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(* Copyright (c) 2012-2013, 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}
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    `{ A, Lookup K A (M A)}
    `{ A, Empty (M A)}
    `{ A, PartialAlter K A (M A)}
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    `{!Merge M}
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    `{ A, FinMapToList K A (M A)}
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    `{ i j : K, Decision (i = j)} := {
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  map_eq {A} (m1 m2 : M A) :
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    ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i :
    ( : M A) !! i = None;
  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;
  lookup_fmap {A B} (f : A  B) (m : M A) i :
    (f <$> m) !! i = f <$> m !! i;
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  map_to_list_nodup {A} (m : M A) :
    NoDup (map_to_list m);
  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
  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}
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  `{Empty M} : Singleton (K * A) M := λ p, <[fst p:=snd p]>.
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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_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)} : Disjoint (M A) :=
  λ A, map_intersection_forall (λ _ _, False).
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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) :=
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  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}.

Global Instance map_subseteq {A} : SubsetEq (M A) := λ m1 m2,
   i x, m1 !! i = Some x  m2 !! i = Some x.
Global Instance: BoundedPreOrder (M A).
Proof. split; [firstorder |]. intros m i x. by rewrite lookup_empty. Qed.
Global Instance : PartialOrder (M A).
Proof.
  split; [apply _ |].
  intros ????. apply map_eq. intros i. apply option_eq. naive_solver.
Qed.

Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
Proof. auto. Qed.
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.
  rewrite eq_None_not_Some. intros Hm2 Hm1m2.
  specialize (Hm1m2 i). destruct (m1 !! i); naive_solver.
Qed.

Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
  m1 !! i = Some x 
  m1  m2 
  m2 !! i = Some y 
  x = y.
Proof.
  intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto.
  congruence.
Qed.

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  .
Proof. intros [? []]. intros i x. by rewrite lookup_empty. Qed.

(** ** Properties of the [partial_alter] operation *)
Lemma partial_alter_compose {A} (m : M A) i f g :
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply map_eq. intros ii. case (decide (i = ii)).
  * intros. subst. by rewrite !lookup_partial_alter.
  * intros. by rewrite !lookup_partial_alter_ne.
Qed.
Lemma partial_alter_commute {A} (m : M A) i j f g :
  i  j 
  partial_alter f i (partial_alter g j m) =
    partial_alter g j (partial_alter f i m).
Proof.
  intros. apply map_eq. intros jj.
  destruct (decide (jj = j)).
  * subst. by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne.
  * destruct (decide (jj = i)).
    + subst. by rewrite lookup_partial_alter,
       !lookup_partial_alter_ne, lookup_partial_alter by congruence.
    + by rewrite !lookup_partial_alter_ne by congruence.
Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
  intros. apply map_eq. intros ii.
  destruct (decide (i = ii)).
  * subst. by rewrite lookup_partial_alter.
  * by rewrite lookup_partial_alter_ne.
Qed.
Lemma partial_alter_self {A} (m : M A) i :
  partial_alter (λ _, m !! i) i m = m.
Proof. by apply partial_alter_self_alt. Qed.

Lemma partial_alter_subseteq {A} (m : M A) i f :
  m !! i = None 
  m  partial_alter f i m.
Proof.
  intros Hi j x Hj. rewrite lookup_partial_alter_ne; congruence.
Qed.
Lemma partial_alter_subset {A} (m : M A) i f :
  m !! i = None 
  is_Some (f (m !! i)) 
  m  partial_alter f i m.
Proof.
  intros Hi Hfi. split.
  * by apply partial_alter_subseteq.
  * inversion Hfi as [x Hx]. intros Hm.
    apply (Some_ne_None x). rewrite <-(Hm i x); [done|].
    by rewrite lookup_partial_alter.
Qed.

(** ** Properties of the [alter] operation *)
Lemma lookup_alter {A} (f : A  A) m i :
  alter f i m !! i = f <$> m !! i.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_alter_ne {A} (f : A  A) m i j :
  i  j  alter f i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.

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.
  destruct (decide (i = j)); subst.
  * 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.
  destruct (decide (i = j)); subst.
  * by rewrite lookup_alter, fmap_None.
  * by rewrite lookup_alter_ne.
Qed.

Lemma alter_None {A} (f : A  A) m i :
  m !! i = None  alter f i m = m.
Proof.
  intros Hi. apply map_eq. intros j. destruct (decide (i = j)); subst.
  * by rewrite lookup_alter, !Hi.
  * by rewrite lookup_alter_ne.
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.
  * destruct (decide (i = j)); subst;
      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.
  destruct (decide (i = j)).
  * subst. rewrite lookup_delete. tauto.
  * rewrite lookup_delete_ne; tauto.
Qed.

Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
Lemma delete_singleton {A} i (x : A) : delete i {[(i, x)]} = .
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.

Lemma delete_notin {A} (m : M A) i :
  m !! i = None  delete i m = m.
Proof.
  intros. apply map_eq. intros j.
  destruct (decide (i = j)).
  * subst. by rewrite lookup_delete.
  * by apply lookup_delete_ne.
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.

Lemma delete_subseteq {A} (m : M A) i :
  delete i m  m.
Proof. intros j x. rewrite lookup_delete_Some. tauto. Qed.
Lemma delete_subseteq_compat {A} (m1 m2 : M A) i :
  m1  m2 
  delete i m1  delete i m2.
Proof. intros ? j x. rewrite !lookup_delete_Some. intuition eauto. Qed.
Lemma delete_subset_alt {A} (m : M A) i x :
  m !! i = Some x 
  delete i m  m.
Proof.
  split.
  * apply delete_subseteq.
  * intros Hi. apply (None_ne_Some x).
    by rewrite <-(lookup_delete m i), (Hi i x).
Qed.
Lemma delete_subset {A} (m : M A) i :
  is_Some (m !! i) 
  delete i m  m.
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.
Lemma lookup_insert_rev {A}  (m : M A) i x y :
  <[i:=x]>m !! i = Some y  x = y.
Proof. rewrite lookup_insert. congruence. Qed.
Lemma lookup_insert_ne {A} (m : M A) i j x :
  i  j  <[i:=x]>m !! j = m !! j.
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.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [[??]|[??]].
    + subst. apply lookup_insert.
    + by rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [??]. by rewrite lookup_insert_ne.
Qed.

