fin_maps.v

(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* 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
induction principles for finite maps and implements the tactic
[simplify_map_equality] to simplify goals involving finite maps. *)
From Coq Require Import Permutation.
From stdpp Require Export relations vector orders.

(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
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. *)

(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
prove well founded recursion on finite maps. *)

(** 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]. *)

Class FinMapToList K A M := map_to_list: M → list (K * A).

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)} := {
  map_eq {A} (m1 m2 : M A) : (∀ 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;
  NoDup_map_to_list {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_omap {A B} (f : A → option B) m i : omap f m !! i = m !! i ≫= f;
  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)
}.

(** * 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
significant performance loss to make including them in the finite map interface
worthwhile. *)
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 :=
  partial_alter (λ _, None).
Instance map_singleton `{PartialAlter K A M, Empty M} :
  SingletonM K A M := λ i x, <[i:=x]> ∅.

Definition map_of_list `{Insert K A M, Empty M} : list (K * A) → M :=
  fold_right (λ p, <[p.1:=p.2]>) ∅.
Definition map_of_collection `{Elements K C, Insert K A M, Empty M}
    (f : K → option A) (X : C) : M :=
  map_of_list (omap (λ i, (i,) <$> f i) (elements X)).

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).

Instance map_equiv `{∀ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 18 :=
  λ m1 m2, ∀ i, m1 !! i ≡ m2 !! i.

(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
Definition map_Forall `{Lookup K A M} (P : K → A → Prop) : M → Prop :=
  λ m, ∀ i x, m !! i = Some x → P i x.
Definition map_relation `{∀ 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,
  option_relation R P Q (m1 !! i) (m2 !! i).
Definition map_included `{∀ A, Lookup K A (M A)} {A}
  (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
Definition map_disjoint `{∀ A, Lookup K A (M A)} {A} : relation (M A) :=
  map_relation (λ _ _, False) (λ _, True) (λ _, True).
Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope.
Hint Extern 0 (_ ⊥ₘ _) => symmetry; eassumption.
Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
Notation "(.⊥ₘ m )" := (λ m2, m2 ⊥ₘ m) (only parsing) : C_scope.
Instance map_subseteq `{∀ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
  map_included (=).

(** 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. *)
Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

(** The difference operation removes all values from the first map whose
index contains a value in the second map as well. *)
Instance map_difference `{Merge M} {A} : Difference (M A) :=
  difference_with (λ _ _, None).

(** A stronger variant of map that allows the mapped function to use the index
of the elements. Implemented by conversion to lists, so not very efficient. *)
Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A),
    ∀ A, FinMapToList K A (M A)} {A B} (f : K → A → option B) (m : M A) : M B :=
  map_of_list (omap (λ ix, (fst ix,) <$> curry f ix) (map_to_list m)).

