Library iris.prelude.fin_maps

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_eq to simplify goals involving finite maps.

From Coq Require Import Permutation.
From iris.prelude Require Export relations orders vector.

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) `{!DiagNone f} 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; trans (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 `{!DiagNone f, !DiagNone g} :
    ((≡) ==> (≡) ==> (≡))%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_l (m1 m2 : M A) i x :
    m1 ≡ m2 → m1 !! i = Some x → ∃ y, m2 !! i = Some y ∧ x ≡ y.
  Proof. generalize (equiv_Some_inv_l (m1 !! i) (m2 !! i) x); naive_solver. Qed.
  Global Instance map_fmap_proper `{Equiv B} (f : A → B) :
    Proper ((≡) ==> (≡)) f → Proper ((≡) ==> (≡)) (fmap (M:=M) f).
  Proof.
    intros ? m m' ? k; rewrite !lookup_fmap. by apply option_fmap_proper.
  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: ∀ {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_eq/=;
    done || etrans; 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_eq; 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 : <[i :=x ]>(delete i m) = <[i :=x ]> m.
Proof. symmetry; apply (partial_alter_compose (λ _, Some x)). 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_insert {A} (m : M A) i x y : <[i :=x ]>(<[i :=y ]>m) = <[i :=x ]>m.
Proof. unfold insert, map_insert. by rewrite <-partial_alter_compose. 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. ∃ (delete i m2). split_and?.
  - rewrite insert_delete, insert_id. done.
    eapply lookup_weaken, strict_include; eauto. by rewrite lookup_insert.
  - eauto using insert_delete_subset.
  - by rewrite lookup_delete.
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.

Properties of the map operations

Lemma fmap_empty {A B} (f : A → B) : f <$> ∅ = ∅.
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A → option B) : omap f ∅ = ∅.
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
Lemma fmap_insert {A B} (f: A → B) m i x: f <$> <[i :=x ]>m = <[i :=f x ]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  - by rewrite lookup_fmap, !lookup_insert.
  - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
Qed.
Lemma fmap_delete {A B} (f: A → B) m i: f <$> delete i m = delete i (f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  - by rewrite lookup_fmap, !lookup_delete.
  - by rewrite lookup_fmap, !lookup_delete_ne, lookup_fmap by done.
Qed.
Lemma omap_insert {A B} (f : A → option B) m i x y :
  f x = Some y → omap f (<[i :=x ]>m) = <[i :=y ]>(omap f m).
Proof.
  intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  - by rewrite lookup_omap, !lookup_insert.
  - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
Qed.
Lemma map_fmap_singleton {A B} (f : A → B) i x : f <$> {[i := x ]} = {[i := f x ]}.
Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.
Lemma omap_singleton {A B} (f : A → option B) i x y :
  f x = Some y → omap f {[ i := x ]} = {[ i := y ]}.
Proof.
  intros. unfold singletonM, map_singleton.
  by erewrite omap_insert, omap_empty by eauto.
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.
Lemma omap_ext {A B} (f1 f2 : A → option B) m :
  (∀ i x, m !! i = Some x → f1 x = f2 x ) → omap f1 m = omap f2 m.
Proof.
  intros Hi; apply map_eq; intros i; rewrite !lookup_omap.
  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_eq; [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.
  ∃ (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_eq.
  - 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_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_of_list_fmap {A B} (f : A → B) l :
  map_of_list (prod_map id f <$> l) = f <$> map_of_list l.
Proof.
  induction l as [|[i x] l IH]; csimpl; rewrite ?fmap_empty; auto.
  rewrite <-map_of_list_cons; simpl. by rewrite IH, <-fmap_insert.
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_to_list_contains {A} (m1 m2 : M A) :
  m1 ⊆ m2 → map_to_list m1 `contains ` map_to_list m2.
Proof.
  intros; apply NoDup_contains; auto using NoDup_map_to_list.
  intros [i x]. rewrite !elem_of_map_to_list; eauto using lookup_weaken.
Qed.
Lemma map_to_list_fmap {A B} (f : A → B) m :
  map_to_list (f <$> m) ≡ₚ prod_map id f <$> map_to_list m.
Proof.
  assert (NoDup ((prod_map id f <$> map_to_list m).*1)).
  { erewrite <-list_fmap_compose, (list_fmap_ext _ fst) by done.
    apply NoDup_fst_map_to_list. }
  rewrite <-(map_of_to_list m) at 1.
  by rewrite <-map_of_list_fmap, map_to_of_list.
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. }
  ∃ 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_eq/=.
    apply elem_of_map_of_list_1_help.
    { apply elem_of_list_omap; ∃ (i,x); split;
        [by apply elem_of_map_to_list|by simplify_option_eq]. }
    intros y'; rewrite elem_of_list_omap; intros ([i' x']&Hi'&?).
    by rewrite elem_of_map_to_list in Hi'; simplify_option_eq.
  - apply not_elem_of_map_of_list; rewrite elem_of_list_fmap.
    intros ([i' x]&->&Hi'); simplify_eq/=.
    rewrite elem_of_list_omap in Hi'; destruct Hi' as ([j y]&Hj&?).
    rewrite elem_of_map_to_list in Hj; simplify_option_eq.
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_eq. }
  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_eq; eauto.
  - intros [??]; ∃ i; simplify_option_eq; eauto.
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 ? Hins. cut (∀ l, NoDup (l .*1) → ∀ m, map_to_list m ≡ₚ l → P m).
  { intros help m.
    apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
  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 m. 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_inv, 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 (strict (@subseteq (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.
Lemma map_Forall_empty : map_Forall P ∅.
Proof. intros i x. by rewrite lookup_empty. Qed.
Lemma map_Forall_impl (Q : K → A → Prop) m :
  map_Forall P m → (∀ i x, P i x → Q i x ) → map_Forall Q m.
Proof. unfold map_Forall; naive_solver. Qed.
Lemma map_Forall_insert_11 m i x : map_Forall P (<[i :=x ]>m) → P i x.
Proof. intros Hm. by apply Hm; rewrite lookup_insert. Qed.
Lemma map_Forall_insert_12 m i x :
  m !! i = None → map_Forall P (<[i :=x ]>m) → map_Forall P m.
Proof.
  intros ? Hm j y ?; apply Hm. by rewrite lookup_insert_ne by congruence.
Qed.
Lemma map_Forall_insert_2 m i x :
  P i x → map_Forall P m → map_Forall P (<[i :=x ]>m).
Proof. intros ?? j y; rewrite lookup_insert_Some; naive_solver. Qed.
Lemma map_Forall_insert m i x :
  m !! i = None → map_Forall P (<[i :=x ]>m) ↔ P i x ∧ map_Forall P m.
Proof.
  naive_solver eauto using map_Forall_insert_11,
    map_Forall_insert_12, map_Forall_insert_2.
Qed.
Lemma map_Forall_ind (Q : M A → Prop) :
  Q ∅ →
  (∀ m i x, m !! i = None → P i x → map_Forall P m → Q m → Q (<[i :=x ]>m)) →
  ∀ m, map_Forall P m → Q m.
Proof.
  intros Hnil Hinsert m. induction m using map_ind; auto.
  rewrite map_Forall_insert by done; intros [??]; eauto.
Qed.