Lemma insert_subseteq {A} (m : M A) i x :
  m !! i = None 
  m  <[i:=x]>m.
Proof. apply partial_alter_subseteq. Qed.
Lemma insert_subset {A} (m : M A) i x :
  m !! i = None 
  m  <[i:=x]>m.
Proof. intro. apply partial_alter_subset; eauto. Qed.
Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
  m1 !! i = None 
  m1  m2 
  m1  <[i:=x]>m2.
Proof.
  intros ?? j ?. destruct (decide (j = i)); subst.
  * congruence.
  * rewrite lookup_insert_ne; auto.
Qed.

Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = None 
  <[i:=x]> m1  m2 
  m1  delete i m2.
Proof.
  intros Hi Hix j y Hj. destruct (decide (i = j)); subst.
  * congruence.
  * rewrite lookup_delete_ne by done. apply Hix.
    by rewrite lookup_insert_ne by done.
Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
  m1 !! i = Some x 
  delete i m1  m2 
  m1  <[i:=x]> m2.
Proof.
  intros Hix Hi j y Hj. destruct (decide (i = j)); subst.
  * rewrite lookup_insert. congruence.
  * rewrite lookup_insert_ne by done. apply Hi.
    by rewrite lookup_delete_ne.
Qed.

Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
  m1 !! i = None 
  <[i:=x]> m1  m2 
  m1  delete i m2.
Proof.
  intros ? [Hm12 Hm21]. split.
  * eauto using insert_delete_subseteq.
  * contradict Hm21. apply delete_insert_subseteq; auto.
    apply Hm12. by rewrite lookup_insert.
Qed.

Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
  m1 !! i = None 
  <[i:=x]> m1  m2 
   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
  intros Hi Hm1m2. exists (delete i m2). split_ands.
  * rewrite insert_delete. done.
    eapply lookup_weaken, subset_subseteq; eauto.
    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) :
  {[(i, x)]} !! j = Some y  i = j  x = y.
Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_Some, lookup_empty. simpl.
  intuition congruence.
Qed.
Lemma lookup_singleton_None {A} i j (x : A) :
  {[(i, x)]} !! j = None  i  j.
Proof.
  unfold singleton, map_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.
Lemma lookup_singleton {A} i (x : A) : {[(i, x)]} !! i = Some x.
Proof. by rewrite lookup_singleton_Some. Qed.
Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[(i, x)]} !! j = None.
Proof. by rewrite lookup_singleton_None. Qed.

Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[(i, x)]} = {[(i, y)]}.
Proof.
  unfold singleton, map_singleton, insert, map_insert.
  by rewrite <-partial_alter_compose.
Qed.
Lemma alter_singleton {A} (f : A  A) i x :
  alter f i {[ (i,x) ]} = {[ (i, f x) ]}.
Proof.
  intros. apply map_eq. intros i'. destruct (decide (i = i')); subst.
  * 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 :
  i  j  alter f i {[ (j,x) ]} = {[ (j, x) ]}.
Proof.
  intros. apply map_eq. intros i'. destruct (decide (i = i')); subst.
  * by rewrite lookup_alter, lookup_singleton_ne.
  * by rewrite lookup_alter_ne.
Qed.

(** ** Properties of conversion to lists *)
Lemma map_to_list_unique {A} (m : M A) i x y :
  (i,x)  map_to_list m 
  (i,y)  map_to_list m 
  x = y.
Proof. rewrite !elem_of_map_to_list. congruence. Qed.
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.

Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
  NoDup (fst <$> l) 
  (i,x)  l 
  map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; simpl.
  { by rewrite elem_of_nil. }
  rewrite NoDup_cons, elem_of_cons, elem_of_list_fmap.
  intros [Hl ?] [?|?]; simplify_equality.
  { by rewrite lookup_insert. }
  destruct (decide (i = j)); simplify_equality.
  * destruct Hl. by exists (j,x).
  * rewrite lookup_insert_ne; auto.
Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
  map_of_list l !! i = Some x 
  (i,x)  l.
Proof.
  induction l as [|[j y] l IH]; simpl.
  { by rewrite lookup_empty. }
  rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
  * rewrite lookup_insert; intuition congruence.
  * rewrite lookup_insert_ne; intuition congruence.
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
  NoDup (fst <$> 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.

Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
  i  fst <$> l 
  map_of_list l !! i = None.
Proof.
  rewrite elem_of_list_fmap, eq_None_not_Some, is_Some_alt.
  intros Hi [x ?]. destruct Hi. exists (i,x). simpl.
  auto using elem_of_map_of_list_2.
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
  map_of_list l !! i = None 
  i  fst <$> l.
Proof.
  induction l as [|[j y] l IH]; simpl.
  { rewrite elem_of_nil. tauto. }
  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.
Proof.
  split; auto using not_elem_of_map_of_list_1,
    not_elem_of_map_of_list_2.
Qed.

Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
  NoDup (fst <$> l1) 
  Permutation l1 l2 
  map_of_list l1 = map_of_list l2.
Proof.
  intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
Qed.
Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
  NoDup (fst <$> l1) 
  NoDup (fst <$> l2) 
  map_of_list l1 = map_of_list l2 
  Permutation l1 l2.
Proof.
  intros ?? Hl1l2.
  apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
  intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
Qed.
Lemma map_of_to_list {A} (m : M A) :
  map_of_list (map_to_list m) = m.
Proof.
  apply map_eq. intros i. apply option_eq. intros x.
  by rewrite <-elem_of_map_of_list, elem_of_map_to_list
    by auto using map_to_list_key_nodup.
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
  NoDup (fst <$> l) 
  Permutation (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.
Lemma map_to_list_inj {A} (m1 m2 : M A) :
  Permutation (map_to_list m1) (map_to_list m2) 
  m1 = m2.
Proof.
  intros.
  rewrite <-(map_of_to_list m1), <-(map_of_to_list m2).
  auto using map_of_list_proper, map_to_list_key_nodup.
Qed.

Lemma map_to_list_empty {A} :
  map_to_list  = @nil (K * A).
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 :
  m !! i = None 
  Permutation (map_to_list (<[i:=x]>m)) ((i,x) :: map_to_list m).
Proof.
  intros. apply map_of_list_inj; simpl.
  * apply map_to_list_key_nodup.
  * constructor; auto using map_to_list_key_nodup.
    rewrite elem_of_list_fmap.
    intros [[??] [? Hlookup]]; subst; simpl in *.
    rewrite elem_of_map_to_list in Hlookup. congruence.
  * by rewrite !map_of_to_list.
Qed.

Lemma map_of_list_nil {A} :
  map_of_list (@nil (K * A)) = .
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.

