The Coq Proof Assistant

# Library Coq.FSets.FMapPositive

# FMapPositive : an implementation of FMapInterface for positive keys.

Require Import Bool OrderedType ZArith OrderedType OrderedTypeEx FMapInterface.
Set Implicit Arguments.
Local Open Scope positive_scope.
This file is an adaptation to the FMap framework of a work by Xavier Leroy and Sandrine Blazy (used for building certified compilers). Keys are of type positive, and maps are binary trees: the sequence of binary digits of a positive number corresponds to a path in such a tree. This is quite similar to the IntMap library, except that no path compression is implemented, and that the current file is simple enough to be self-contained.
First, some stuff about positive
Fixpoint append (i j : positive) : positive :=
match i with
| xH => j
| xI ii => xI (append ii j)
| xO ii => xO (append ii j)
end.
Lemma append_assoc_0 :
forall (i j : positive), append i (xO j) = append (append i (xO xH)) j.
Lemma append_assoc_1 :
forall (i j : positive), append i (xI j) = append (append i (xI xH)) j.
Lemma append_neutral_r : forall (i : positive), append i xH = i.
Lemma append_neutral_l : forall (i : positive), append xH i = i.
The module of maps over positive keys
Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits.
Module E:=PositiveOrderedTypeBits.
Module ME:=KeyOrderedType E.
Definition key := positive : Type.
#[universes(template)]
Inductive tree (A : Type) :=
| Leaf : tree A
| Node : tree A -> option A -> tree A -> tree A.
Scheme tree_ind := Induction for tree Sort Prop.
Definition t := tree.
Section A.
Variable A:Type.
Arguments Leaf {A}.
Definition empty : t A := Leaf.
Fixpoint is_empty (m : t A) : bool :=
match m with
| Leaf => true
| Node l None r => (is_empty l) && (is_empty r)
| _ => false
end.
Fixpoint find (i : key) (m : t A) : option A :=
match m with
| Leaf => None
| Node l o r =>
match i with
| xH => o
| xO ii => find ii l
| xI ii => find ii r
end
end.
Fixpoint mem (i : key) (m : t A) : bool :=
match m with
| Leaf => false
| Node l o r =>
match i with
| xH => match o with None => false | _ => true end
| xO ii => mem ii l
| xI ii => mem ii r
end
end.
Fixpoint add (i : key) (v : A) (m : t A) : t A :=
match m with
| Leaf =>
match i with
| xH => Node Leaf (Some v) Leaf
| xO ii => Node (add ii v Leaf) None Leaf
| xI ii => Node Leaf None (add ii v Leaf)
end
| Node l o r =>
match i with
| xH => Node l (Some v) r
| xO ii => Node (add ii v l) o r
| xI ii => Node l o (add ii v r)
end
end.
Fixpoint remove (i : key) (m : t A) : t A :=
match i with
| xH =>
match m with
| Leaf => Leaf
| Node Leaf _ Leaf => Leaf
| Node l _ 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.
elements
Fixpoint xelements (m : t A) (i : key) : list (key * A) :=
match m with
| Leaf => nil
| Node l None r =>
(xelements l (append i (xO xH))) ++ (xelements r (append i (xI xH)))
| Node l (Some x) r =>
(xelements l (append i (xO xH)))
++ ((i, x) :: xelements r (append i (xI xH)))
end.
Definition elements (m : t A) := xelements m xH.
cardinal
Fixpoint cardinal (m : t A) : nat :=
match m with
| Leaf => 0%nat
| Node l None r => (cardinal l + cardinal r)%nat
| Node l (Some _) r => S (cardinal l + cardinal r)
end.
Section CompcertSpec.
Theorem gempty:
forall (i: key), find i empty = None.
Theorem gss:
forall (i: key) (x: A) (m: t A), find i (add i x m) = Some x.
Lemma gleaf : forall (i : key), find i (Leaf : t A) = None.
Theorem gso:
forall (i j: key) (x: A) (m: t A),
i <> j -> find i (add j x m) = find i m.
Lemma rleaf : forall (i : key), remove i Leaf = Leaf.
