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(* Copyright (c) 2012-2015, Robbert Krebbers. *)
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(* This file is distributed under the terms of the BSD license. *)
(** Finite maps associate data to keys. This file defines an interface for
finite maps and collects some theory on it. Most importantly, it proves useful
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induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
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Require Import Permutation.
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Require Export ars vector orders.

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(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
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course limits the space of finite map implementations, but since we are mainly
interested in finite maps with numbers as indexes, we do not consider this to
be a serious limitation. The main application of finite maps is to implement
the memory, where extensionality of Leibniz equality is very important for a
convenient use in the assertions of our axiomatic semantics. *)
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(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
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prove well founded recursion on finite maps. *)
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(** Finite map implementations are required to implement the [merge] function
which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)
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Class FinMapToList K A M := map_to_list: M  list (K * A).
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Class FinMap K M `{FMap M,  A, Lookup K A (M A),  A, Empty (M A),  A,
    PartialAlter K A (M A), OMap M, Merge M,  A, FinMapToList K A (M A),
     i j : K, Decision (i = j)} := {
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  map_eq {A} (m1 m2 : M A) : ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i : ( : M A) !! i = None;
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  lookup_partial_alter {A} f (m : M A) i :
    partial_alter f i m !! i = f (m !! i);
  lookup_partial_alter_ne {A} f (m : M A) i j :
    i  j  partial_alter f i m !! j = m !! j;
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  lookup_fmap {A B} (f : A  B) (m : M A) i : (f <$> m) !! i = f <$> m !! i;
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  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
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  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
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  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
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  lookup_merge {A B C} (f : option A  option B  option C)
      `{!PropHolds (f None None = None)} m1 m2 i :
    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
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}.

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(** * Derived operations *)
(** All of the following functions are defined in a generic way for arbitrary
finite map implementations. These generic implementations do not cause a
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significant performance loss to make including them in the finite map interface
worthwhile. *)
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Instance map_insert `{PartialAlter K A M} : Insert K A M :=
  λ i x, partial_alter (λ _, Some x) i.
Instance map_alter `{PartialAlter K A M} : Alter K A M :=
  λ f, partial_alter (fmap f).
Instance map_delete `{PartialAlter K A M} : Delete K M :=
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  partial_alter (λ _, None).
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Instance map_singleton `{PartialAlter K A M, Empty M} :
  Singleton (K * A) M := λ p, <[p.1:=p.2]> .
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Definition map_of_list `{Insert K A M, Empty M} : list (K * A)  M :=
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  fold_right (λ p, <[p.1:=p.2]>) .
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Definition map_of_collection `{Elements K C, Insert K A M, Empty M}
    (f : K  option A) (X : C) : M :=
  map_of_list (omap (λ i, (i,) <$> f i) (elements X)).
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Instance map_union_with `{Merge M} {A} : UnionWith A (M A) :=
  λ f, merge (union_with f).
Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
  λ f, merge (intersection_with f).
Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
  λ f, merge (difference_with f).
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(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
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Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
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  λ m,  i x, m !! i = Some x  P i x.
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Definition map_Forall2 `{ A, Lookup K A (M A)} {A B}
    (R : A  B  Prop) (P : A  Prop) (Q : B  Prop)
    (m1 : M A) (m2 : M B) : Prop :=  i,
  match m1 !! i, m2 !! i with
  | Some x, Some y => R x y
  | Some x, None => P x
  | None, Some y => Q y
  | None, None => True
  end.
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Definition map_included `{ A, Lookup K A (M A)} {A}
  (R : relation A) : relation (M A) := map_Forall2 R (λ _, False) (λ _, True).
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Instance map_disjoint `{ A, Lookup K A (M A)} {A} : Disjoint (M A) :=
  map_Forall2 (λ _ _, False) (λ _, True) (λ _, True).
Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
  map_Forall2 (=) (λ _, False) (λ _, True).
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(** The union of two finite maps only has a meaningful definition for maps
that are disjoint. However, as working with partial functions is inconvenient
in Coq, we define the union as a total function. In case both finite maps
have a value at the same index, we take the value of the first map. *)
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Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
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Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

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(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
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Instance map_difference `{Merge M} {A} : Difference (M A) :=
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  difference_with (λ _ _, None).
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(** A stronger variant of map that allows the mapped function to use the index
of the elements. Implemented by conversion to lists, so not very efficient. *)
Definition map_imap `{ A, Insert K A (M A),  A, Empty (M A),
     A, FinMapToList K A (M A)} {A B} (f : K  A  option B) (m : M A) : M B :=
  map_of_list (omap (λ ix, (fst ix,) <$> curry f ix) (map_to_list m)).