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

(** ** Setoids *)
Section setoid.
  Context `{Equiv A} `{!Equivalence ((≡) : relation A)}.
  Global Instance map_equivalence : Equivalence ((≡) : relation (M A)).
  Proof.
    split.
    * by intros m i.
    * by intros m1 m2 ? i.
    * by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
  Qed.
  Global Instance lookup_proper (i : K) :
    Proper ((≡) ==> (≡)) (lookup (M:=M A) i).
  Proof. by intros m1 m2 Hm. Qed.
  Global Instance partial_alter_proper :
    Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (partial_alter (M:=M A)).
  Proof.
    by intros f1 f2 Hf i ? <- m1 m2 Hm j; destruct (decide (i = j)) as [->|];
      rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne by done;
      try apply Hf; apply lookup_proper.
  Qed.
  Global Instance insert_proper (i : K) :
    Proper ((≡) ==> (≡) ==> (≡)) (insert (M:=M A) i).
  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
  Global Instance singleton_proper k :
    Proper ((≡) ==> (≡)) (singletonM k : A → M A).
  Proof. by intros ???; apply insert_proper. Qed.
  Global Instance delete_proper (i : K) :
    Proper ((≡) ==> (≡)) (delete (M:=M A) i).
  Proof. by apply partial_alter_proper; [constructor|]. Qed.
  Global Instance alter_proper :
    Proper (((≡) ==> (≡)) ==> (=) ==> (≡) ==> (≡)) (alter (A:=A) (M:=M A)).
  Proof.
    intros ?? Hf; apply partial_alter_proper.
    by destruct 1; constructor; apply Hf.
  Qed.
  Lemma merge_ext f g
      `{!PropHolds (f None None = None), !PropHolds (g None None = None)} :
    ((≡) ==> (≡) ==> (≡))%signature f g →
    ((≡) ==> (≡) ==> (≡))%signature (merge (M:=M) f) (merge g).
  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
    Proper (((≡) ==> (≡) ==> (≡)) ==> (≡) ==> (≡) ==>(≡)) (union_with (M:=M A)).
  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
  Qed.    
  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
  Proof.
    intros m1 m2 Hm; apply map_eq; intros i.
    by unfold_leibniz; apply lookup_proper.
  Qed.
  Lemma map_equiv_empty (m : M A) : m ≡ ∅ ↔ m = ∅.
  Proof.
    split; [intros Hm; apply map_eq; intros i|by intros ->].
    by rewrite lookup_empty, <-equiv_None, Hm, lookup_empty.
  Qed.
  Lemma map_equiv_lookup (m1 m2 : M A) i x :
    m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y.
  Proof.
    intros Hm ?. destruct (equiv_Some (m1 !! i) (m2 !! i) x) as (y&?&?); eauto.
  Qed.
End setoid.

(** ** General properties *)
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_relation. split; intros Hm i;
    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Global Instance: EmptySpec (M A).
Proof.
  intros A m. rewrite !map_subseteq_spec.
  intros i x. by rewrite lookup_empty.
Qed.
Global Instance: ∀ {A} (R : relation A), PreOrder R → PreOrder (map_included R).
Proof.
  split; [intros m i; by destruct (m !! i); simpl|].
  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_equality';
    done || etransitivity; eauto.
Qed.
Global Instance: PartialOrder ((⊆) : relation (M A)).
Proof.
  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  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. rewrite !map_subseteq_spec. 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 map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
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 [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
Lemma map_fmap_empty {A B} (f : A → B) : f <$> (∅ : M A) = ∅.
Proof. by apply map_eq; intros i; rewrite lookup_fmap, !lookup_empty. Qed.

(** ** Properties of the [partial_alter] operation *)
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.
  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
  partial_alter (f ∘ g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
Qed.
Lemma partial_alter_commute {A} f g (m : M A) i j :
  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)) 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.
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. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?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} f (m : M A) i :
  m !! i = None → m ⊆ partial_alter f i m.
Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
Lemma partial_alter_subset {A} f (m : M A) i :
  m !! i = None → is_Some (f (m !! i)) → m ⊂ partial_alter f i m.
Proof.
  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.
Qed.