Theorem grs:
forall (i: key) (m: t A), find i (remove i m) = None.
Theorem gro:
forall (i j: key) (m: t A),
i <> j -> find i (remove j m) = find i m.
Lemma xelements_correct:
forall (m: t A) (i j : key) (v: A),
find i m = Some v -> List.In (append j i, v) (xelements m j).
Theorem elements_correct:
forall (m: t A) (i: key) (v: A),
find i m = Some v -> List.In (i, v) (elements m).
Fixpoint xfind (i j : key) (m : t A) : option A :=
match i, j with
| _, xH => find i m
| xO ii, xO jj => xfind ii jj m
| xI ii, xI jj => xfind ii jj m
| _, _ => None
end.
Lemma xfind_left :
forall (j i : key) (m1 m2 : t A) (o : option A) (v : A),
xfind i (append j (xO xH)) m1 = Some v -> xfind i j (Node m1 o m2) = Some v.
Lemma xelements_ii :
forall (m: t A) (i j : key) (v: A),
List.In (xI i, v) (xelements m (xI j)) -> List.In (i, v) (xelements m j).
Lemma xelements_io :
forall (m: t A) (i j : key) (v: A),
~List.In (xI i, v) (xelements m (xO j)).
Lemma xelements_oo :
forall (m: t A) (i j : key) (v: A),
List.In (xO i, v) (xelements m (xO j)) -> List.In (i, v) (xelements m j).
Lemma xelements_oi :
forall (m: t A) (i j : key) (v: A),
~List.In (xO i, v) (xelements m (xI j)).
Lemma xelements_ih :
forall (m1 m2: t A) (o: option A) (i : key) (v: A),
List.In (xI i, v) (xelements (Node m1 o m2) xH) -> List.In (i, v) (xelements m2 xH).
Lemma xelements_oh :
forall (m1 m2: t A) (o: option A) (i : key) (v: A),
List.In (xO i, v) (xelements (Node m1 o m2) xH) -> List.In (i, v) (xelements m1 xH).
Lemma xelements_hi :
forall (m: t A) (i : key) (v: A),
~List.In (xH, v) (xelements m (xI i)).
Lemma xelements_ho :
forall (m: t A) (i : key) (v: A),
~List.In (xH, v) (xelements m (xO i)).
Lemma find_xfind_h :
forall (m: t A) (i: key), find i m = xfind i xH m.
Lemma xelements_complete:
forall (i j : key) (m: t A) (v: A),
List.In (i, v) (xelements m j) -> xfind i j m = Some v.
Theorem elements_complete:
forall (m: t A) (i: key) (v: A),
List.In (i, v) (elements m) -> find i m = Some v.
Lemma cardinal_1 :
forall (m: t A), cardinal m = length (elements m).
End CompcertSpec.
Definition MapsTo (i:key)(v:A)(m:t A) := find i m = Some v.
Definition In (i:key)(m:t A) := exists e:A, MapsTo i e m.
Definition Empty m := forall (a : key)(e:A) , ~ MapsTo a e m.
Definition eq_key (p p':key*A) := E.eq (fst p) (fst p').
Definition eq_key_elt (p p':key*A) :=
E.eq (fst p) (fst p') /\ (snd p) = (snd p').
Definition lt_key (p p':key*A) := E.lt (fst p) (fst p').
Global Instance eqk_equiv : Equivalence eq_key := _.
Global Instance eqke_equiv : Equivalence eq_key_elt := _.
Global Instance ltk_strorder : StrictOrder lt_key := _.
Lemma mem_find :
forall m x, mem x m = match find x m with None => false | _ => true end.
Lemma Empty_alt : forall m, Empty m <-> forall a, find a m = None.
Lemma Empty_Node : forall l o r, Empty (Node l o r) <-> o=None /\ Empty l /\ Empty r.
Section FMapSpec.
Lemma mem_1 : forall m x, In x m -> mem x m = true.
Lemma mem_2 : forall m x, mem x m = true -> In x m.
Variable m m' m'' : t A.
Variable x y z : key.