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(** * Theorems *)
Section theorems.
Context `{FinMap K M}.

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Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2   i, m1 !! i = m2 !! i.
Proof. split. by intros ->. apply map_eq. Qed.
Lemma map_subseteq_spec {A} (m1 m2 : M A) :
  m1  m2   i x, m1 !! i = Some x  m2 !! i = Some x.
Proof.
  unfold subseteq, map_subseteq, map_Forall2. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
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Global Instance: EmptySpec (M A).
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Proof.
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  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
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Qed.
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Global Instance:  {A} (R : relation A), PreOrder R  PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i)|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
  destruct (m1 !! i), (m2 !! i), (m3 !! i); try done; etransitivity; eauto.
Qed.
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Global Instance: PartialOrder (() : relation (M A)).
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Proof.
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  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
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Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
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Proof. rewrite !map_subseteq_spec. auto. Qed.
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Lemma lookup_weaken_is_Some {A} (m1 m2 : M A) i :
  is_Some (m1 !! i)  m1  m2  is_Some (m2 !! i).
Proof. inversion 1. eauto using lookup_weaken. Qed.
Lemma lookup_weaken_None {A} (m1 m2 : M A) i :
  m2 !! i = None  m1  m2  m1 !! i = None.
Proof.
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  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
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Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
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  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
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Lemma lookup_ne {A} (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.
Lemma map_empty {A} (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed.
Lemma lookup_empty_is_Some {A} i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. by inversion 1. Qed.
Lemma lookup_empty_Some {A} i (x : A) : ¬ !! i = Some x.
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m  .
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Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
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(** ** Properties of the [partial_alter] operation *)
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Lemma partial_alter_ext {A} (f g : option A  option A) (m : M A) i :
  ( x, m !! i = x  f x = g x)  partial_alter f i m = partial_alter g i m.
Proof.
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  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
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Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
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  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
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  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
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Qed.
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Lemma partial_alter_commute {A} f g (m : M A) i j :
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  i  j  partial_alter f i (partial_alter g j m) =
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    partial_alter g j (partial_alter f i m).
Proof.
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  intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?].
  { by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne. }
  destruct (decide (jj = i)) as [->|?].
  * by rewrite lookup_partial_alter,
     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
  * by rewrite !lookup_partial_alter_ne by congruence.
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Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
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  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
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Qed.
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Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
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Proof. by apply partial_alter_self_alt. Qed.
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Lemma partial_alter_subseteq {A} f (m : M A) i :
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  m !! i = None  m  partial_alter f i m.
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Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
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Lemma partial_alter_subset {A} f (m : M A) i :
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  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
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Proof.
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  intros Hi Hfi. split; [by apply partial_alter_subseteq|].
  rewrite !map_subseteq_spec. inversion Hfi as [x Hx]. intros Hm.
  apply (Some_ne_None x). rewrite <-(Hm i x); [done|].
  by rewrite lookup_partial_alter.
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Qed.

(** ** Properties of the [alter] operation *)
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Lemma alter_ext {A} (f g : A  A) (m : M A) i :
  ( x, m !! i = Some x  f x = g x)  alter f i m = alter g i m.
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Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
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Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
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Proof. unfold alter. apply lookup_partial_alter. Qed.
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Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
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Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
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Lemma alter_compose {A} (f g : A  A) (m : M A) i:
  alter (f  g) i m = alter f i (alter g i m).
Proof.
  unfold alter, map_alter. rewrite <-partial_alter_compose.
  apply partial_alter_ext. by intros [?|].
Qed.
Lemma alter_commute {A} (f g : A  A) (m : M A) i j :
  i  j  alter f i (alter g j m) = alter g j (alter f i m).
Proof. apply partial_alter_commute. Qed.
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Lemma lookup_alter_Some {A} (f : A  A) m i j y :
  alter f i m !! j = Some y 
    (i = j   x, m !! j = Some x  y = f x)  (i  j  m !! j = Some y).
Proof.
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  destruct (decide (i = j)) as [->|?].
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  * rewrite lookup_alter. naive_solver (simplify_option_equality; eauto).
  * rewrite lookup_alter_ne by done. naive_solver.
Qed.
Lemma lookup_alter_None {A} (f : A  A) m i j :
  alter f i m !! j = None  m !! j = None.
Proof.
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  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
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Qed.
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Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
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Proof.
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  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
  { rewrite lookup_alter; destruct (m !! j); f_equal'; auto. }
  by rewrite lookup_alter_ne by done.
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Qed.

