | rrright a x b y c : rrspec f g (Rd a x (Rd b y c)) (f a x b y c)
 | rrelse t : notredred t -> rrspec f g t (g t).

Fact rrmatch {A} f g t : rrspec (A:=A) f g t (rrcase f g t).

Definition rrcase' {A} f g t : A :=
 match t with
 | Rd a x (Rd b y c) => f a x b y c
 | Rd (Rd a x b) y c => f a x b y c
 | _ => g t
 end.

Fact rrmatch' {A} f g t : rrspec (A:=A) f g t (rrcase' f g t).

Balancing operations are instances of generic match

Fact lbal_match l k r :
 rrspec
   (fun a x b y c => Rd (Bk a x b) y (Bk c k r))
   (fun l => Bk l k r)
   l
   (lbal l k r).

Fact rbal_match l k r :
 rrspec
   (fun a x b y c => Rd (Bk l k a) x (Bk b y c))
   (fun r => Bk l k r)
   r
   (rbal l k r).

Fact rbal'_match l k r :
 rrspec
   (fun a x b y c => Rd (Bk l k a) x (Bk b y c))
   (fun r => Bk l k r)
   r
   (rbal' l k r).

Fact lbalS_match l x r :
 rspec
  (fun a y b => Rd (Bk a y b) x r)
  (fun l =>
    match r with
    | Bk a y b => rbal' l x (Rd a y b)
    | Rd (Bk a y b) z c => Rd (Bk l x a) y (rbal' b z (makeRed c))
    | _ => Rd l x r
    end)
  l
  (lbalS l x r).

Fact rbalS_match l x r :
 rspec
  (fun a y b => Rd l x (Bk a y b))
  (fun r =>
    match l with
    | Bk a y b => lbal (Rd a y b) x r
    | Rd a y (Bk b z c) => Rd (lbal (makeRed a) y b) z (Bk c x r)
    | _ => Rd l x r
    end)
  r
  (rbalS l x r).

Balancing for insertion


Lemma lbal_spec l x r y :
   InT y (lbal l x r) <-> X.eq y x \/ InT y l \/ InT y r.

Instance lbal_ok l x r `(Ok l, Ok r, lt_tree x l, gt_tree x r) :
 Ok (lbal l x r).

Lemma rbal_spec l x r y :
   InT y (rbal l x r) <-> X.eq y x \/ InT y l \/ InT y r.

Instance rbal_ok l x r `(Ok l, Ok r, lt_tree x l, gt_tree x r) :
 Ok (rbal l x r).

Lemma rbal'_spec l x r y :
   InT y (rbal' l x r) <-> X.eq y x \/ InT y l \/ InT y r.

Instance rbal'_ok l x r `(Ok l, Ok r, lt_tree x l, gt_tree x r) :
 Ok (rbal' l x r).

Hint Rewrite In_node_iff In_leaf_iff
 makeRed_spec makeBlack_spec lbal_spec rbal_spec rbal'_spec : rb.

Ltac descolor := destruct_all Color.t.
Ltac destree t := destruct t as [|[|] ? ? ?].
Ltac autorew := autorewrite with rb.
Tactic Notation "autorew" "in" ident(H) := autorewrite with rb in H.

Insertion


Lemma ins_spec : forall s x y,
 InT y (ins x s) <-> X.eq y x \/ InT y s.
Hint Rewrite ins_spec : rb.

Instance ins_ok s x `{Ok s} : Ok (ins x s).

Lemma add_spec' s x y :
 InT y (add x s) <-> X.eq y x \/ InT y s.

Hint Rewrite add_spec' : rb.

Lemma add_spec s x y `{Ok s} :
 InT y (add x s) <-> X.eq y x \/ InT y s.

Instance add_ok s x `{Ok s} : Ok (add x s).

Balancing for deletion


Lemma lbalS_spec l x r y :
  InT y (lbalS l x r) <-> X.eq y x \/ InT y l \/ InT y r.