Variable e e' : A.
Lemma MapsTo_1 : E.eq x y -> MapsTo x e m -> MapsTo y e m.
Lemma find_1 : MapsTo x e m -> find x m = Some e.
Lemma find_2 : find x m = Some e -> MapsTo x e m.
Lemma empty_1 : Empty empty.
Lemma is_empty_1 : Empty m -> is_empty m = true.
Lemma is_empty_2 : is_empty m = true -> Empty m.
Lemma add_1 : E.eq x y -> MapsTo y e (add x e m).
Lemma add_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (add x e' m).
Lemma add_3 : ~ E.eq x y -> MapsTo y e (add x e' m) -> MapsTo y e m.
Lemma remove_1 : E.eq x y -> ~ In y (remove x m).
Lemma remove_2 : ~ E.eq x y -> MapsTo y e m -> MapsTo y e (remove x m).
Lemma remove_3 : MapsTo y e (remove x m) -> MapsTo y e m.
Lemma elements_1 :
MapsTo x e m -> InA eq_key_elt (x,e) (elements m).
Lemma elements_2 :
InA eq_key_elt (x,e) (elements m) -> MapsTo x e m.
Lemma xelements_bits_lt_1 : forall p p0 q m v,
List.In (p0,v) (xelements m (append p (xO q))) -> E.bits_lt p0 p.
Lemma xelements_bits_lt_2 : forall p p0 q m v,
List.In (p0,v) (xelements m (append p (xI q))) -> E.bits_lt p p0.
Lemma xelements_sort : forall p, sort lt_key (xelements m p).
Lemma elements_3 : sort lt_key (elements m).
Lemma elements_3w : NoDupA eq_key (elements m).
End FMapSpec.
map and mapi
Variable B : Type.
Section Mapi.
Variable f : key -> A -> B.
Fixpoint xmapi (m : t A) (i : key) : t B :=
match m with
| Leaf => @Leaf B
| Node l o r => Node (xmapi l (append i (xO xH)))
(option_map (f i) o)
(xmapi r (append i (xI xH)))
end.
Definition mapi m := xmapi m xH.
End Mapi.
Definition map (f : A -> B) m := mapi (fun _ => f) m.
End A.
Lemma xgmapi:
forall (A B: Type) (f: key -> A -> B) (i j : key) (m: t A),
find i (xmapi f m j) = option_map (f (append j i)) (find i m).
Theorem gmapi:
forall (A B: Type) (f: key -> A -> B) (i: key) (m: t A),
find i (mapi f m) = option_map (f i) (find i m).
Lemma mapi_1 :
forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:key->elt->elt'),
MapsTo x e m ->
exists y, E.eq y x /\ MapsTo x (f y e) (mapi f m).
Lemma mapi_2 :
forall (elt elt':Type)(m: t elt)(x:key)(f:key->elt->elt'),
In x (mapi f m) -> In x m.
Lemma map_1 : forall (elt elt':Type)(m: t elt)(x:key)(e:elt)(f:elt->elt'),
MapsTo x e m -> MapsTo x (f e) (map f m).
Lemma map_2 : forall (elt elt':Type)(m: t elt)(x:key)(f:elt->elt'),
In x (map f m) -> In x m.
Section map2.
Variable A B C : Type.
Variable f : option A -> option B -> option C.
Arguments Leaf {A}.
Fixpoint xmap2_l (m : t A) : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node (xmap2_l l) (f o None) (xmap2_l r)
end.
Lemma xgmap2_l : forall (i : key) (m : t A),
f None None = None -> find i (xmap2_l m) = f (find i m) None.
Fixpoint xmap2_r (m : t B) : t C :=
match m with
| Leaf => Leaf
| Node l o r => Node (xmap2_r l) (f None o) (xmap2_r r)
end.
Lemma xgmap2_r : forall (i : key) (m : t B),
f None None = None -> find i (xmap2_r m) = f None (find i m).