(** ** Properties of the [delete] operation *)
Lemma lookup_delete {A} (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne {A} (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.
Lemma lookup_delete_Some {A} (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
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  * destruct (decide (i = j)) as [->|?];
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      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. by rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
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  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
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Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
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Lemma delete_singleton {A} i (x : A) : delete i {[i, x]} = .
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Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_commute {A} (m : M A) i j :
  delete i (delete j m) = delete j (delete i m).
Proof. destruct (decide (i = j)). by subst. by apply partial_alter_commute. Qed.
Lemma delete_insert_ne {A} (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. by apply partial_alter_commute. Qed.
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Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
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Proof.
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  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
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Qed.
Lemma delete_partial_alter {A} (m : M A) i f :
  m !! i = None  delete i (partial_alter f i m) = m.
Proof.
  intros. unfold delete, map_delete. rewrite <-partial_alter_compose.
  unfold compose. by apply partial_alter_self_alt.
Qed.
Lemma delete_insert {A} (m : M A) i x :
  m !! i = None  delete i (<[i:=x]>m) = m.
Proof. apply delete_partial_alter. Qed.
Lemma insert_delete {A} (m : M A) i x :
  m !! i = Some x  <[i:=x]>(delete i m) = m.
Proof.
  intros Hmi. unfold delete, map_delete, insert, map_insert.
  rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
  by apply partial_alter_self_alt.
Qed.
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Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
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Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
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Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
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  m1  m2  delete i m1  delete i m2.
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Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
Qed.
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Lemma delete_subset_alt {A} (m : M A) i x : m !! i = Some x  delete i m  m.
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Proof.
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  split; [apply delete_subseteq|].
  rewrite !map_subseteq_spec. intros Hi. apply (None_ne_Some x).
  by rewrite <-(lookup_delete m i), (Hi i x).
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Qed.
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Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
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Proof. inversion 1. eauto using delete_subset_alt. Qed.

(** ** Properties of the [insert] operation *)
Lemma lookup_insert {A} (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
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Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
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Proof. rewrite lookup_insert. congruence. Qed.
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Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
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Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
Lemma insert_commute {A} (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_insert_Some {A} (m : M A) i j x y :
  <[i:=x]>m !! j = Some y  (i = j  x = y)  (i  j  m !! j = Some y).
Proof.
  split.
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  * destruct (decide (i = j)) as [->|?];
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      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
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Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
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  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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Qed.
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Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
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Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|];
    by rewrite ?lookup_insert, ?lookup_insert_ne by done.
Qed.
Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
  ( y, m !! i = Some y  R y x)  map_included R m (<[i:=x]>m).
Proof.
  intros ? j; destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert. destruct (m !! j); eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j).
Qed.
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Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
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Proof. apply partial_alter_subseteq. Qed.
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Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
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Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
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  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
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Proof.
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  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [congruence|].
  rewrite lookup_insert_ne; auto.
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Qed.
Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
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Proof.
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  rewrite !map_subseteq_spec. intros Hi Hix j y Hj.
  destruct (decide (i = j)) as [->|]; [congruence|].
  rewrite lookup_delete_ne by done.
  apply Hix; by rewrite lookup_insert_ne by done.
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Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
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  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
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Proof.
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  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
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  * rewrite lookup_insert. congruence.
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  * rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
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Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
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Proof.
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  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
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Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2 
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   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
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  * rewrite insert_delete. done. eapply lookup_weaken, strict_include; eauto.
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    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
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Lemma fmap_insert {A B} (f : A  B) (m : M A) i x :
  f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  * by rewrite lookup_fmap, !lookup_insert.
  * by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
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Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i,x]}.
Proof. done. Qed.
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(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
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  {[i, x]} !! j = Some y  i = j  x = y.
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Proof.
  unfold singleton, map_singleton.
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  rewrite lookup_insert_Some, lookup_empty. simpl. intuition congruence.
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Qed.
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Lemma lookup_singleton_None {A} i j (x : A) : {[i, x]} !! j = None  i  j.
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Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.
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Lemma lookup_singleton {A} i (x : A) : {[i, x]} !! i = Some x.
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Proof. by rewrite lookup_singleton_Some. Qed.
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Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i, x]} !! j = None.
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Proof. by rewrite lookup_singleton_None. Qed.
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Lemma map_non_empty_singleton {A} i (x : A) : {[i,x]}  .
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Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
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Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i, x]} = {[i, y]}.
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Proof.
  unfold singleton, map_singleton, insert, map_insert.
  by rewrite <-partial_alter_compose.
Qed.
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Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i,x]} = {[i, f x]}.
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Proof.
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  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
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  * by rewrite lookup_alter, !lookup_singleton.
  * by rewrite lookup_alter_ne, !lookup_singleton_ne.
Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
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  i  j  alter f i {[j,x]} = {[j,x]}.
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Proof.
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  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
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Qed.