Instance lbalS_ok l x r :
 forall `(Ok l, Ok r, lt_tree x l, gt_tree x r), Ok (lbalS l x r).

Lemma rbalS_spec l x r y :
  InT y (rbalS l x r) <-> X.eq y x \/ InT y l \/ InT y r.

Instance rbalS_ok l x r :
 forall `(Ok l, Ok r, lt_tree x l, gt_tree x r), Ok (rbalS l x r).

Hint Rewrite lbalS_spec rbalS_spec : rb.

Append for deletion


Ltac append_tac l r :=
 induction l as [| lc ll _ lx lr IHlr];
 [intro r; simpl
 |induction r as [| rc rl IHrl rx rr _];
   [simpl
   |destruct lc, rc;
     [specialize (IHlr rl); clear IHrl
     |simpl;
      assert (Hr:notred (Bk rl rx rr)) by (simpl; trivial);
      set (r:=Bk rl rx rr) in *; clearbody r; clear IHrl rl rx rr;
      specialize (IHlr r)
     |change (append _ _) with (Rd (append (Bk ll lx lr) rl) rx rr);
      assert (Hl:notred (Bk ll lx lr)) by (simpl; trivial);
      set (l:=Bk ll lx lr) in *; clearbody l; clear IHlr ll lx lr
     |specialize (IHlr rl); clear IHrl]]].

Fact append_rr_match ll lx lr rl rx rr :
 rspec
  (fun a x b => Rd (Rd ll lx a) x (Rd b rx rr))
  (fun t => Rd ll lx (Rd t rx rr))
  (append lr rl)
  (append (Rd ll lx lr) (Rd rl rx rr)).

Fact append_bb_match ll lx lr rl rx rr :
 rspec
  (fun a x b => Rd (Bk ll lx a) x (Bk b rx rr))
  (fun t => lbalS ll lx (Bk t rx rr))
  (append lr rl)
  (append (Bk ll lx lr) (Bk rl rx rr)).

Lemma append_spec l r x :
 InT x (append l r) <-> InT x l \/ InT x r.

Hint Rewrite append_spec : rb.

Lemma append_ok : forall x l r `{Ok l, Ok r},
 lt_tree x l -> gt_tree x r -> Ok (append l r).

Deletion


Lemma del_spec : forall s x y `{Ok s},
 InT y (del x s) <-> InT y s /\ ~X.eq y x.

Hint Rewrite del_spec : rb.

Instance del_ok s x `{Ok s} : Ok (del x s).

Lemma remove_spec s x y `{Ok s} :
 InT y (remove x s) <-> InT y s /\ ~X.eq y x.

Hint Rewrite remove_spec : rb.

Instance remove_ok s x `{Ok s} : Ok (remove x s).

Removing the minimal element


Lemma delmin_spec l y r c x s' `{O : Ok (Node c l y r)} :
 delmin l y r = (x,s') ->
  min_elt (Node c l y r) = Some x /\ del x (Node c l y r) = s'.

Lemma remove_min_spec1 s x s' `{Ok s}:
 remove_min s = Some (x,s') ->
  min_elt s = Some x /\ remove x s = s'.

Lemma remove_min_spec2 s : remove_min s = None -> Empty s.

Lemma remove_min_ok (s:t) `{Ok s}:
 match remove_min s with
 | Some (_,s') => Ok s'
 | None => True
 end.

Treeify


Notation ifpred p n := (if p then pred n else n%nat).

Definition treeify_invariant size (f:treeify_t) :=
 forall acc,
 size <= length acc ->
 let (t,acc') := f acc in
 cardinal t = size /\ acc = elements t ++ acc'.

Lemma treeify_zero_spec : treeify_invariant 0 treeify_zero.

Lemma treeify_one_spec : treeify_invariant 1 treeify_one.

Lemma treeify_cont_spec f g size1 size2 size :
 treeify_invariant size1 f ->
 treeify_invariant size2 g ->
 size = S (size1 + size2) ->
 treeify_invariant size (treeify_cont f g).