Lemma map_to_list_empty_inv_alt {A}  (m : M A) :
  Permutation (map_to_list m) []  m = .
Proof. rewrite <-map_to_list_empty. apply map_to_list_inj. Qed.
Lemma map_to_list_empty_inv {A} (m : M A) :
  map_to_list m = []  m = .
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 :
  Permutation (map_to_list m) ((i,x) :: l) 
  m = <[i:=x]>(map_of_list l).
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) :
  P  
  ( i x m, m !! i = None  P m  P (<[i:=x]>m)) 
   m, P m.
Proof.
  intros Hemp Hins.
  cut ( l, NoDup (fst <$> l)   m, Permutation (map_to_list m) l  P m).
  { 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.
  apply map_to_list_insert_inv in Hml. subst. apply Hins.
  * 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) :
  m1  m2 
  length (map_to_list m1) < length (map_to_list m2).
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.
    apply nil_length, map_to_list_empty_inv in Hlen.
    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.

Lemma map_wf {A} : wf (@subset (M A) _).
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 *)
Section map_forall.
Context {A} (P : K  A  Prop).

Lemma map_forall_to_list m :
  map_forall P m  Forall (curry P) (map_to_list m).
Proof.
  rewrite Forall_forall. split.
  * 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)).
Qed.

Context `{ i x, Decision (P i x)}.
Global Instance map_forall_dec m : Decision (map_forall P m).
Proof.
  refine (cast_if (decide (Forall (curry P) (map_to_list m))));
    by rewrite map_forall_to_list.
Defined.

Lemma map_not_forall (m : M A) :
  ¬map_forall P m   i x, m !! i = Some x  ¬P i x.
Proof.
  split.
  * 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.
  * intros (i&x&?&?) Hm. specialize (Hm i x). tauto.
Qed.
End map_forall.

(** ** 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.
  split; [| intro; subst; apply (lookup_merge _) ].
  intros Hlookup. apply map_eq. intros. rewrite Hlookup.
  apply (lookup_merge _).
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.
  by rewrite !(lookup_merge f), lookup_empty, (left_id None f).
Qed.
Global Instance: RightId (=) None f  RightId (=)  (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (right_id None f).
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.
  intros ????. apply merge_associative. intros. by apply (associative f).
Qed.
Lemma merge_idempotent m1 :
  ( i, f (m1 !! i) (m1 !! i) = m1 !! i) 
  merge f m1 m1 = m1.
Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
Proof.
  intros ??. apply merge_idempotent. intros. by apply (idempotent f).
Qed.

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.

(** ** Properties on the [map_intersection_forall] relation *)
Section intersection_forall.
Context {A} (R : relation A).

Global Instance map_intersection_forall_sym:
  Symmetric R  Symmetric (map_intersection_forall R).
Proof. firstorder auto. Qed.
Lemma map_intersection_forall_empty_l (m : M A) :
  map_intersection_forall R  m.
Proof. intros ???. by rewrite lookup_empty. Qed.
Lemma map_intersection_forall_empty_r (m : M A) :
  map_intersection_forall R m .
Proof. intros ???. by rewrite lookup_empty. Qed.

Lemma map_intersection_forall_alt (m1 m2 : M A) :
  map_intersection_forall R m1 m2 
    map_forall (λ _, curry R) (merge (λ x y,
      match x, y with
      | Some x, Some y => Some (x,y)
      | _, _ => None
      end) m1 m2).
Proof.
  split.
  * intros Hm12 i [x y]. rewrite lookup_merge by done. intros.
    destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simplify_equality.
    eapply Hm12; eauto.
  * intros Hm12 i x y ??. apply (Hm12 i (x,y)).
    rewrite lookup_merge by done. by simplify_option_equality.
Qed.

(** Due to the finiteness of finite maps, we can extract a witness if the
relation does not hold. *)
Lemma map_not_intersection_forall `{ x y, Decision (R x y)} (m1 m2 : M A) :
  ¬map_intersection_forall R m1 m2
      i x1 x2, m1 !! i = Some x1  m2 !! i = Some x2  ¬R x1 x2.
Proof.
  split.
  * rewrite map_intersection_forall_alt, (map_not_forall _).
    intros (i & [x y] & Hm12 & ?). rewrite lookup_merge in Hm12 by done.
    exists i x y. by destruct (m1 !! i), (m2 !! i); simplify_equality.
  * intros (i & x1 & x2 & Hx1 & Hx2 & Hx1x2) Hm12.
    by apply Hx1x2, (Hm12 i x1 x2).
Qed.
End intersection_forall.

(** ** Properties on the disjoint maps *)
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.
  unfold disjoint, map_disjoint.
  rewrite map_not_intersection_forall.
  * naive_solver.
  * right. auto.
Qed.

Global Instance: Symmetric (@disjoint (M A) _).
Proof. intro. apply map_intersection_forall_sym. auto. Qed.
Lemma map_disjoint_empty_l {A} (m : M A) :   m.
Proof. apply map_intersection_forall_empty_l. Qed.
Lemma map_disjoint_empty_r {A} (m : M A) : m  .
Proof. apply map_intersection_forall_empty_r. Qed.

Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
  m1'  m2' 
  m1  m1'  m2  m2' 
  m1  m2.
Proof.
  intros Hdisjoint Hm1 Hm2 i x1 x2 Hx1 Hx2.
  destruct (Hdisjoint i x1 x2); auto.
Qed.
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  m2'  m2  m2'  m1  m2.
Proof. eauto using map_disjoint_weaken. Qed.

Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
  m1  m2 
  m1 !! i = Some x 
  m2 !! i = None.
Proof.
  intros Hdisjoint ?. rewrite eq_None_not_Some, is_Some_alt.
  intros [x2 ?]. by apply (Hdisjoint i x x2).
Qed.
Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
  m1  m2 
  m2 !! i = Some x 
  m1 !! i = None.
Proof. rewrite (symmetry_iff ()). apply map_disjoint_Some_l. Qed.

Lemma map_disjoint_singleton_l {A} (m : M A) i x :
  {[(i, x)]}  m  m !! i = None.
Proof.
  split.
  * intro. apply (map_disjoint_Some_l {[(i, x)]} _ _ x);
      auto using lookup_singleton.
  * intros ? j y1 y2. destruct (decide (i = j)); subst.
    + rewrite lookup_singleton. intuition congruence.
    + by rewrite lookup_singleton_ne.
Qed.
Lemma map_disjoint_singleton_r {A} (m : M A) i x :
  m  {[(i, x)]}  m !! i = None.
Proof. by rewrite (symmetry_iff ()), map_disjoint_singleton_l. Qed.

Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x :
  m !! i = None  {[(i, x)]}  m.
Proof. by rewrite map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_r_2 {A} (m : M A) i x :
  m !! i = None  m  {[(i, x)]}.
Proof. by rewrite map_disjoint_singleton_r. Qed.

Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i :
  m1  m2  delete i m1  m2.
Proof.
  rewrite !map_disjoint_alt.
  intros Hdisjoint j. destruct (Hdisjoint j); auto.
  rewrite lookup_delete_None. tauto.
Qed.
Lemma map_disjoint_delete_r {A} (m1 m2 : M A) i :
  m1  m2  m1  delete i m2.
Proof. symmetry. by apply map_disjoint_delete_l. Qed.

(** ** Properties of the [union_with] operation *)
Section union_with.
Context {A} (f : A  A  option A).

Lemma lookup_union_with m1 m2 i z :
  union_with f m1 m2 !! i = z 
    (m1 !! i = None  m2 !! i = None  z = None) 
    ( x, m1 !! i = Some x  m2 !! i = None  z = Some x) 
    ( y, m1 !! i = None  m2 !! i = Some y  z = Some y) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  z = f x y).
Proof.
  unfold union_with, map_union_with. rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma lookup_union_with_Some m1 m2 i z :
  union_with f m1 m2 !! i = Some z 
    (m1 !! i = Some z  m2 !! i = None) 
    (m1 !! i = None  m2 !! i = Some z) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof. rewrite lookup_union_with. naive_solver. Qed.
Lemma lookup_union_with_None m1 m2 i :
  union_with f m1 m2 !! i = None 
    (m1 !! i = None  m2 !! i = None) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = None).
Proof. rewrite lookup_union_with. naive_solver. Qed.

Lemma lookup_union_with_Some_lr m1 m2 i x y z :
  m1 !! i = Some x 
  m2 !! i = Some y 
  f x y = Some z 
  union_with f m1 m2 !! i = Some z.
Proof. rewrite lookup_union_with. naive_solver. Qed.
Lemma lookup_union_with_Some_l m1 m2 i x :
  m1 !! i = Some x 
  m2 !! i = None 
  union_with f m1 m2 !! i = Some x.
Proof. rewrite lookup_union_with. naive_solver. Qed.
Lemma lookup_union_with_Some_r m1 m2 i y :
  m1 !! i = None 
  m2 !! i = Some y 
  union_with f m1 m2 !! i = Some y.
Proof. rewrite lookup_union_with. naive_solver. Qed.

Global Instance: LeftId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.
Global Instance: RightId (@eq (M A))  (union_with f).
Proof. unfold union_with, map_union_with. apply _. Qed.

Lemma union_with_commutative m1 m2 :
  ( i x y, m1 !! i = Some x  m2 !! i = Some y  f x y = f y x) 
  union_with f m1 m2 = union_with f m2 m1.
Proof.
  intros. apply (merge_commutative _). intros i.
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Global Instance: Commutative (=) f  Commutative (@eq (M A)) (union_with f).
Proof. intros ???. apply union_with_commutative. eauto. Qed.

Lemma union_with_idempotent m :
  ( i x, m !! i = Some x  f x x = Some x) 
  union_with f m m = m.
Proof.
  intros. apply (merge_idempotent _). intros i.
  destruct (m !! i) eqn:?; simpl; eauto.
Qed.

Lemma alter_union_with (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) (g y)) 
  alter g i (union_with f m1 m2) =
    union_with f (alter g i m1) (alter g i m2).
Proof.
  intros. apply (partial_alter_merge _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Lemma alter_union_with_l (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f (g x) y) 
  ( y, m1 !! i = None  m2 !! i = Some y  g y = y) 
  alter g i (union_with f m1 m2) = union_with f (alter g i m1) m2.
Proof.
  intros. apply (partial_alter_merge_l _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto using f_equal.
Qed.
Lemma alter_union_with_r (g : A  A) m1 m2 i :
  ( x y, m1 !! i = Some x  m2 !! i = Some y  g <$> f x y = f x (g y)) 
  ( x, m1 !! i = Some x  m2 !! i = None  g x = x) 
  alter g i (union_with f m1 m2) = union_with f m1 (alter g i m2).
Proof.
  intros. apply (partial_alter_merge_r _).
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto using f_equal.
Qed.

Lemma delete_union_with m1 m2 i :
  delete i (union_with f m1 m2) = union_with f (delete i m1) (delete i m2).
Proof. by apply (partial_alter_merge _). Qed.

Lemma delete_list_union_with (m1 m2 : M A) is :
  delete_list is (union_with f m1 m2) =
    union_with f (delete_list is m1) (delete_list is m2).
Proof. induction is; simpl. done. by rewrite IHis, delete_union_with. Qed.

Lemma insert_union_with m1 m2 i x :
  ( x, f x x = Some x) 
  <[i:=x]>(union_with f m1 m2) = union_with f (<[i:=x]>m1) (<[i:=x]>m2).
Proof. intros. apply (partial_alter_merge _). simpl. auto. Qed.
Lemma insert_union_with_l m1 m2 i x :
  m2 !! i = None 
  <[i:=x]>(union_with f m1 m2) = union_with f (<[i:=x]>m1) m2.
Proof.
  intros Hm2. unfold union_with, map_union_with.
  rewrite (insert_merge_l _). done. by rewrite Hm2.
Qed.
Lemma insert_union_with_r m1 m2 i x :
  m1 !! i = None 
  <[i:=x]>(union_with f m1 m2) = union_with f m1 (<[i:=x]>m2).
Proof.
  intros Hm1. unfold union_with, map_union_with.
  rewrite (insert_merge_r _). done. by rewrite Hm1.
Qed.
End union_with.

(** ** Properties of the [union] operation *)
Global Instance: LeftId (@eq (M A))  () := _.
Global Instance: RightId (@eq (M A))  () := _.
Global Instance: Associative (@eq (M A)) ().
Proof.
  intros A m1 m2 m3. unfold union, map_union, union_with, map_union_with.
  apply (merge_associative _). intros i.
  by destruct (m1 !! i), (m2 !! i), (m3 !! i).
Qed.
Global Instance: Idempotent (@eq (M A)) ().
Proof. intros A ?. by apply union_with_idempotent. Qed.

Lemma lookup_union_Some_raw {A} (m1 m2 : M A) i x :
  (m1  m2) !! i = Some x 
    m1 !! i = Some x  (m1 !! i = None  m2 !! i = Some x).
Proof.
  unfold union, map_union, union_with, map_union_with.
  rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
Lemma lookup_union_None {A} (m1 m2 : M A) i :
  (m1  m2) !! i = None  m1 !! i = None  m2 !! i = None.
Proof.
  unfold union, map_union, union_with, map_union_with.
  rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.