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(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
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Lemma omap_singleton {A B} (f : A  option B) i x y :
  f x = Some y  omap f {[ i,x ]} = {[ i,y ]}.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|].
  * by rewrite lookup_omap, !lookup_singleton.
  * by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
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(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
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  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
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Proof. rewrite !elem_of_map_to_list. congruence. Qed.
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Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
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Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
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Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
  (i,x)  l  ( y, (i,y)  l  y = x)  map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
  intros [?|?] Hdup; simplify_equality; [by rewrite lookup_insert|].
  destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert; f_equal; eauto.
  * rewrite lookup_insert_ne by done; eauto.
Qed.
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Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
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  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
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Proof.
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  intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst.
  intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
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  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
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    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
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Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
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  map_of_list l !! i = Some x  (i,x)  l.
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Proof.
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  induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|].
  rewrite elem_of_cons. destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
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  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
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Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
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Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
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  i  l.*1  map_of_list l !! i = None.
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Proof.
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  rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi.
  exists (i,x); simpl; auto using elem_of_map_of_list_2.
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Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
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  map_of_list l !! i = None  i  l.*1.
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Proof.
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  induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
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  rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
  * by rewrite lookup_insert.
  * by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
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  i  l.*1  map_of_list l !! i = None.
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Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
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Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
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  NoDup (l1.*1)  l1  l2  map_of_list l1 = map_of_list l2.
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Proof.
  intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
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  NoDup (l1.*1)  NoDup (l2.*1)  map_of_list l1 = map_of_list l2  l1  l2.
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Proof.
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  intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
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  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
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Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
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Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
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    by auto using NoDup_fst_map_to_list.
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Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
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  NoDup (l.*1)  map_to_list (map_of_list l)  l.
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Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
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Lemma map_to_list_inj {A} (m1 m2 : M A) :
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  map_to_list m1  map_to_list m2  m1 = m2.
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Proof.
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  intros. rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
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  auto using map_of_list_proper, NoDup_fst_map_to_list.
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Qed.
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Lemma map_to_of_list_flip {A} (m1 : M A) l2 :
  map_to_list m1  l2  m1 = map_of_list l2.
Proof.
  intros. rewrite <-(map_of_to_list m1).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
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Lemma map_to_list_empty {A} : map_to_list  = @nil (K * A).
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Proof.
  apply elem_of_nil_inv. intros [i x].
  rewrite elem_of_map_to_list. apply lookup_empty_Some.
Qed.
Lemma map_to_list_insert {A} (m : M A) i x :
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  m !! i = None  map_to_list (<[i:=x]>m)  (i,x) :: map_to_list m.
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Proof.
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  intros. apply map_of_list_inj; csimpl.
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  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
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    rewrite elem_of_list_fmap. intros [[??] [? Hlookup]]; subst; simpl in *.
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    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.
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Lemma map_of_list_nil {A} : map_of_list (@nil (K * A)) = .
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Proof. done. Qed.
Lemma map_of_list_cons {A} (l : list (K * A)) i x :
  map_of_list ((i, x) :: l) = <[i:=x]>(map_of_list l).
Proof. done. Qed.
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Lemma map_to_list_empty_inv_alt {A}  (m : M A) : map_to_list m  []  m = .
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Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
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Lemma map_to_list_empty_inv {A} (m : M A) : map_to_list m = []  m = .
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Proof. intros Hm. apply map_to_list_empty_inv_alt. by rewrite Hm. Qed.
Lemma map_to_list_insert_inv {A} (m : M A) l i x :
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  map_to_list m  (i,x) :: l  m = <[i:=x]>(map_of_list l).
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Proof.
  intros Hperm. apply map_to_list_inj.
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  assert (i  l.*1  NoDup (l.*1)) as [].
  { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm.
    auto using NoDup_fst_map_to_list. }
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  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
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Lemma map_choose {A} (m : M A) : m     i x, m !! i = Some x.
Proof.
  intros Hemp. destruct (map_to_list m) as [|[i x] l] eqn:Hm.
  { destruct Hemp; eauto using map_to_list_empty_inv. }
  exists i x. rewrite <-elem_of_map_to_list, Hm. by left.
Qed.
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(** Properties of the imap function *)
Lemma lookup_imap {A B} (f : K  A  option B) m i :
  map_imap f m !! i = m !! i = f i.
Proof.
  unfold map_imap; destruct (m !! i = f i) as [y|] eqn:Hi; simpl.
  * destruct (m !! i) as [x|] eqn:?; simplify_equality'.
    apply elem_of_map_of_list_1_help.
    { apply elem_of_list_omap; exists (i,x); split;
        [by apply elem_of_map_to_list|by simplify_option_equality]. }
    intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
    by rewrite elem_of_map_to_list in Hi'; simplify_option_equality.
  * apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
    intros ([i' x]&->&Hi'); simplify_equality'.
    rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
    rewrite elem_of_map_to_list in Hj; simplify_option_equality.
Qed.