Lemma treeify_aux_spec n (p:bool) :
 treeify_invariant (ifpred p (Pos.to_nat n)) (treeify_aux p n).

Lemma plength_aux_spec l p :
  Pos.to_nat (plength_aux l p) = length l + Pos.to_nat p.

Lemma plength_spec l : Pos.to_nat (plength l) = S (length l).

Lemma treeify_elements l : elements (treeify l) = l.

Lemma treeify_spec x l : InT x (treeify l) <-> InA X.eq x l.

Lemma treeify_ok l : sort X.lt l -> Ok (treeify l).

Filter


Lemma filter_app A f (l l':list A) :
 List.filter f (l ++ l') = List.filter f l ++ List.filter f l'.

Lemma filter_aux_elements s f acc :
 filter_aux f s acc = List.filter f (elements s) ++ acc.

Lemma filter_elements s f :
 elements (filter f s) = List.filter f (elements s).

Lemma filter_spec s x f :
 Proper (X.eq==>Logic.eq) f ->
 (InT x (filter f s) <-> InT x s /\ f x = true).

Instance filter_ok s f `(Ok s) : Ok (filter f s).

Partition


Lemma partition_aux_spec s f acc1 acc2 :
 partition_aux f s acc1 acc2 =
  (filter_aux f s acc1, filter_aux (fun x => negb (f x)) s acc2).

Lemma partition_spec s f :
 partition f s = (filter f s, filter (fun x => negb (f x)) s).

Lemma partition_spec1 s f :
 Proper (X.eq==>Logic.eq) f ->
 Equal (fst (partition f s)) (filter f s).

Lemma partition_spec2 s f :
 Proper (X.eq==>Logic.eq) f ->
 Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).

Instance partition_ok1 s f `(Ok s) : Ok (fst (partition f s)).

Instance partition_ok2 s f `(Ok s) : Ok (snd (partition f s)).

An invariant for binary list functions with accumulator.


Ltac inA :=
 rewrite ?InA_app_iff, ?InA_cons, ?InA_nil, ?InA_rev in *; auto_tc.

Record INV l1 l2 acc : Prop := {
 l1_sorted : sort X.lt (rev l1);
 l2_sorted : sort X.lt (rev l2);
 acc_sorted : sort X.lt acc;
 l1_lt_acc x y : InA X.eq x l1 -> InA X.eq y acc -> X.lt x y;
 l2_lt_acc x y : InA X.eq x l2 -> InA X.eq y acc -> X.lt x y}.
Local Hint Resolve l1_sorted l2_sorted acc_sorted.

Lemma INV_init s1 s2 `(Ok s1, Ok s2) :
 INV (rev_elements s1) (rev_elements s2) nil.

Lemma INV_sym l1 l2 acc : INV l1 l2 acc -> INV l2 l1 acc.

Lemma INV_drop x1 l1 l2 acc :
  INV (x1 :: l1) l2 acc -> INV l1 l2 acc.

Lemma INV_eq x1 x2 l1 l2 acc :
  INV (x1 :: l1) (x2 :: l2) acc -> X.eq x1 x2 ->
  INV l1 l2 (x1 :: acc).

Lemma INV_lt x1 x2 l1 l2 acc :
  INV (x1 :: l1) (x2 :: l2) acc -> X.lt x1 x2 ->
  INV (x1 :: l1) l2 (x2 :: acc).

Lemma INV_rev l1 l2 acc :
 INV l1 l2 acc -> Sorted X.lt (rev_append l1 acc).

union


Lemma union_list_ok l1 l2 acc :
 INV l1 l2 acc -> sort X.lt (union_list l1 l2 acc).

Instance linear_union_ok s1 s2 `(Ok s1, Ok s2) :
 Ok (linear_union s1 s2).

Instance fold_add_ok s1 s2 `(Ok s1, Ok s2) :
 Ok (fold add s1 s2).