Lemma lookup_union_Some {A} (m1 m2 : M A) i x :
  m1  m2 
  (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = Some x.
Proof.
  intros Hdisjoint. rewrite lookup_union_Some_raw.
  intuition eauto using map_disjoint_Some_r.
Qed.

Lemma lookup_union_Some_l {A} (m1 m2 : M A) i x :
  m1 !! i = Some x 
  (m1  m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some_raw; intuition. Qed.
Lemma lookup_union_Some_r {A} (m1 m2 : M A) i x :
  m1  m2 
  m2 !! i = Some x 
  (m1  m2) !! i = Some x.
Proof. intro. rewrite lookup_union_Some; intuition. Qed.

Lemma map_union_commutative {A} (m1 m2 : M A) :
  m1  m2 
  m1  m2 = m2  m1.
Proof.
  intros Hdisjoint. apply (merge_commutative (union_with (λ x _, Some x))).
  intros i. specialize (Hdisjoint i).
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

Lemma map_subseteq_union {A} (m1 m2 : M A) :
  m1  m2 
  m1  m2 = m2.
Proof.
  intros Hm1m2.
  apply map_eq. intros i. apply option_eq. intros x.
  rewrite lookup_union_Some_raw. split; [by intuition |].
  intros Hm2. specialize (Hm1m2 i).
  destruct (m1 !! i) as [y|]; [| by auto].
  rewrite (Hm1m2 y eq_refl) in Hm2. intuition congruence.
Qed.

Lemma map_union_subseteq_l {A} (m1 m2 : M A) :
  m1  m1  m2.
Proof. intros ? i x. rewrite lookup_union_Some_raw. intuition. Qed.
Lemma map_union_subseteq_r {A} (m1 m2 : M A) :
  m1  m2 
  m2  m1  m2.
Proof.
  intros. rewrite map_union_commutative by done.
  by apply map_union_subseteq_l.
Qed.

Lemma map_union_subseteq_l_alt {A} (m1 m2 m3 : M A) :
  m1  m2 
  m1  m2  m3.
Proof. intros. transitivity m2; auto using map_union_subseteq_l. Qed.
Lemma map_union_subseteq_r_alt {A} (m1 m2 m3 : M A) :
  m2  m3 
  m1  m3 
  m1  m2  m3.
Proof. intros. transitivity m3; auto using map_union_subseteq_r. Qed.

Lemma map_union_preserving_l {A} (m1 m2 m3 : M A) :
  m1  m2 
  m3  m1  m3  m2.
Proof. intros ???. rewrite !lookup_union_Some_raw. naive_solver. Qed.
Lemma map_union_preserving_r {A} (m1 m2 m3 : M A) :
  m2  m3 
  m1  m2 
  m1  m3  m2  m3.
Proof.
  intros. rewrite !(map_union_commutative _ m3)
    by eauto using map_disjoint_weaken_l.
  by apply map_union_preserving_l.
Qed.

Lemma map_union_reflecting_l {A} (m1 m2 m3 : M A) :
  m3  m1 
  m3  m2 
  m3  m1  m3  m2 
  m1  m2.
Proof.
  intros Hm3m1 Hm3m2 E b x ?.
  specialize (E b x). rewrite !lookup_union_Some in E by done.
  destruct E; auto. by destruct (Hm3m1 b x x).
Qed.
Lemma map_union_reflecting_r {A} (m1 m2 m3 : M A) :
  m1  m3 
  m2  m3 
  m1  m3  m2  m3 
  m1  m2.
Proof.
  intros ??. rewrite !(map_union_commutative _ m3) by done.
  by apply map_union_reflecting_l.
Qed.

Lemma map_union_cancel_l {A} (m1 m2 m3 : M A) :
  m1  m3 
  m2  m3 
  m3  m1 = m3  m2 
  m1 = m2.
Proof.
  intros. by apply (anti_symmetric _);
    apply map_union_reflecting_l with m3; auto with congruence.
Qed.
Lemma map_union_cancel_r {A} (m1 m2 m3 : M A) :
  m1  m3 
  m2  m3 
  m1  m3 = m2  m3 
  m1 = m2.
Proof.
  intros. apply (anti_symmetric _);
    apply map_union_reflecting_r with m3; auto with congruence.
Qed.

Lemma map_disjoint_union_l {A} (m1 m2 m3 : M A) :
  m1  m2  m3  m1  m3  m2  m3.
Proof.
  rewrite !map_disjoint_alt.
  setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_r {A} (m1 m2 m3 : M A) :
  m1  m2  m3  m1  m2  m1  m3.
Proof.
  rewrite !map_disjoint_alt.
  setoid_rewrite lookup_union_None. naive_solver.
Qed.
Lemma map_disjoint_union_l_2 {A} (m1 m2 m3 : M A) :
  m1  m3  m2  m3  m1  m2  m3.
Proof. by rewrite map_disjoint_union_l. Qed.
Lemma map_disjoint_union_r_2 {A} (m1 m2 m3 : M A) :
  m1  m2  m1  m3  m1  m2  m3.
Proof. by rewrite map_disjoint_union_r. Qed.

Lemma insert_union_singleton_l {A} (m : M A) i x :
  <[i:=x]>m = {[(i,x)]}  m.
Proof.
  apply map_eq. intros j. apply option_eq. intros y.
  rewrite lookup_union_Some_raw.
  destruct (decide (i = j)); subst.
  * rewrite !lookup_singleton, lookup_insert. intuition congruence.
  * rewrite !lookup_singleton_ne, lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_union_singleton_r {A} (m : M A) i x :
  m !! i = None 
  <[i:=x]>m = m  {[(i,x)]}.
Proof.
  intro. rewrite insert_union_singleton_l, map_union_commutative; [done |].
  by apply map_disjoint_singleton_l.
Qed.

Lemma map_disjoint_insert_l {A} (m1 m2 : M A) i x :
  <[i:=x]>m1  m2  m2 !! i = None  m1  m2.
Proof.
  rewrite insert_union_singleton_l.
  by rewrite map_disjoint_union_l, map_disjoint_singleton_l.
Qed.
Lemma map_disjoint_insert_r {A} (m1 m2 : M A) i x :
  m1  <[i:=x]>m2  m1 !! i = None  m1  m2.
Proof.
  rewrite insert_union_singleton_l.
  by rewrite map_disjoint_union_r, map_disjoint_singleton_r.
Qed.

