Commit 097645aa authored by Robbert Krebbers's avatar Robbert Krebbers

Add finite map with integer keys Z.

parent 716f554d
(* Copyright (c) 2012-2013, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This files extends the implementation of finite over [positive] to finite
maps whose keys range over Coq's data type of binary naturals [Z]. *)
Require Import pmap mapset.
Require Export prelude fin_maps.
Local Open Scope Z_scope.
Record Zmap A :=
ZMap { Zmap_0 : option A; Zmap_pos : Pmap A; Zmap_neg : Pmap A }.
Arguments Zmap_0 {_} _.
Arguments Zmap_pos {_} _.
Arguments Zmap_neg {_} _.
Arguments ZMap {_} _ _ _.
Instance Zmap_eq_dec `{ x y : A, Decision (x = y)} (t1 t2 : Zmap A) :
Decision (t1 = t2).
Proof.
refine
match t1, t2 with
| ZMap x t1 t1', ZMap y t2 t2' =>
cast_if_and3 (decide (x = y)) (decide (t1 = t2)) (decide (t1' = t2'))
end; abstract congruence.
Defined.
Instance Zempty {A} : Empty (Zmap A) := ZMap None .
Instance Zlookup {A} : Lookup Z A (Zmap A) := λ i t,
match i with
| Z0 => Zmap_0 t | Zpos p => Zmap_pos t !! p | Zneg p => Zmap_neg t !! p
end.
Instance Zpartial_alter {A} : PartialAlter Z A (Zmap A) := λ f i t,
match i, t with
| Z0, ZMap o t t' => ZMap (f o) t t'
| Zpos p, ZMap o t t' => ZMap o (partial_alter f p t) t'
| Zneg p, ZMap o t t' => ZMap o t (partial_alter f p t')
end.
Instance Zto_list {A} : FinMapToList Z A (Zmap A) := λ t,
match t with
| ZMap o t t' => default [] o (λ x, [(0,x)]) ++
(prod_map Zpos id <$> map_to_list t) ++
(prod_map Zneg id <$> map_to_list t')
end.
Instance Zomap: OMap Zmap := λ A B f t,
match t with ZMap o t t' => ZMap (o = f) (omap f t) (omap f t') end.
Instance Zmerge: Merge Zmap := λ A B C f t1 t2,
match t1, t2 with
| ZMap o1 t1 t1', ZMap o2 t2 t2' => ZMap (f o1 o2) (merge f t1 t2) (merge f t1' t2')
end.
Instance Nfmap: FMap Zmap := λ A B f t,
match t with ZMap o t t' => ZMap (f <$> o) (f <$> t) (f <$> t') end.
Instance: FinMap Z Zmap.
Proof.
split.
* intros ? [??] [??] H. f_equal.
+ apply (H 0).
+ apply map_eq. intros i. apply (H (Zpos i)).
+ apply map_eq. intros i. apply (H (Zneg i)).
* by intros ? [].
* intros ? f [] [|?|?]; simpl; [done| |]; apply lookup_partial_alter.
* intros ? f [] [|?|?] [|?|?]; simpl; intuition congruence ||
intros; apply lookup_partial_alter_ne; congruence.
* intros ??? [??] []; simpl; [done| |]; apply lookup_fmap.
* intros ? [o t t']; unfold map_to_list; simpl.
assert (NoDup ((prod_map Z.pos id <$> map_to_list t) ++
prod_map Z.neg id <$> map_to_list t')).
{ apply NoDup_app; split_ands.
- apply (fmap_nodup _). apply map_to_list_nodup.
- intro. rewrite !elem_of_list_fmap. naive_solver.
- apply (fmap_nodup _). apply map_to_list_nodup. }
destruct o; simpl; auto. constructor; auto.
rewrite elem_of_app, !elem_of_list_fmap. naive_solver.
* intros ? t i x. unfold map_to_list. split.
+ destruct t as [[y|] t t']; simpl.
- rewrite elem_of_cons, elem_of_app, !elem_of_list_fmap.
intros [?|[[[??][??]]|[[??][??]]]]; simplify_equality; simpl; [done| |];
by apply elem_of_map_to_list.
- rewrite elem_of_app, !elem_of_list_fmap. intros [[[??][??]]|[[??][??]]];
simplify_equality'; by apply elem_of_map_to_list.
+ destruct t as [[y|] t t']; simpl.
- rewrite elem_of_cons, elem_of_app, !elem_of_list_fmap.
destruct i as [|i|i]; simpl; [intuition congruence| |].
{ right; left. exists (i, x). by rewrite elem_of_map_to_list. }
right; right. exists (i, x). by rewrite elem_of_map_to_list.
- rewrite elem_of_app, !elem_of_list_fmap.
destruct i as [|i|i]; simpl; [done| |].
{ left; exists (i, x). by rewrite elem_of_map_to_list. }
right; exists (i, x). by rewrite elem_of_map_to_list.
* intros ?? f [??] [|?|?]; simpl; [done| |]; apply (lookup_omap f).
* intros ??? f ? [??] [??] [|?|?]; simpl; [done| |]; apply (lookup_merge f).
Qed.
(** * Finite sets *)
(** We construct sets of [Z]s satisfying extensional equality. *)
Notation Zset := (mapset Zmap).
Instance Zmap_dom {A} : Dom (Zmap A) Zset := mapset_dom.
Instance: FinMapDom Z Zmap Zset := mapset_dom_spec.
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