Instance union_ok s1 s2 `(Ok s1, Ok s2) : Ok (union s1 s2).

Lemma union_list_spec x l1 l2 acc :
 InA X.eq x (union_list l1 l2 acc) <->
  InA X.eq x l1 \/ InA X.eq x l2 \/ InA X.eq x acc.

Lemma linear_union_spec s1 s2 x :
 InT x (linear_union s1 s2) <-> InT x s1 \/ InT x s2.

Lemma fold_add_spec s1 s2 x :
 InT x (fold add s1 s2) <-> InT x s1 \/ InT x s2.

Lemma union_spec' s1 s2 x :
 InT x (union s1 s2) <-> InT x s1 \/ InT x s2.

Lemma union_spec : forall s1 s2 y `{Ok s1, Ok s2},
 (InT y (union s1 s2) <-> InT y s1 \/ InT y s2).

inter


Lemma inter_list_ok l1 l2 acc :
 INV l1 l2 acc -> sort X.lt (inter_list l1 l2 acc).

Instance linear_inter_ok s1 s2 `(Ok s1, Ok s2) :
 Ok (linear_inter s1 s2).

Instance inter_ok s1 s2 `(Ok s1, Ok s2) : Ok (inter s1 s2).

Lemma inter_list_spec x l1 l2 acc :
 sort X.lt (rev l1) ->
 sort X.lt (rev l2) ->
 (InA X.eq x (inter_list l1 l2 acc) <->
   (InA X.eq x l1 /\ InA X.eq x l2) \/ InA X.eq x acc).

Lemma linear_inter_spec s1 s2 x `(Ok s1, Ok s2) :
 InT x (linear_inter s1 s2) <-> InT x s1 /\ InT x s2.


Lemma inter_spec s1 s2 y `{Ok s1, Ok s2} :
 InT y (inter s1 s2) <-> InT y s1 /\ InT y s2.

difference


Lemma diff_list_ok l1 l2 acc :
 INV l1 l2 acc -> sort X.lt (diff_list l1 l2 acc).

Instance diff_inter_ok s1 s2 `(Ok s1, Ok s2) :
 Ok (linear_diff s1 s2).

Instance fold_remove_ok s1 s2 `(Ok s2) :
 Ok (fold remove s1 s2).

Instance diff_ok s1 s2 `(Ok s1, Ok s2) : Ok (diff s1 s2).

Lemma diff_list_spec x l1 l2 acc :
 sort X.lt (rev l1) ->
 sort X.lt (rev l2) ->
 (InA X.eq x (diff_list l1 l2 acc) <->
   (InA X.eq x l1 /\ ~InA X.eq x l2) \/ InA X.eq x acc).

Lemma linear_diff_spec s1 s2 x `(Ok s1, Ok s2) :
 InT x (linear_diff s1 s2) <-> InT x s1 /\ ~InT x s2.

Lemma fold_remove_spec s1 s2 x `(Ok s2) :
  InT x (fold remove s1 s2) <-> InT x s2 /\ ~InT x s1.

Lemma diff_spec s1 s2 y `{Ok s1, Ok s2} :
 InT y (diff s1 s2) <-> InT y s1 /\ ~InT y s2.

End MakeRaw.

Balancing properties

We now prove that all operations preserve a red-black invariant, and that trees have hence a logarithmic depth.

Module BalanceProps(X:Orders.OrderedType)(Import M : MakeRaw X).

Import M.MX.

Red-Black invariants

In a red-black tree :
  • a red node has no red children
  • the black depth at each node is the same along all paths.
The black depth is here an argument of the predicate.

Inductive rbt : nat -> tree -> Prop :=
 | RB_Leaf : rbt 0 Leaf
 | RB_Rd n l k r :
   notred l -> notred r -> rbt n l -> rbt n r -> rbt n (Rd l k r)
 | RB_Bk n l k r : rbt n l -> rbt n r -> rbt (S n) (Bk l k r).