(** ** Properties of the [alter] operation *)
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.
Proof. intro. apply partial_alter_ext. intros [x|] ?; f_equal'; auto. Qed.
Lemma lookup_alter {A} (f : A → A) m i : alter f i m !! i = f <$> m !! i.
Proof. unfold alter. 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. unfold alter. apply lookup_partial_alter_ne. Qed.
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.
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)) as [->|?].
  * 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.
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
Qed.
Lemma alter_id {A} (f : A → A) m i :
  (∀ x, m !! i = Some x → f x = x) → alter f i m = m.
Proof.
  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
  { rewrite lookup_alter; destruct (m !! j); f_equal'; auto. }
  by rewrite lookup_alter_ne by done.
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)) as [->|?];
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. by rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_is_Some {A} (m : M A) i j :
  is_Some (delete i m !! j) ↔ i ≠ j ∧ is_Some (m !! j).
Proof. unfold is_Some; setoid_rewrite lookup_delete_Some; naive_solver. 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)) as [->|?];
    rewrite ?lookup_delete, ?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. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?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.
  rewrite !map_subseteq_spec. 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.
  rewrite !map_subseteq_spec. 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|].
  rewrite !map_subseteq_spec. 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)) as [->|?];
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_is_Some {A} (m : M A) i j x :
  is_Some (<[i:=x]>m !! j) ↔ i = j ∨ i ≠ j ∧ is_Some (m !! j).
Proof. unfold is_Some; setoid_rewrite lookup_insert_Some; naive_solver. Qed.
Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None ↔ m !! j = None ∧ i ≠ j.
Proof.
  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma insert_id {A} (m : M A) i x : m !! i = Some x → <[i:=x]>m = m.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|];
    by rewrite ?lookup_insert, ?lookup_insert_ne by done.
Qed.
Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
  (∀ y, m !! i = Some y → R y x) → map_included R m (<[i:=x]>m).
Proof.
  intros ? j; destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  * rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
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.
  rewrite !map_subseteq_spec. intros ?? j ?.
  destruct (decide (j = i)) as [->|?]; [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.
  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.
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.
  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
  * 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.
  eapply lookup_weaken, 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, strict_include; eauto.
    by rewrite lookup_insert.
  * eauto using insert_delete_subset.
  * by rewrite lookup_delete.
Qed.
Lemma fmap_insert {A B} (f : A → B) (m : M A) i x :
  f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  * by rewrite lookup_fmap, !lookup_insert.
  * by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
Lemma insert_empty {A} i (x : A) : <[i:=x]>∅ = {[i ↦ x]}.
Proof. done. 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.
  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
Qed.
Lemma lookup_singleton_None {A} i j (x : A) : {[i ↦ x]} !! j = None ↔ i ≠ j.
Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; 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 map_non_empty_singleton {A} i (x : A) : {[i ↦ x]} ≠ ∅.
Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i ↦ x]} = {[i ↦ y]}.
Proof.
  unfold singletonM, 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')) as [->|?].
  * 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'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
Qed.
Lemma map_fmap_singleton {A B} (f : A → B) i x : f <$> {[i ↦ x]} = {[i ↦ f x]}.
Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.

