(* Copyright (c) 2012-2015, 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]. *) From stdpp Require Import pmap mapset. From stdpp Require Export prelude fin_maps. Local Open Scope Z_scope. Record Zmap (A : Type) : Type := 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 : Lookup Z Zmap := λ A 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 : PartialAlter Z Zmap := λ A 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 : FinMapToList Z Zmap := λ A 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_and?. - apply (NoDup_fmap_2 _), NoDup_map_to_list. - intro. rewrite !elem_of_list_fmap. naive_solver. - apply (NoDup_fmap_2 _), NoDup_map_to_list. } 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_eq/=; [done| |]; by apply elem_of_map_to_list. * rewrite elem_of_app, !elem_of_list_fmap. intros [[[??][??]]|[[??][??]]]; simplify_eq/=; 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 : Dom Zmap Zset := mapset_dom. Instance: FinMapDom Z Zmap Zset := mapset_dom_spec.