Commit 8da8fbc7 authored by Robbert Krebbers's avatar Robbert Krebbers

Finite maps and finite sets over strings.

parent 18e47df6
(* Copyright (c) 2012-2014, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This files implements an efficient implementation of finite maps whose keys
range over Coq's data type of strings [string]. The implementation uses radix-2
search trees (uncompressed Patricia trees) as implemented in the file [pmap]
and guarantees logarithmic-time operations. *)
Require Export fin_maps.
Require Import Ascii String list pmap mapset.
Instance assci_eq_dec (a1 a2 : ascii) : Decision (a1 = a2).
Proof. solve_decision. Defined.
Instance string_eq_dec (s1 s2 : string) : Decision (s1 = s2).
Proof. solve_decision. Defined.
(** * Encoding and decoding *)
(** In order to reuse or existing implementation of radix-2 search trees over
positive binary naturals [positive], we define an injection [string_to_pos]
from [string] into [positive]. *)
Fixpoint digits_to_pos (βs : list bool) : positive :=
match βs with
| [] => xH
| false :: βs => (digits_to_pos βs)~0
| true :: βs => (digits_to_pos βs)~1
end%positive.
Definition ascii_to_digits (a : Ascii.ascii) : list bool :=
match a with
| Ascii.Ascii β1 β2 β3 β4 β5 β6 β7 β8 => [β1;β2;β3;β4;β5;β6;β7;β8]
end.
Fixpoint string_to_pos (s : string) : positive :=
match s with
| EmptyString => xH
| String a s => string_to_pos s ++ digits_to_pos (ascii_to_digits a)
end%positive.
Fixpoint digits_of_pos (p : positive) : list bool :=
match p with
| xH => []
| p~0 => false :: digits_of_pos p
| p~1 => true :: digits_of_pos p
end%positive.
Fixpoint ascii_of_digits (βs : list bool) : ascii :=
match βs with
| [] => zero
| β :: βs => Ascii.shift β (ascii_of_digits βs)
end.
Fixpoint string_of_digits (βs : list bool) : string :=
match βs with
| β1 :: β2 :: β3 :: β4 :: β5 :: β6 :: β7 :: β8 :: βs =>
String (ascii_of_digits [β1;β2;β3;β4;β5;β6;β7;β8]) (string_of_digits βs)
| _ => EmptyString
end.
Definition string_of_pos (p : positive) : string :=
string_of_digits (digits_of_pos p).
Lemma string_of_to_pos s : string_of_pos (string_to_pos s) = s.
Proof.
unfold string_of_pos.
by induction s as [|[[] [] [] [] [] [] [] []]]; simpl; f_equal.
Qed.
Instance: Injective (=) (=) string_to_pos.
Proof.
intros s1 s2 Hs. by rewrite <-(string_of_to_pos s1), Hs, string_of_to_pos.
Qed.
(** * The data structure *)
(** We pack a [Pmap] together with a proof that ensures that all keys correspond
to actual strings. *)
Definition stringmap_wf {A} : Pmap A Prop :=
map_Forall (λ p _, string_to_pos (string_of_pos p) = p).
Record stringmap A := StringMap {
stringmap_car : Pmap A;
stringmap_prf : bool_decide (stringmap_wf stringmap_car)
}.
Arguments StringMap {_} _ _.
Arguments stringmap_car {_} _.
Lemma stringmap_eq {A} (m1 m2 : stringmap A) :
m1 = m2 stringmap_car m1 = stringmap_car m2.
Proof.
split; [by intros ->|intros]. destruct m1, m2; simplify_equality'.
f_equal; apply proof_irrel.
Qed.
Instance stringmap_eq_eq {A} `{ x y : A, Decision (x = y)}
(m1 m2 : stringmap A) : Decision (m1 = m2).
Proof.
refine (cast_if (decide (stringmap_car m1 = stringmap_car m2)));
abstract (by rewrite stringmap_eq).
Defined.
(** * Operations on the data structure *)
Instance stringmap_lookup {A} : Lookup string A (stringmap A) := λ s m,
let (m,_) := m in m !! string_to_pos s.
Instance stringmap_empty {A} : Empty (stringmap A) := StringMap I.
Lemma stringmap_partial_alter_wf {A} (f : option A option A) m s :
stringmap_wf m stringmap_wf (partial_alter f (string_to_pos s) m).
Proof.
intros Hm p x. destruct (decide (string_to_pos s = p)) as [<-|?].
* by rewrite string_of_to_pos.
* rewrite lookup_partial_alter_ne by done. by apply Hm.
Qed.