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(** ** Properties of conversion from collections *)
Lemma lookup_map_of_collection {A} `{FinCollection K C}
    (f : K  option A) X i x :
  map_of_collection f X !! i = Some x  i  X  f i = Some x.
Proof.
  assert (NoDup (fst <$> omap (λ i, (i,) <$> f i) (elements X))).
  { induction (NoDup_elements X) as [|i' l]; csimpl; [constructor|].
    destruct (f i') as [x'|]; csimpl; auto; constructor; auto.
    rewrite elem_of_list_fmap. setoid_rewrite elem_of_list_omap.
    by intros (?&?&?&?&?); simplify_option_equality. }
  unfold map_of_collection; rewrite <-elem_of_map_of_list by done.
  rewrite elem_of_list_omap. setoid_rewrite elem_of_elements; split.
  * intros (?&?&?); simplify_option_equality; eauto.
  * intros [??]; exists i; simplify_option_equality; eauto.
Qed.

(** ** Induction principles *)
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Lemma map_ind {A} (P : M A  Prop) :
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  P   ( i x m, m !! i = None  P m  P (<[i:=x]>m))   m, P m.
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Proof.
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  intros ? Hins. cut ( l, NoDup (l.*1)   m, map_to_list m  l  P m).
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  { intros help m.
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    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
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  induction l as [|[i x] l IH]; intros Hnodup m Hml.
  { apply map_to_list_empty_inv_alt in Hml. by subst. }
  inversion_clear Hnodup.
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  apply map_to_list_insert_inv in Hml; subst m. apply Hins.
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  * by apply not_elem_of_map_of_list_1.
  * apply IH; auto using map_to_of_list.
Qed.
Lemma map_to_list_length {A} (m1 m2 : M A) :
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  m1  m2  length (map_to_list m1) < length (map_to_list m2).
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Proof.
  revert m2. induction m1 as [|i x m ? IH] using map_ind.
  { intros m2 Hm2. rewrite map_to_list_empty. simpl.
    apply neq_0_lt. intros Hlen. symmetry in Hlen.
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    apply nil_length_inv, map_to_list_empty_inv in Hlen.
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    rewrite Hlen in Hm2. destruct (irreflexivity ()  Hm2). }
  intros m2 Hm2.
  destruct (insert_subset_inv m m2 i x) as (m2'&?&?&?); auto; subst.
  rewrite !map_to_list_insert; simpl; auto with arith.
Qed.
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Lemma map_wf {A} : wf (strict (@subseteq (M A) _)).
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Proof.
  apply (wf_projected (<) (length  map_to_list)).
  * by apply map_to_list_length.
  * by apply lt_wf.
Qed.