A red-red tree is almost a red-black tree, except that it has a red root node which may have red children. Note that a red-red tree is hence non-empty, and all its strict subtrees are red-black.

Inductive rrt (n:nat) : tree -> Prop :=
 | RR_Rd l k r : rbt n l -> rbt n r -> rrt n (Rd l k r).

An almost-red-black tree is almost a red-black tree, except that it's permitted to have two red nodes in a row at the very root (only). We implement this notion by saying that a quasi-red-black tree is either a red-black tree or a red-red tree.

Inductive arbt (n:nat)(t:tree) : Prop :=
 | ARB_RB : rbt n t -> arbt n t
 | ARB_RR : rrt n t -> arbt n t.

The main exported invariant : being a red-black tree for some black depth.

Class Rbt (t:tree) := RBT : exists d, rbt d t.

Basic tactics and results about red-black


Scheme rbt_ind := Induction for rbt Sort Prop.
Local Hint Constructors rbt rrt arbt.
Local Hint Extern 0 (notred _) => (exact I).
Ltac invrb := intros; invtree rrt; invtree rbt; try contradiction.
Ltac desarb := match goal with H:arbt _ _ |- _ => destruct H end.
Ltac nonzero n := destruct n as [|n]; [try split; invrb|].

Lemma rr_nrr_rb n t :
 rrt n t -> notredred t -> rbt n t.

Local Hint Resolve rr_nrr_rb.

Lemma arb_nrr_rb n t :
 arbt n t -> notredred t -> rbt n t.

Lemma arb_nr_rb n t :
 arbt n t -> notred t -> rbt n t.

Local Hint Resolve arb_nrr_rb arb_nr_rb.

A Red-Black tree has indeed a logarithmic depth


Definition redcarac s := rcase (fun _ _ _ => 1) (fun _ => 0) s.

Lemma rb_maxdepth s n : rbt n s -> maxdepth s <= 2*n + redcarac s.

Lemma rb_mindepth s n : rbt n s -> n + redcarac s <= mindepth s.

Lemma maxdepth_upperbound s : Rbt s ->
 maxdepth s <= 2 * Nat.log2 (S (cardinal s)).

Lemma maxdepth_lowerbound s : s<>Leaf ->
 Nat.log2 (cardinal s) < maxdepth s.

Singleton


Lemma singleton_rb x : Rbt (singleton x).

makeBlack and makeRed


Lemma makeBlack_rb n t : arbt n t -> Rbt (makeBlack t).

Lemma makeRed_rr t n :
 rbt (S n) t -> notred t -> rrt n (makeRed t).

Balancing


Lemma lbal_rb n l k r :
 arbt n l -> rbt n r -> rbt (S n) (lbal l k r).

Lemma rbal_rb n l k r :
 rbt n l -> arbt n r -> rbt (S n) (rbal l k r).

Lemma rbal'_rb n l k r :
 rbt n l -> arbt n r -> rbt (S n) (rbal' l k r).

Lemma lbalS_rb n l x r :
 arbt n l -> rbt (S n) r -> notred r -> rbt (S n) (lbalS l x r).

Lemma lbalS_arb n l x r :
 arbt n l -> rbt (S n) r -> arbt (S n) (lbalS l x r).

Lemma rbalS_rb n l x r :
 rbt (S n) l -> notred l -> arbt n r -> rbt (S n) (rbalS l x r).

Lemma rbalS_arb n l x r :
 rbt (S n) l -> arbt n r -> arbt (S n) (rbalS l x r).

Insertion

The next lemmas combine simultaneous results about rbt and arbt. A first solution here: statement with if ... then ... else

Definition ifred s (A B:Prop) := rcase (fun _ _ _ => A) (fun _ => B) s.

Lemma ifred_notred s A B : notred s -> (ifred s A B <-> B).

Lemma ifred_or s A B : ifred s A B -> A\/B.