(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A → B) : f <$> ∅ = ∅.
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A → option B) : omap f ∅ = ∅.
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
Lemma omap_singleton {A B} (f : A → option B) i x y :
  f x = Some y → omap f {[ i ↦ x ]} = {[ i ↦ y ]}.
Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|].
  * by rewrite lookup_omap, !lookup_singleton.
  * by rewrite lookup_omap, !lookup_singleton_ne.
Qed.
Lemma map_fmap_id {A} (m : M A) : id <$> m = m.
Proof. apply map_eq; intros i; by rewrite lookup_fmap, option_fmap_id. Qed.
Lemma map_fmap_compose {A B C} (f : A → B) (g : B → C) (m : M A) :
  g ∘ f <$> m = g <$> f <$> m.
Proof. apply map_eq; intros i; by rewrite !lookup_fmap,option_fmap_compose. Qed.
Lemma map_fmap_setoid_ext `{Equiv A, Equiv B} (f1 f2 : A → B) m :
  (∀ i x, m !! i = Some x → f1 x ≡ f2 x) → f1 <$> m ≡ f2 <$> m.
Proof.
  intros Hi i; rewrite !lookup_fmap.
  destruct (m !! i) eqn:?; constructor; eauto.
Qed.
Lemma map_fmap_ext {A B} (f1 f2 : A → B) m :
  (∀ i x, m !! i = Some x → f1 x = f2 x) → f1 <$> m = f2 <$> m.
Proof.
  intros Hi; apply map_eq; intros i; rewrite !lookup_fmap.
  by destruct (m !! i) eqn:?; simpl; erewrite ?Hi by eauto.
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 NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
  (i,x) ∈ l → (∀ y, (i,y) ∈ l → y = x) → map_of_list l !! i = Some x.
Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
  intros [?|?] Hdup; simplify_equality; [by rewrite lookup_insert|].
  destruct (decide (i = j)) as [->|].
  * rewrite lookup_insert; f_equal; eauto.
  * rewrite lookup_insert_ne by done; eauto.
Qed.
Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
  NoDup (l.*1) → (i,x) ∈ l → map_of_list l !! i = Some x.
Proof.
  intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst.
  intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
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)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
Qed.
Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
  NoDup (l.*1) → (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 ∉ l.*1 → map_of_list l !! i = None.
Proof.
  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.
Qed.
Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
  map_of_list l !! i = None → i ∉ l.*1.
Proof.
  induction l as [|[j y] l IH]; csimpl; [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 ∉ l.*1 ↔ map_of_list l !! i = None.
Proof. red; 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 (l1.*1) → l1 ≡ₚ 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 (l1.*1) → NoDup (l2.*1) → map_of_list l1 = map_of_list l2 → l1 ≡ₚ 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 NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list {A} (l : list (K * A)) :
  NoDup (l.*1) → map_to_list (map_of_list l) ≡ₚ l.
Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
Lemma map_to_list_inj {A} (m1 m2 : M A) :
  map_to_list m1 ≡ₚ 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, NoDup_fst_map_to_list.
Qed.
Lemma map_to_of_list_flip {A} (m1 : M A) l2 :
  map_to_list m1 ≡ₚ l2 → m1 = map_of_list l2.
Proof.
  intros. rewrite <-(map_of_to_list m1).
  auto using map_of_list_proper, NoDup_fst_map_to_list.
Qed.
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 → map_to_list (<[i:=x]>m) ≡ₚ (i,x) :: map_to_list m.
Proof.
  intros. apply map_of_list_inj; csimpl.
  * apply NoDup_fst_map_to_list.
  * constructor; auto using NoDup_fst_map_to_list.
    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) : map_to_list m ≡ₚ [] → 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 :
  map_to_list m ≡ₚ (i,x) :: l → m = <[i:=x]>(map_of_list l).
Proof.
  intros Hperm. apply map_to_list_inj.
  assert (i ∉ l.*1 ∧ NoDup (l.*1)) as [].
  { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm.
    auto using NoDup_fst_map_to_list. }
  rewrite Hperm, map_to_list_insert, map_to_of_list;
    auto using not_elem_of_map_of_list_1.
Qed.
Lemma map_choose {A} (m : M A) : m ≠ ∅ → ∃ i x, m !! i = Some x.
Proof.
  intros Hemp. destruct (map_to_list m) as [|[i x] l] eqn:Hm.
  { destruct Hemp; eauto using map_to_list_empty_inv. }
  exists i, x. rewrite <-elem_of_map_to_list, Hm. by left.
Qed.

(** Properties of the imap function *)
Lemma lookup_imap {A B} (f : K → A → option B) m i :
  map_imap f m !! i = m !! i ≫= f i.
Proof.
  unfold map_imap; destruct (m !! i ≫= f i) as [y|] eqn:Hi; simpl.
  * destruct (m !! i) as [x|] eqn:?; simplify_equality'.
    apply elem_of_map_of_list_1_help.
    { apply elem_of_list_omap; exists (i,x); split;
        [by apply elem_of_map_to_list|by simplify_option_equality]. }
    intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
    by rewrite elem_of_map_to_list in Hi'; simplify_option_equality.
  * apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
    intros ([i' x]&->&Hi'); simplify_equality'.
    rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
    rewrite elem_of_map_to_list in Hj; simplify_option_equality.
Qed.

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

(** ** Induction principles *)
Lemma map_ind {A} (P : M A → Prop) :