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(* Copyright (c) 2012-2014, 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;
  map_to_list_nodup {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}
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  (l : list (K * A)) : M := insert_list l .
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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.
(*
Definition map_intersection_Forall `{Lookup K A M}
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    (R : relation A) : relation M := λ m1 m2,
  ∀ i x1 x2, m1 !! i = Some x1 → m2 !! i = Some x2 → R x1 x2.
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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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(** * 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: BoundedPreOrder (M A).
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Proof.
  repeat split.
  * intros m. by rewrite map_subseteq_spec.
  * intros m1 m2 m3. rewrite !map_subseteq_spec. naive_solver.
  * intros m. rewrite !map_subseteq_spec. intros i x. by rewrite lookup_empty.
Qed.
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Global Instance : PartialOrder (@subseteq (M A) _).
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Proof.
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  split; [apply _ |]. intros ??. 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. 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. 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_None {A} (f : A  A) m i : m !! i = None  alter f i m = m.
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Proof.
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  intros Hi. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?Hi, ?lookup_alter_ne.
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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_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.

(** ** 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 singleton_ne_empty {A} i (x : A) : {[i,x]}  .
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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(** ** 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 map_to_list_key_nodup {A} (m : M A) : NoDup (fst <$> map_to_list m).
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, map_to_list_nodup. Qed.
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Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
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  NoDup (fst <$> l)  (i,x)  l  map_of_list l !! i = Some x.
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Proof.
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  induction l as [|[j y] l IH]; simpl; [by rewrite elem_of_nil|].
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  rewrite NoDup_cons, elem_of_cons, elem_of_list_fmap.
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  intros [Hl ?] [?|?]; simplify_equality; [by rewrite lookup_insert|].
  destruct (decide (i = j)) as [->|]; [|rewrite lookup_insert_ne; auto].
  destruct Hl. by exists (j,x).
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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 (fst <$> l)  (i,x)  l  map_of_list l !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
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Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
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  i  fst <$> l  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  fst <$> l.
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Proof.
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  induction l as [|[j y] l IH]; simpl; [rewrite elem_of_nil; tauto|].
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  rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
  * by rewrite lookup_insert.
  * by rewrite lookup_insert_ne; intuition.
Qed.
Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
  i  fst <$> l  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 (fst <$> l1)  l1  l2  map_of_list l1 = map_of_list l2.
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Proof.
  intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
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  NoDup (fst <$> l1)  NoDup (fst <$> l2) 
  map_of_list l1 = map_of_list l2  l1  l2.
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Proof.
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  intros ?? Hl1l2. apply NoDup_Permutation; auto using (fmap_nodup_1 fst).
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  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
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Lemma map_of_to_list {A} (m : M A) : map_of_list (map_to_list m) = m.
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Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
    by auto using map_to_list_key_nodup.
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
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  NoDup (fst <$> l)  map_to_list (map_of_list l)  l.
Proof. auto using map_of_list_inj, map_to_list_key_nodup, 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, map_to_list_key_nodup.
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.
  intros. apply map_of_list_inj; simpl.
  * apply map_to_list_key_nodup.
  * constructor; auto using map_to_list_key_nodup.
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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.
  assert (NoDup (fst <$> (i, x) :: l)) as Hnodup.
  { rewrite <-Hperm. auto using map_to_list_key_nodup. }
  simpl in Hnodup. rewrite NoDup_cons in Hnodup. destruct Hnodup.
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.

(** * Induction principles *)
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 (fst <$> l)   m, map_to_list m  l  P m).
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  { intros help m.
    apply (help (map_to_list m)); auto using map_to_list_key_nodup. }
  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.

(** ** 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.

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.
  split.
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  * rewrite map_Forall_to_list. intros Hm.
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    apply (not_Forall_Exists _), Exists_exists in Hm.
    destruct Hm as ([i x]&?&?). exists i x. by rewrite <-elem_of_map_to_list.
  * intros (i&x&?&?) Hm. specialize (Hm i x). tauto.
Qed.
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End map_Forall.
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(** ** Properties of the [merge] operation *)
Lemma merge_Some {A B C} (f : option A  option B  option C)
    `{!PropHolds (f None None = None)} m1 m2 m :
  ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
Proof.
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  split; [|intros <-; apply (lookup_merge _) ].
  intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _).
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Qed.

Section merge.
Context {A} (f : option A  option A  option A).

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.

Context `{!PropHolds (f None None = None)}.