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(** ** Properties of the [map_Forall] predicate *)
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Section map_Forall.
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Context {A} (P : K  A  Prop).

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Lemma map_Forall_to_list m : map_Forall P m  Forall (curry P) (map_to_list m).
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Proof.
  rewrite Forall_forall. split.
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  * intros Hforall [i x]. rewrite elem_of_map_to_list. by apply (Hforall i x).
  * intros Hforall i x. rewrite <-elem_of_map_to_list. by apply (Hforall (i,x)).
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Qed.
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Lemma map_Forall_empty : map_Forall P .
Proof. intros i x. by rewrite lookup_empty. Qed.
Lemma map_Forall_impl (Q : K  A  Prop) m :
  map_Forall P m  ( i x, P i x  Q i x)  map_Forall Q m.
Proof. unfold map_Forall; naive_solver. Qed.
Lemma map_Forall_insert_11 m i x : map_Forall P (<[i:=x]>m)  P i x.
Proof. intros Hm. by apply Hm; rewrite lookup_insert. Qed.
Lemma map_Forall_insert_12 m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  map_Forall P m.
Proof.
  intros ? Hm j y ?; apply Hm. by rewrite lookup_insert_ne by congruence.
Qed.
Lemma map_Forall_insert_2 m i x :
  P i x  map_Forall P m  map_Forall P (<[i:=x]>m).
Proof. intros ?? j y; rewrite lookup_insert_Some; naive_solver. Qed.
Lemma map_Forall_insert m i x :
  m !! i = None  map_Forall P (<[i:=x]>m)  P i x  map_Forall P m.
Proof.
  naive_solver eauto using map_Forall_insert_11,
    map_Forall_insert_12, map_Forall_insert_2.
Qed.
Lemma map_Forall_ind (Q : M A  Prop) :
  Q  
  ( m i x, m !! i = None  P i x  map_Forall P m  Q m  Q (<[i:=x]>m)) 
   m, map_Forall P m  Q m.
Proof.
  intros Hnil Hinsert m. induction m using map_ind; auto.
  rewrite map_Forall_insert by done; intros [??]; eauto.
Qed.
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Context `{ i x, Decision (P i x)}.
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Global Instance map_Forall_dec m : Decision (map_Forall P m).
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Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
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    by rewrite map_Forall_to_list.
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Defined.
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Lemma map_not_Forall (m : M A) :
  ¬map_Forall P m   i x, m !! i = Some x  ¬P i x.
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Proof.
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  split; [|intros (i&x&?&?) Hm; specialize (Hm i x); tauto].
  rewrite map_Forall_to_list. intros Hm.
  apply (not_Forall_Exists _), Exists_exists in Hm.
  destruct Hm as ([i x]&?&?). exists i x. by rewrite <-elem_of_map_to_list.
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Qed.
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End map_Forall.
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(** ** Properties of the [merge] operation *)
Section merge.
Context {A} (f : option A  option A  option A).
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Context `{!PropHolds (f None None = None)}.
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Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
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  by rewrite !(lookup_merge f), lookup_empty, (left_id_L None f).
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Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
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  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
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Qed.
Lemma merge_commutative m1 m2 :
  ( i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) 
  merge f m1 m2 = merge f m2 m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Commutative (=) f  Commutative (=) (merge f).
Proof.
  intros ???. apply merge_commutative. intros. by apply (commutative f).
Qed.
Lemma merge_associative m1 m2 m3 :
  ( i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
        f (f (m1 !! i) (m2 !! i)) (m3 !! i)) 
  merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Associative (=) f  Associative (=) (merge f).
Proof.