Module nmap

Require Import pmap mapset.
Require Export prelude fin_maps.

Local Open Scope N_scope.

Record Nmap A := NMap { Nmap_0 : option A; Nmap_pos : Pmap A }.
Arguments Nmap_0 {_} _.
Arguments Nmap_pos {_} _.
Arguments NMap {_} _ _.

Instance Nmap_eq_dec `{∀ x y : A, Decision (x = y)} (t1 t2 : Nmap A) :
  Decision (t1 = t2).
Proof.
 refine
  match t1, t2 with
  | NMap x t1, NMap y t2 => cast_if_and (decide (x = y)) (decide (t1 = t2))
  end; abstract congruence.
Defined.
Instance Nempty {A} : Empty (Nmap A) := NMap None ∅.
Global Opaque Nempty.
Instance Nlookup {A} : Lookup N A (Nmap A) := λ i t,
  match i with
  | N0 => Nmap_0 t
  | Npos p => Nmap_pos t !! p
  end.
Instance Npartial_alter {A} : PartialAlter N A (Nmap A) := λ f i t,
  match i, t with
  | N0, NMap o t => NMap (f o) t
  | Npos p, NMap o t => NMap o (partial_alter f p t)
  end.
Instance Nto_list {A} : FinMapToList N A (Nmap A) := λ t,
  match t with
  | NMap o t =>
     default [] o (λ x, [(0,x)]) ++ (prod_map Npos id <$> map_to_list t)
  end.
Instance Nomap: OMap Nmap := λ A B f t,
  match t with NMap o t => NMap (o ≫= f) (omap f t) end.
Instance Nmerge: Merge Nmap := λ A B C f t1 t2,
  match t1, t2 with
  | NMap o1 t1, NMap o2 t2 => NMap (f o1 o2) (merge f t1 t2)
  end.
Instance Nfmap: FMap Nmap := λ A B f t,
  match t with NMap o t => NMap (f <$> o) (f <$> t) end.

Instance: FinMap N Nmap.
Proof.
  split.
  * intros ? [??] [??] H. f_equal; [apply (H 0)|].
    apply map_eq. intros i. apply (H (Npos i)).
  * by intros ? [|?].
  * intros ? f [? t] [|i]; simpl; [done |]. apply lookup_partial_alter.
  * intros ? f [? t] [|i] [|j]; simpl; try intuition congruence.
    intros. apply lookup_partial_alter_ne. congruence.
  * intros ??? [??] []; simpl. done. apply lookup_fmap.
  * intros ? [[x|] t]; unfold map_to_list; simpl.
    + constructor.
      - rewrite elem_of_list_fmap. by intros [[??] [??]].
      - by apply (NoDup_fmap _), NoDup_map_to_list.
    + apply (NoDup_fmap _), NoDup_map_to_list.
  * intros ? t i x. unfold map_to_list. split.
    + destruct t as [[y|] t]; simpl.
      - rewrite elem_of_cons, elem_of_list_fmap.
        intros [? | [[??] [??]]]; simplify_equality'; [done |].
        by apply elem_of_map_to_list.
      - rewrite elem_of_list_fmap; intros [[??] [??]]; simplify_equality'.
        by apply elem_of_map_to_list.
    + destruct t as [[y|] t]; simpl.
      - rewrite elem_of_cons, elem_of_list_fmap.
        destruct i as [|i]; simpl; [intuition congruence |].
        intros. right. exists (i, x). by rewrite elem_of_map_to_list.
      - rewrite elem_of_list_fmap.
        destruct i as [|i]; simpl; [done |].
        intros. 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

Notation Nset := (mapset (Nmap unit)).
Instance Nmap_dom {A} : Dom (Nmap A) Nset := mapset_dom.
Instance: FinMapDom N Nmap Nset := mapset_dom_spec.

Fresh numbers

Definition Nfresh {A} (m : Nmap A) : N :=
  match m with NMap None _ => 0 | NMap _ m => Npos (Pfresh m) end.
Lemma Nfresh_fresh {A} (m : Nmap A) : m !! Nfresh m = None.
Proof. destruct m as [[]]. apply Pfresh_fresh. done. Qed.

Instance Nset_fresh : Fresh N Nset := λ X,
  let (m) := X in Nfresh m.
Instance Nset_fresh_spec : FreshSpec N Nset.
Proof.
  split.
  * apply _.
  * intros X Y; rewrite <-elem_of_equiv_L. by intros ->.
  * unfold elem_of, mapset_elem_of, fresh; intros [m]; simpl.
    by rewrite Nfresh_fresh.
Qed.