Instance stringmap_partial_alter {A} :
PartialAlter string A (stringmap A) := λ f s m,
let (m,Hm) := m in StringMap (partial_alter f (string_to_pos s) m)
(bool_decide_pack _ (stringmap_partial_alter_wf f m s
(bool_decide_unpack _ Hm))).
Lemma stringmap_fmap_wf {A B} (f : A B) m :
stringmap_wf m stringmap_wf (f <$> m).
Proof. intros ? p x. rewrite lookup_fmap, fmap_Some; intros (?&?&?); eauto. Qed.
Instance stringmap_fmap : FMap stringmap := λ A B f m,
let (m,Hm) := m in StringMap (f <$> m)
(bool_decide_pack _ (stringmap_fmap_wf f m (bool_decide_unpack _ Hm))).
Lemma stringmap_omap_wf {A B} (f : A option B) m :
stringmap_wf m stringmap_wf (omap f m).
Proof. intros ? p x; rewrite lookup_omap, bind_Some; intros (?&?&?); eauto. Qed.
Instance stringmap_omap : OMap stringmap := λ A B f m,
let (m,Hm) := m in StringMap (omap f m)
(bool_decide_pack _ (stringmap_omap_wf f m (bool_decide_unpack _ Hm))).
Lemma stringmap_merge_wf {A B C} (f : option A option B option C) m1 m2 :
let f' o1 o2 := match o1, o2 with None, None => None | _, _ => f o1 o2 end in
stringmap_wf m1 stringmap_wf m2 stringmap_wf (merge f' m1 m2).
Proof.
intros f' Hm1 Hm2 p z; rewrite lookup_merge by done; intros.
destruct (m1 !! _) eqn:?, (m2 !! _) eqn:?; naive_solver.
Qed.
Instance stringmap_merge : Merge stringmap := λ A B C f m1 m2,
let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
let f' o1 o2 := match o1, o2 with None, None => None | _, _ => f o1 o2 end in
StringMap (merge f' m1 m2) (bool_decide_pack _ (stringmap_merge_wf f _ _
(bool_decide_unpack _ Hm1) (bool_decide_unpack _ Hm2))).
Instance stringmap_to_list {A} : FinMapToList string A (stringmap A) := λ m,
let (m,_) := m in prod_map string_of_pos id <$> map_to_list m.
(** * Instantiation of the finite map interface *)
Instance: FinMap string stringmap.
Proof.
split.
* unfold lookup; intros A [m1 Hm1] [m2 Hm2] H.
apply stringmap_eq, map_eq; intros i; simpl in *.
apply bool_decide_unpack in Hm1; apply bool_decide_unpack in Hm2.
apply option_eq; intros x; split; intros Hi.
+ generalize Hi. rewrite <-(Hm1 i x) by done; eauto using option_eq_1.
+ generalize Hi. rewrite <-(Hm2 i x) by done; eauto using option_eq_1.
* done.
* intros A f [m Hm] s; apply (lookup_partial_alter f m).
* intros A f [m Hm] s s' Hs; apply (lookup_partial_alter_ne f m).
by contradict Hs; apply (injective string_to_pos).
* intros A B f [m Hm] s; apply (lookup_fmap f m).
* intros A [m Hm]; unfold map_to_list; simpl.
apply bool_decide_unpack, map_Forall_to_list in Hm; revert Hm.
induction (NoDup_map_to_list m) as [|[p x] l Hpx];
inversion 1 as [|??? Hm']; simplify_equality'; constructor; eauto.
rewrite elem_of_list_fmap; intros ([p' x']&?&?); simplify_equality.
cut (string_to_pos (string_of_pos p') = p'); [congruence|].
rewrite Forall_forall in Hm'. eapply (Hm' (_,_)); eauto.
* intros A [m Hm] s x; unfold map_to_list, lookup; simpl.
apply bool_decide_unpack in Hm; rewrite elem_of_list_fmap; split.
+ intros ([p' x']&?&Hp'); simplify_equality'.
apply elem_of_map_to_list in Hp'. by rewrite (Hm p' x').
+ intros. exists (string_to_pos s,x); simpl.
by rewrite elem_of_map_to_list, string_of_to_pos.
* intros A B f [m Hm] s; apply (lookup_omap f m).
* intros A B C f ? [m1 Hm1] [m2 Hm2] s; unfold merge, lookup; simpl.
set (f' o1 o2 := match o1, o2 with None, None => None | _, _ => f o1 o2 end).
rewrite lookup_merge by done. by destruct (m1 !! _), (m2 !! _).
Qed.
(** * Finite sets *)
(** We construct sets of [strings]s satisfying extensional equality. *)
Notation stringset := (mapset (stringmap unit)).
Instance stringmap_dom {A} : Dom (stringmap A) stringset := mapset_dom.
Instance: FinMapDom positive Pmap Pset := mapset_dom_spec.
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