Lemma map_disjoint_insert_l_2 {A} (m1 m2 : M A) i x :
  m2 !! i = None  m1  m2  <[i:=x]>m1  m2.
Proof. by rewrite map_disjoint_insert_l. Qed.
Lemma map_disjoint_insert_r_2 {A} (m1 m2 : M A) i x :
  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
Proof. by rewrite map_disjoint_insert_r. Qed.

Lemma insert_union_l {A} (m1 m2 : M A) i x :
  <[i:=x]>(m1  m2) = <[i:=x]>m1  m2.
Proof. by rewrite !insert_union_singleton_l, (associative ()). Qed.
Lemma insert_union_r {A} (m1 m2 : M A) i x :
  m1 !! i = None 
  <[i:=x]>(m1  m2) = m1  <[i:=x]>m2.
Proof.
  intro. rewrite !insert_union_singleton_l, !(associative ()).
  rewrite (map_union_commutative m1); [done |].
  by apply map_disjoint_singleton_r.
Qed.

Lemma insert_list_union {A} (m : M A) l :
  insert_list l m = map_of_list l  m.
Proof.
  induction l; simpl.
  * by rewrite (left_id _ _).
  * by rewrite IHl, insert_union_l.
Qed.

Lemma delete_union {A} (m1 m2 : M A) i :
  delete i (m1  m2) = delete i m1  delete i m2.
Proof. apply delete_union_with. Qed.

(** ** Properties of the [union_list] operation *)
Lemma map_disjoint_union_list_l {A} (ms : list (M A)) (m : M A) :
   ms  m  Forall ( m) ms.
Proof.
  split.
  * induction ms; simpl; rewrite ?map_disjoint_union_l; intuition.
  * induction 1; simpl.
    + apply map_disjoint_empty_l.
    + by rewrite map_disjoint_union_l.
Qed.
Lemma map_disjoint_union_list_r {A} (ms : list (M A)) (m : M A) :
  m   ms  Forall ( m) ms.
Proof. by rewrite (symmetry_iff ()), map_disjoint_union_list_l. Qed.

Lemma map_disjoint_union_list_l_2 {A} (ms : list (M A)) (m : M A) :
  Forall ( m) ms   ms  m.
Proof. by rewrite map_disjoint_union_list_l. Qed.
Lemma map_disjoint_union_list_r_2 {A} (ms : list (M A)) (m : M A) :
  Forall ( m) ms  m   ms.
Proof. by rewrite map_disjoint_union_list_r. Qed.

Lemma map_union_sublist {A} (ms1 ms2 : list (M A)) :
  list_disjoint ms2 
  sublist ms1 ms2 
   ms1   ms2.
Proof.
  intros Hms2. revert ms1.
  induction Hms2 as [|m2 ms2]; intros ms1; [by inversion 1|].
  rewrite sublist_cons_r. intros [?|(ms1' &?&?)]; subst; simpl.
  * transitivity ( ms2); auto. by apply map_union_subseteq_r.
  * apply map_union_preserving_l; auto.
Qed.

(** ** Properties of the [intersection] operation *)
Lemma lookup_intersection_Some {A} (m1 m2 : M A) i x :
  (m1  m2) !! i = Some x  m1 !! i = Some x  is_Some (m2 !! i).
Proof.
  unfold intersection, map_intersection,
    intersection_with, map_intersection_with.
  rewrite (lookup_merge _).
  rewrite is_Some_alt.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

(** ** Properties of the [delete_list] function *)
Lemma lookup_delete_list {A} (m : M A) is j :
  j  is  delete_list is m !! j = None.
Proof.
  induction 1 as [|i j is]; simpl.
  * by rewrite lookup_delete.
  * destruct (decide (i = j)).
    + subst. by rewrite lookup_delete.
    + rewrite lookup_delete_ne; auto.
Qed.
Lemma lookup_delete_list_not_elem_of {A} (m : M A) is j :
  j  is  delete_list is m !! j = m !! j.
Proof.
  induction is; simpl; [done |].
  rewrite elem_of_cons. intros.
  intros. rewrite lookup_delete_ne; intuition.
Qed.
Lemma delete_list_notin {A} (m : M A) is :
  Forall (λ i, m !! i = None) is  delete_list is m = m.
Proof.
  induction 1; simpl; [done |].
  rewrite delete_notin; congruence.
Qed.

Lemma delete_list_insert_ne {A} (m : M A) is j x :
  j  is  delete_list is (<[j:=x]>m) = <[j:=x]>(delete_list is m).
Proof.
  induction is; simpl; [done |].
  rewrite elem_of_cons. intros.
  rewrite IHis, delete_insert_ne; intuition.
Qed.

Lemma map_disjoint_delete_list_l {A} (m1 m2 : M A) is :
  m1  m2  delete_list is m1  m2.
Proof. induction is; simpl; auto using map_disjoint_delete_l. Qed.
Lemma map_disjoint_delete_list_r {A} (m1 m2 : M A) is :
  m1  m2  m1  delete_list is m2.
Proof. induction is; simpl; auto using map_disjoint_delete_r. Qed.

Lemma delete_list_union {A} (m1 m2 : M A) is :
  delete_list is (m1  m2) = delete_list is m1  delete_list is m2.
Proof. apply delete_list_union_with. Qed.

(** ** Properties on disjointness of conversion to lists *)
Lemma map_disjoint_of_list_l {A} (m : M A) ixs :
  map_of_list ixs  m  Forall (λ ix, m !! fst ix = None) ixs.
Proof.
  split.
  * induction ixs; simpl; rewrite ?map_disjoint_insert_l in *; intuition.
  * induction 1; simpl.
    + apply map_disjoint_empty_l.
    + rewrite map_disjoint_insert_l. auto.
Qed.
Lemma map_disjoint_of_list_r {A} (m : M A) ixs :
  m  map_of_list ixs  Forall (λ ix, m !! fst ix = None) ixs.
Proof. by rewrite (symmetry_iff ()), map_disjoint_of_list_l. Qed.

