Commit 2cb45dad authored by Robbert Krebbers's avatar Robbert Krebbers

Finite maps and finite sets over any countable type.

parent 2cbe04e6
......@@ -14,6 +14,11 @@ Definition encode_nat `{Countable A} (x : A) : nat :=
pred (Pos.to_nat (encode x)).
Definition decode_nat `{Countable A} (i : nat) : option A :=
decode (Pos.of_nat (S i)).
Instance encode_injective `{Countable A} : Injective (=) (=) encode.
Proof.
intros x y Hxy; apply (injective Some).
by rewrite <-(decode_encode x), Hxy, decode_encode.
Qed.
Lemma decode_encode_nat `{Countable A} x : decode_nat (encode_nat x) = Some x.
Proof.
pose proof (Pos2Nat.is_pos (encode x)).
......
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file implements finite maps and finite sets with keys of any countable
type. The implementation is based on [Pmap]s, radix-2 search trees. *)
Require Export prelude.countable prelude.fin_maps prelude.fin_map_dom.
Require Import prelude.pmap prelude.mapset.
(** * The data structure *)
(** We pack a [Pmap] together with a proof that ensures that all keys correspond
to codes of actual elements of the countable type. *)
Definition gmap_wf `{Countable K} {A} : Pmap A Prop :=
map_Forall (λ p _, encode <$> decode p = Some p).
Record gmap K `{Countable K} A := GMap {
gmap_car : Pmap A;
gmap_prf : bool_decide (gmap_wf gmap_car)
}.
Arguments GMap {_ _ _ _} _ _.
Arguments gmap_car {_ _ _ _} _.
Lemma gmap_eq `{Countable K} {A} (m1 m2 : gmap K A) :
m1 = m2 gmap_car m1 = gmap_car m2.
Proof.
split; [by intros ->|intros]. destruct m1, m2; simplify_equality'.
f_equal; apply proof_irrel.
Qed.
Instance gmap_eq_eq `{Countable K} `{ x y : A, Decision (x = y)}
(m1 m2 : gmap K A) : Decision (m1 = m2).
Proof.
refine (cast_if (decide (gmap_car m1 = gmap_car m2)));
abstract (by rewrite gmap_eq).
Defined.
(** * Operations on the data structure *)
Instance gmap_lookup `{Countable K} {A} : Lookup K A (gmap K A) := λ i m,
let (m,_) := m in m !! encode i.
Instance gmap_empty `{Countable K} {A} : Empty (gmap K A) := GMap I.
Lemma gmap_partial_alter_wf `{Countable K} {A} (f : option A option A) m i :
gmap_wf m gmap_wf (partial_alter f (encode i) m).
Proof.
intros Hm p x. destruct (decide (encode i = p)) as [<-|?].
* rewrite decode_encode; eauto.
* rewrite lookup_partial_alter_ne by done. by apply Hm.
Qed.
Instance gmap_partial_alter `{Countable K} {A} :
PartialAlter K A (gmap K A) := λ f i m,
let (m,Hm) := m in GMap (partial_alter f (encode i) m)
(bool_decide_pack _ (gmap_partial_alter_wf f m i
(bool_decide_unpack _ Hm))).
Lemma gmap_fmap_wf `{Countable K} {A B} (f : A B) m :
gmap_wf m gmap_wf (f <$> m).
Proof. intros ? p x. rewrite lookup_fmap, fmap_Some; intros (?&?&?); eauto. Qed.
Instance gmap_fmap `{Countable K} : FMap (gmap K) := λ A B f m,
let (m,Hm) := m in GMap (f <$> m)
(bool_decide_pack _ (gmap_fmap_wf f m (bool_decide_unpack _ Hm))).
Lemma gmap_omap_wf `{Countable K} {A B} (f : A option B) m :
gmap_wf m gmap_wf (omap f m).
Proof. intros ? p x; rewrite lookup_omap, bind_Some; intros (?&?&?); eauto. Qed.
Instance gmap_omap `{Countable K} : OMap (gmap K) := λ A B f m,
let (m,Hm) := m in GMap (omap f m)
(bool_decide_pack _ (gmap_omap_wf f m (bool_decide_unpack _ Hm))).
Lemma gmap_merge_wf `{Countable K} {A B C}
(f : option A option B option C) m1 m2 :
let f' o1 o2 := match o1, o2 with None, None => None | _, _ => f o1 o2 end in
gmap_wf m1 gmap_wf m2 gmap_wf (merge f' m1 m2).
Proof.
intros f' Hm1 Hm2 p z; rewrite lookup_merge by done; intros.
destruct (m1 !! _) eqn:?, (m2 !! _) eqn:?; naive_solver.
Qed.
Instance gmap_merge `{Countable K} : Merge (gmap K) := λ A B C f m1 m2,
let (m1,Hm1) := m1 in let (m2,Hm2) := m2 in
let f' o1 o2 := match o1, o2 with None, None => None | _, _ => f o1 o2 end in
GMap (merge f' m1 m2) (bool_decide_pack _ (gmap_merge_wf f _ _
(bool_decide_unpack _ Hm1) (bool_decide_unpack _ Hm2))).
Instance gmap_to_list `{Countable K} {A} : FinMapToList K A (gmap K A) := λ m,
let (m,_) := m in omap (λ ix : positive * A,
let (i,x) := ix in (,x) <$> decode i) (map_to_list m).
(** * Instantiation of the finite map interface *)
Instance gmap_finmap `{Countable K} : FinMap K (gmap K).
Proof.
split.
* unfold lookup; intros A [m1 Hm1] [m2 Hm2] Hm.
apply gmap_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.
+ pose proof (Hm1 i x Hi); simpl in *.
by destruct (decode i); simplify_equality'; rewrite <-Hm.
+ pose proof (Hm2 i x Hi); simpl in *.
by destruct (decode i); simplify_equality'; rewrite Hm.
* done.
* intros A f [m Hm] i; apply (lookup_partial_alter f m).
* intros A f [m Hm] i j Hs; apply (lookup_partial_alter_ne f m).
by contradict Hs; apply (injective encode).
* intros A B f [m Hm] i; 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'; [by constructor|].
destruct (decode p) as [i|] eqn:?; simplify_equality'; constructor; eauto.
rewrite elem_of_list_omap; intros ([p' x']&?&?); simplify_equality'.
feed pose proof (proj1 (Forall_forall _ _) Hm' (p',x')); simpl in *; auto.
by destruct (decode p') as [i'|]; simplify_equality'.
* intros A [m Hm] i x; unfold map_to_list, lookup; simpl.
apply bool_decide_unpack in Hm; rewrite elem_of_list_omap; split.
+ intros ([p' x']&Hp'&?); apply elem_of_map_to_list in Hp'.
feed pose proof (Hm p' x'); simpl in *; auto.
by destruct (decode p') as [i'|] eqn:?; simplify_equality'.
+ intros; exists (encode i,x); simpl.
by rewrite elem_of_map_to_list, decode_encode.
* intros A B f [m Hm] i; apply (lookup_omap f m).
* intros A B C f ? [m1 Hm1] [m2 Hm2] i; unfold merge, lookup; simpl.
set (f' o1 o2 := match o1, o2 with None,None => None | _, _ => f o1 o2 end).
by rewrite lookup_merge by done; destruct (m1 !! _), (m2 !! _).
Qed.
(** * Finite sets *)
Notation gset K := (mapset (gmap K)).
Instance gset_dom `{Countable K} {A} : Dom (gmap K A) (gset K) := mapset_dom.
Instance gset_dom_spec `{Countable K} :
FinMapDom K (gmap K) (gset K) := mapset_dom_spec.
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