Fixpoint _map2 (m1 : t A)(m2 : t B) : t C :=
match m1 with
| Leaf => xmap2_r m2
| Node l1 o1 r1 =>
match m2 with
| Leaf => xmap2_l m1
| Node l2 o2 r2 => Node (_map2 l1 l2) (f o1 o2) (_map2 r1 r2)
end
end.
Lemma gmap2: forall (i: key)(m1:t A)(m2: t B),
f None None = None ->
find i (_map2 m1 m2) = f (find i m1) (find i m2).
End map2.
Definition map2 (elt elt' elt'':Type)(f:option elt->option elt'->option elt'') :=
_map2 (fun o1 o2 => match o1,o2 with None,None => None | _, _ => f o1 o2 end).
Lemma map2_1 : forall (elt elt' elt'':Type)(m: t elt)(m': t elt')
(x:key)(f:option elt->option elt'->option elt''),
In x m \/ In x m' ->
find x (map2 f m m') = f (find x m) (find x m').
Lemma map2_2 : forall (elt elt' elt'':Type)(m: t elt)(m': t elt')
(x:key)(f:option elt->option elt'->option elt''),
In x (map2 f m m') -> In x m \/ In x m'.
Section Fold.
Variables A B : Type.
Variable f : key -> A -> B -> B.
Fixpoint xfoldi (m : t A) (v : B) (i : key) :=
match m with
| Leaf _ => v
| Node l (Some x) r =>
xfoldi r (f i x (xfoldi l v (append i 2))) (append i 3)
| Node l None r =>
xfoldi r (xfoldi l v (append i 2)) (append i 3)
end.
Lemma xfoldi_1 :
forall m v i,
xfoldi m v i = fold_left (fun a p => f (fst p) (snd p) a) (xelements m i) v.
Definition fold m i := xfoldi m i 1.
End Fold.
Lemma fold_1 :
forall (A:Type)(m:t A)(B:Type)(i : B) (f : key -> A -> B -> B),
fold f m i = fold_left (fun a p => f (fst p) (snd p) a) (elements m) i.
Fixpoint equal (A:Type)(cmp : A -> A -> bool)(m1 m2 : t A) : bool :=
match m1, m2 with
| Leaf _, _ => is_empty m2
| _, Leaf _ => is_empty m1
| Node l1 o1 r1, Node l2 o2 r2 =>
(match o1, o2 with
| None, None => true
| Some v1, Some v2 => cmp v1 v2
| _, _ => false
end)
&& equal cmp l1 l2 && equal cmp r1 r2
end.
Definition Equal (A:Type)(m m':t A) :=
forall y, find y m = find y m'.
Definition Equiv (A:Type)(eq_elt:A->A->Prop) m m' :=
(forall k, In k m <-> In k m') /\
(forall k e e', MapsTo k e m -> MapsTo k e' m' -> eq_elt e e').
Definition Equivb (A:Type)(cmp: A->A->bool) := Equiv (Cmp cmp).
Lemma equal_1 : forall (A:Type)(m m':t A)(cmp:A->A->bool),
Equivb cmp m m' -> equal cmp m m' = true.
Lemma equal_2 : forall (A:Type)(m m':t A)(cmp:A->A->bool),
equal cmp m m' = true -> Equivb cmp m m'.
End PositiveMap.
Here come some additional facts about this implementation. Most are facts that cannot be derivable from the general interface.
Module PositiveMapAdditionalFacts.
Import PositiveMap.
Theorem gsspec:
forall (A:Type)(i j: key) (x: A) (m: t A),
find i (add j x m) = if E.eq_dec i j then Some x else find i m.
Theorem gsident:
forall (A:Type)(i: key) (m: t A) (v: A),
find i m = Some v -> add i v m = m.
Lemma xmap2_lr :
forall (A B : Type)(f g: option A -> option A -> option B)(m : t A),
(forall (i j : option A), f i j = g j i) ->
xmap2_l f m = xmap2_r g m.
Theorem map2_commut:
forall (A B: Type) (f g: option A -> option A -> option B),
(forall (i j: option A), f i j = g j i) ->
forall (m1 m2: t A),
_map2 f m1 m2 = _map2 g m2 m1.
End PositiveMapAdditionalFacts.