Lemma map_disjoint_of_list_zip_l {A} (m : M A) is xs :
  same_length is xs 
  map_of_list (zip is xs)  m  Forall (λ i, m !! i = None) is.
Proof.
  intro. rewrite map_disjoint_of_list_l.
  rewrite <-(zip_fst is xs) at 2 by done.
  by rewrite Forall_fmap.
Qed.
Lemma map_disjoint_of_list_zip_r {A} (m : M A) is xs :
  same_length is xs 
  m  map_of_list (zip is xs)  Forall (λ i, m !! i = None) is.
Proof.
  intro. by rewrite (symmetry_iff ()), map_disjoint_of_list_zip_l.
Qed.
Lemma map_disjoint_of_list_zip_l_2 {A} (m : M A) is xs :
  same_length is xs 
  Forall (λ i, m !! i = None) is 
  map_of_list (zip is xs)  m.
Proof. intro. by rewrite map_disjoint_of_list_zip_l. Qed.
Lemma map_disjoint_of_list_zip_r_2 {A} (m : M A) is xs :
  same_length is xs 
  Forall (λ i, m !! i = None) is 
  m  map_of_list (zip is xs).
Proof. intro. by rewrite map_disjoint_of_list_zip_r. Qed.

(** ** Properties with respect to vectors *)
Lemma union_delete_vec {A n} (ms : vec (M A) n) (i : fin n) :
  list_disjoint ms 
  ms !!! i   delete (fin_to_nat i) (vec_to_list ms) =  ms.
Proof.
  induction ms as [|m ? ms]; inversion_clear 1;
    inv_fin i; simpl; [done | intros i].
  rewrite (map_union_commutative m), (associative_eq _ _), IHms.
  * by rewrite map_union_commutative.
  * done.
  * apply map_disjoint_weaken_r with ( ms); [done |].
    apply map_union_sublist; auto using sublist_delete.
Qed.

Lemma union_insert_vec {A n} (ms : vec (M A) n) (i : fin n) m :
  m   delete (fin_to_nat i) (vec_to_list ms) 
   vinsert i m ms = m   delete (fin_to_nat i) (vec_to_list ms).
Proof.
  induction ms as [|m' ? ms IH];
    inv_fin i; simpl; [done | intros i Hdisjoint].
  rewrite map_disjoint_union_r in Hdisjoint.
  rewrite IH, !(associative_eq ()), (map_union_commutative m); intuition.
Qed.

(** ** Properties of the [difference_with] operation *)
Section difference_with.
Context {A} (f : A  A  option A).

Lemma lookup_difference_with m1 m2 i z :
  difference_with f m1 m2 !! i = z 
    (m1 !! i = None  z = None) 
    ( x, m1 !! i = Some x  m2 !! i = None  z = Some x) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  z = f x y).
Proof.
  unfold difference_with, map_difference_with. rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma lookup_difference_with_Some m1 m2 i z :
  difference_with f m1 m2 !! i = Some z 
    (m1 !! i = Some z  m2 !! i = None) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = Some z).
Proof. rewrite lookup_difference_with. naive_solver. Qed.
Lemma lookup_difference_with_None m1 m2 i :
  difference_with f m1 m2 !! i = None 
    (m1 !! i = None) 
    ( x y, m1 !! i = Some x  m2 !! i = Some y  f x y = None).
Proof. rewrite lookup_difference_with. naive_solver. Qed.

Lemma lookup_difference_with_Some_lr m1 m2 i x y z :
  m1 !! i = Some x 
  m2 !! i = Some y 
  f x y = Some z 
  difference_with f m1 m2 !! i = Some z.
Proof. rewrite lookup_difference_with. naive_solver. Qed.
Lemma lookup_difference_with_None_lr m1 m2 i x y :
  m1 !! i = Some x 
  m2 !! i = Some y 
  f x y = None 
  difference_with f m1 m2 !! i = None.
Proof. rewrite lookup_difference_with. naive_solver. Qed.
Lemma lookup_difference_with_Some_l m1 m2 i x :
  m1 !! i = Some x 
  m2 !! i = None 
  difference_with f m1 m2 !! i = Some x.
Proof. rewrite lookup_difference_with. naive_solver. Qed.
End difference_with.

(** ** Properties of the [difference] operation *)
Lemma lookup_difference_Some {A} (m1 m2 : M A) i x :
  (m1  m2) !! i = Some x  m1 !! i = Some x  m2 !! i = None.
Proof.
  unfold difference, map_difference, difference_with, map_difference_with.
  rewrite (lookup_merge _).
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.

Lemma map_disjoint_difference_l {A} (m1 m2 : M A) :
  m1  m2 
  m2  m1  m1.
Proof.
  intros E i. specialize (E i).
  unfold difference, map_difference. intros x1 x2.
  rewrite lookup_difference_with_Some. intros [?| (?&?&?&?&?)] ?.
  * specialize (E x2). intuition congruence.
  * done.
Qed.
Lemma map_disjoint_difference_r {A} (m1 m2 : M A) :
  m1  m2 
  m1  m2  m1.
Proof. intros. symmetry. by apply map_disjoint_difference_l. Qed.

Lemma map_difference_union {A} (m1 m2 : M A) :
  m1  m2  m1  m2  m1 = m2.
Proof.
  intro Hm1m2. apply map_eq. intros i.
  apply option_eq. intros v. specialize (Hm1m2 i).
  unfold difference, map_difference,
    difference_with, map_difference_with.
  rewrite lookup_union_Some_raw, (lookup_merge _).
  destruct (m1 !! i) as [v'|], (m2 !! i);
    try specialize (Hm1m2 v'); compute; intuition congruence.
Qed.
End theorems.

(** * Tactics *)
(** The tactic [decompose_map_disjoint] simplifies occurrences of [disjoint]
in the hypotheses that involve the empty map [∅], the union [(∪)] or insert
[<[_:=_]>] operation, the singleton [{[ _ ]}] map, and disjointness of lists of
maps. This tactic does not yield any information loss as all simplifications
performed are reversible. *)
Ltac decompose_map_disjoint := repeat
  match goal with
  | H : _  _  _ |- _ =>
    apply map_disjoint_union_l in H; destruct H
  | H : _  _  _ |- _ =>
    apply map_disjoint_union_r in H; destruct H
  | H : {[ _ ]}  _ |- _ => apply map_disjoint_singleton_l in H
  | H : _  {[ _ ]} |- _ =>  apply map_disjoint_singleton_r in H
  | H : <[_:=_]>_  _ |- _ =>
    apply map_disjoint_insert_l in H; destruct H
  | H : _  <[_:=_]>_ |- _ =>
    apply map_disjoint_insert_r in H; destruct H
  | H :  _  _ |- _ => apply map_disjoint_union_list_l in H
  | H : _   _ |- _ => apply map_disjoint_union_list_r in H
  | H :   _ |- _ => clear H
  | H : _   |- _ => clear H
  | H : list_disjoint [] |- _ => clear H
  | H : list_disjoint [_] |- _ => clear H
  | H : list_disjoint (_ :: _) |- _ =>
    apply list_disjoint_cons_inv in H; destruct H
  | H : Forall ( _) _ |- _ => rewrite Forall_vlookup in H
  | H : Forall ( _) [] |- _ => clear H
  | H : Forall ( _) (_ :: _) |- _ =>
    rewrite Forall_cons in H; destruct H
  | H : Forall ( _) (_ :: _) |- _ =>
    rewrite Forall_app in H; destruct H
  end.