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.
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  intros ????. apply merge_associative. intros. by apply (associative_L f).
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Qed.
Lemma merge_idempotent m1 :
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  ( i, f (m1 !! i) (m1 !! i) = m1 !! i)  merge f m1 m1 = m1.
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Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
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Proof. intros ??. apply merge_idempotent. intros. by apply (idempotent f). Qed.
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Lemma partial_alter_merge (g g1 g2 : option A  option A) m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) =
    merge f (partial_alter g1 i m1) (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_l (g g1 : option A  option A) m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (g1 (m1 !! i)) (m2 !! i) 
  partial_alter g i (merge f m1 m2) = merge f (partial_alter g1 i m1) m2.
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.
Lemma partial_alter_merge_r (g g2 : option A  option A) m1 m2 i :
  g (f (m1 !! i) (m2 !! i)) = f (m1 !! i) (g2 (m2 !! i)) 
  partial_alter g i (merge f m1 m2) = merge f m1 (partial_alter g2 i m2).
Proof.
  intro. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite (lookup_merge _), !lookup_partial_alter, !(lookup_merge _).
  * by rewrite (lookup_merge _), !lookup_partial_alter_ne, (lookup_merge _).
Qed.

Lemma insert_merge_l m1 m2 i x :
  f (Some x) (m2 !! i) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f (<[i:=x]>m1) m2.
Proof.
  intros. unfold insert, map_insert, alter, map_alter.
  by apply partial_alter_merge_l.
Qed.
Lemma insert_merge_r m1 m2 i x :
  f (m1 !! i) (Some x) = Some x 
  <[i:=x]>(merge f m1 m2) = merge f m1 (<[i:=x]>m2).
Proof.
  intros. unfold insert, map_insert, alter, map_alter.
  by apply partial_alter_merge_r.
Qed.
End merge.

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(** ** Properties on the [map_Forall2] relation *)
Section Forall2.
Context {A B} (R : A  B  Prop) (P : A  Prop) (Q : B  Prop).
Context `{ x y, Decision (R x y),  x, Decision (P x),  y, Decision (Q y)}.

Let f (mx : option A) (my : option B) : option bool :=
  match mx, my with
  | Some x, Some y => Some (bool_decide (R x y))
  | Some x, None => Some (bool_decide (P x))
  | None, Some y => Some (bool_decide (Q y))
  | None, None => None
  end.
Lemma map_Forall2_alt (m1 : M A) (m2 : M B) :
  map_Forall2 R P Q m1 m2  map_Forall (λ _ P, Is_true P) (merge f m1 m2).
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Proof.
  split.
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  * intros Hm i P'; rewrite lookup_merge by done; intros.
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
      simplify_equality; auto using bool_decide_pack.
  * intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); simplify_equality'; auto;
      by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_Forall2_dec `{ x y, Decision (R x y),  x, Decision (P x),
   y, Decision (Q y)} m1 m2 : Decision (map_Forall2 R P Q m1 m2).
Proof.
  refine (cast_if (decide (map_Forall (λ _ P, Is_true P) (merge f m1 m2))));
    abstract by rewrite map_Forall2_alt.
Defined.
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(** Due to the finiteness of finite maps, we can extract a witness if the
relation does not hold. *)
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Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
  ¬map_Forall2 R P Q m1 m2   i,
    ( x y, m1 !! i = Some x  m2 !! i = Some y  ¬R x y)
     ( x, m1 !! i = Some x  m2 !! i = None  ¬P x)
     ( y, m1 !! i = None  m2 !! i = Some y  ¬Q y).
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Proof.
  split.
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  * rewrite map_Forall2_alt, (map_not_Forall _). intros (i&?&Hm&?); exists i.
    rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
  * by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
      specialize (Hm i); simplify_option_equality.
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Qed.
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End Forall2.
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(** ** Properties on the disjoint maps *)
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Lemma map_disjoint_spec {A} (m1 m2 : M A) :
  m1  m2   i x y, m1 !! i = Some x  m2 !! i = Some y  False.
Proof.
  split; intros Hm i; specialize (Hm i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
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Lemma map_disjoint_alt {A} (m1 m2 : M A) :
  m1  m2   i, m1 !! i = None  m2 !! i = None.
Proof.
  split; intros Hm1m2 i; specialize (Hm1m2 i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_not_disjoint {A} (m1 m2 : M A) :
  ¬m1  m2   i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2.
Proof.
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  unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
  split; [|naive_solver].
  * intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
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Qed.
Global Instance: Symmetric (@disjoint (M A) _).
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Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
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Lemma map_disjoint_empty_l {A} (m : M A) :   m.
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Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
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Lemma map_disjoint_empty_r {A} (m : M A) : m  .
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Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
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Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
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  m1'  m2'  m1  m1'  m2  m2'  m1  m2.
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Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
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Lemma map_disjoint_weaken_l {A} (m1 m1' m2  : M A) :
  m1'  m2  m1  m1'  m1  m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
  m1