Lemma ins_rr_rb x s n : rbt n s ->
 ifred s (rrt n (ins x s)) (rbt n (ins x s)).

Lemma ins_arb x s n : rbt n s -> arbt n (ins x s).

Instance add_rb x s : Rbt s -> Rbt (add x s).

Deletion

A second approach here: statement with ... /\ ...

Lemma append_arb_rb n l r : rbt n l -> rbt n r ->
 (arbt n (append l r)) /\
 (notred l -> notred r -> rbt n (append l r)).

A third approach : Lemma ... with ...

Lemma del_arb s x n : rbt (S n) s -> isblack s -> arbt n (del x s)
with del_rb s x n : rbt n s -> notblack s -> rbt n (del x s).
Instance remove_rb s x : Rbt s -> Rbt (remove x s).

Treeify


Definition treeify_rb_invariant size depth (f:treeify_t) :=
 forall acc,
 size <= length acc ->
  rbt depth (fst (f acc)) /\
  size + length (snd (f acc)) = length acc.

Lemma treeify_zero_rb : treeify_rb_invariant 0 0 treeify_zero.

Lemma treeify_one_rb : treeify_rb_invariant 1 0 treeify_one.

Lemma treeify_cont_rb f g size1 size2 size d :
 treeify_rb_invariant size1 d f ->
 treeify_rb_invariant size2 d g ->
 size = S (size1 + size2) ->
 treeify_rb_invariant size (S d) (treeify_cont f g).

Lemma treeify_aux_rb n :
 exists d, forall (b:bool),
  treeify_rb_invariant (ifpred b (Pos.to_nat n)) d (treeify_aux b n).

The black depth of treeify l is actually a log2, but we don't need to mention that.

Instance treeify_rb l : Rbt (treeify l).

Filtering


Instance filter_rb f s : Rbt (filter f s).

Instance partition_rb1 f s : Rbt (fst (partition f s)).

Instance partition_rb2 f s : Rbt (snd (partition f s)).

Union, intersection, difference


Instance fold_add_rb s1 s2 : Rbt s2 -> Rbt (fold add s1 s2).

Instance fold_remove_rb s1 s2 : Rbt s2 -> Rbt (fold remove s1 s2).

Lemma union_rb s1 s2 : Rbt s1 -> Rbt s2 -> Rbt (union s1 s2).

Lemma inter_rb s1 s2 : Rbt s1 -> Rbt s2 -> Rbt (inter s1 s2).

Lemma diff_rb s1 s2 : Rbt s1 -> Rbt s2 -> Rbt (diff s1 s2).

End BalanceProps.

Final Encapsulation

Now, in order to really provide a functor implementing S, we need to encapsulate everything into a type of binary search trees. They also happen to be well-balanced, but this has no influence on the correctness of operations, so we won't state this here, see BalanceProps if you need more than just the MSet interface.

Module Type MSetInterface_S_Ext := MSetInterface.S <+ MSetRemoveMin.

Module Make (X: Orders.OrderedType) <:
 MSetInterface_S_Ext with Module E := X.
 Module Raw. Include MakeRaw X. End Raw.
 Include MSetInterface.Raw2Sets X Raw.

 Definition opt_ok (x:option (elt * Raw.t)) :=
  match x with Some (_,s) => Raw.Ok s | None => True end.

 Definition mk_opt_t (x: option (elt * Raw.t))(P: opt_ok x) :
   option (elt * t) :=
 match x as o return opt_ok o -> option (elt * t) with
 | Some (k,s') => fun P : Raw.Ok s' => Some (k, Mkt s')
 | None => fun _ => None
 end P.

 Definition remove_min s : option (elt * t) :=
  mk_opt_t (Raw.remove_min (this s)) (Raw.remove_min_ok s).

 Lemma remove_min_spec1 s x s' :
  remove_min s = Some (x,s') ->
   min_elt s = Some x /\ Equal (remove x s) s'.

 Lemma remove_min_spec2 s : remove_min s = None -> Empty s.

End Make.