(** To prove a disjointness property, we first decompose all hypotheses, and
then use an auto database to prove the required property. *)
Create HintDb map_disjoint.
Ltac solve_map_disjoint :=
  solve [decompose_map_disjoint; auto with map_disjoint].

(** We declare these hints using [Hint Extern] instead of [Hint Resolve] as
[eauto] works badly with hints parametrized by type class constraints. *)
Hint Extern 1 (_  _) => done : map_disjoint.
Hint Extern 2 (  _) => apply map_disjoint_empty_l : map_disjoint.
Hint Extern 2 (_  ) => apply map_disjoint_empty_r : map_disjoint.
Hint Extern 2 ({[ _ ]}  _) =>
  apply map_disjoint_singleton_l_2 : map_disjoint.
Hint Extern 2 (_  {[ _ ]}) =>
  apply map_disjoint_singleton_r_2 : map_disjoint.
Hint Extern 2 (list_disjoint []) => apply disjoint_nil : map_disjoint.
Hint Extern 2 (list_disjoint (_ :: _)) => apply disjoint_cons : map_disjoint.
Hint Extern 2 (_  _  _) => apply map_disjoint_union_l_2 : map_disjoint.
Hint Extern 2 (_  _  _) => apply map_disjoint_union_r_2 : map_disjoint.
Hint Extern 2 (<[_:=_]>_  _) => apply map_disjoint_insert_l_2 : map_disjoint.
Hint Extern 2 (_  <[_:=_]>_) => apply map_disjoint_insert_r_2 : map_disjoint.
Hint Extern 2 (delete _ _  _) => apply map_disjoint_delete_l : map_disjoint.
Hint Extern 2 (_  delete _ _) => apply map_disjoint_delete_r : map_disjoint.
Hint Extern 2 (map_of_list _  _) =>
  apply map_disjoint_of_list_zip_l_2 : mem_disjoint.
Hint Extern 2 (_  map_of_list _) =>
  apply map_disjoint_of_list_zip_r_2 : mem_disjoint.
Hint Extern 2 ( _  _) => apply map_disjoint_union_list_l_2 : mem_disjoint.
Hint Extern 2 (_   _) => apply map_disjoint_union_list_r_2 : mem_disjoint.
Hint Extern 2 (delete_list _ _  _) =>
  apply map_disjoint_delete_list_l : map_disjoint.
Hint Extern 2 (_  delete_list _ _) =>
  apply map_disjoint_delete_list_r : map_disjoint.

(** The tactic [simpl_map by tac] simplifies occurrences of finite map look 
ups. It uses [tac] to discharge generated inequalities. Look ups in unions do
not have nice equational properties, hence it invokes [tac] to prove that such
look ups yield [Some]. *)
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Tactic Notation "simpl_map" "by" tactic3(tac) := repeat
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  match goal with
  | H : context[  !! _ ] |- _ => rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert in H
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  | H : context[ (<[_:=_]>_) !! _ ] |- _ =>
   rewrite lookup_insert_ne in H by tac
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  | H : context[ (delete _ _) !! _] |- _ => rewrite lookup_delete in H
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  | H : context[ (delete _ _) !! _] |- _ =>
    rewrite lookup_delete_ne in H by tac
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  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton in H
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  | H : context[ {[ _ ]} !! _ ] |- _ =>
    rewrite lookup_singleton_ne in H by tac
  | H : context[ lookup (A:=?A) ?i (?m1  ?m2) ] |- _ =>
    let x := fresh in evar (x:A);
    let x' := eval unfold x in x in clear x;
    let E := fresh in
    assert ((m1  m2) !! i = Some x') as E by (clear H; by tac);
    rewrite E in H; clear E
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  | |- context[  !! _ ] => rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert
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  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert_ne by tac
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  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete
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  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete_ne by tac
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  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton
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  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton_ne by tac
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  | |- context [ lookup (A:=?A) ?i ?m ] =>
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    let x := fresh in evar (x:A);
    let x' := eval unfold x in x in clear x;
    let E := fresh in
    assert (m !! i = Some x') as E by tac;
    rewrite E; clear E
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  end.
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Create HintDb simpl_map.
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Tactic Notation "simpl_map" := simpl_map by eauto with simpl_map map_disjoint.
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Hint Extern 80 ((_  _) !! _ = Some _) =>
  apply lookup_union_Some_l : simpl_map.
Hint Extern 81 ((_  _) !! _ = Some _) =>
  apply lookup_union_Some_r : simpl_map.
Hint Extern 80 ({[ _ ]} !! _ = Some _) =>
  apply lookup_singleton : simpl_map.
Hint Extern 80 (<[_:=_]> _ !! _ = Some _) =>
  apply lookup_insert : simpl_map.

(** Now we take everything together and also discharge conflicting look ups,
simplify overlapping look ups, and perform cancellations of equalities
involving unions. *)
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Tactic Notation "simplify_map_equality" "by" tactic3(tac) := repeat
  match goal with
  | _ => progress simpl_map by tac
  | _ => progress simplify_equality
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  | H : {[ _ ]} !! _ = None |- _ => rewrite lookup_singleton_None in H
  | H : {[ _ ]} !! _ = Some _ |- _ =>
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    rewrite lookup_singleton_Some in H; destruct H
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  | H1 : ?m1 !! ?i = Some ?x, H2 : ?m2 !! ?i = Some ?y |- _ =>
    let H3 := fresh in
    feed pose proof (lookup_weaken_inv m1 m2 i x y) as H3;
      [done | by tac | done | ];
    clear H2; symmetry in H3
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  | H1 : ?m1 !! ?i = Some ?x, H2 : ?m2 !! ?i = None |- _ =>
    let H3 := fresh in
    assert (m1  m2) as H3 by tac;
    apply H3 in H1; congruence
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  | H : ?m  _ = ?m  _ |- _ =>
    apply map_union_cancel_l in H; [| solve[tac] | solve [tac]]
  | H :