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; [|intros (i&x&?&?) Hm; specialize (Hm i x); tauto].
  rewrite map_Forall_to_list. intros Hm.
  apply (not_Forall_Exists _), Exists_exists in Hm.
  destruct Hm as ([i x]&?&?). ∃ i, x. by rewrite <-elem_of_map_to_list.
Qed.
End map_Forall.

Properties of the merge operation

Section merge.
Context {A} (f : option A → option A → option A) `{!DiagNone f}.
Global Instance: LeftId (=) None f → LeftId (=) ∅ (merge f).
Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (left_id_L 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_L None f).
Qed.
Lemma merge_comm 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 merge_comm' : Comm (=) f → Comm (=) (merge f).
Proof. intros ???. apply merge_comm. intros. by apply (comm f). Qed.
Lemma merge_assoc 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 merge_assoc' : Assoc (=) f → Assoc (=) (merge f).
Proof. intros ????. apply merge_assoc. intros. by apply (assoc_L f). Qed.
Lemma merge_idemp 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 merge_idemp' : IdemP (=) f → IdemP (=) (merge f).
Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed.
End merge.

Section more_merge.
Context {A B C} (f : option A → option B → option C) `{!DiagNone f}.

Lemma merge_Some m1 m2 m :
  (∀ i, m !! i = f (m1 !! i) (m2 !! i)) ↔ merge f m1 m2 = m.
Proof.
  split; [|intros <-; apply (lookup_merge _) ].
  intros Hlookup. apply map_eq; intros. rewrite Hlookup. apply (lookup_merge _).
Qed.
Lemma merge_empty : merge f ∅ ∅ = ∅.
Proof. apply map_eq. intros. by rewrite !(lookup_merge f), !lookup_empty. Qed.
Lemma partial_alter_merge g g1 g2 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 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 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 m1 m2 i x y z :
  f (Some y) (Some z) = Some x →
  <[i :=x ]>(merge f m1 m2) = merge f (<[i :=y ]>m1) (<[i :=z ]>m2).
Proof. by intros; apply partial_alter_merge. Qed.
Lemma merge_singleton i x y z :
  f (Some y) (Some z) = Some x → merge f {[i := y ]} {[i := z ]} = {[i := x ]}.
Proof.
  intros. by erewrite <-!insert_empty, <-insert_merge, merge_empty by eauto.
Qed.
Lemma insert_merge_l m1 m2 i x y :
  f (Some y) (m2 !! i) = Some x →
  <[i :=x ]>(merge f m1 m2) = merge f (<[i :=y ]>m1) m2.
Proof. by intros; apply partial_alter_merge_l. Qed.
Lemma insert_merge_r m1 m2 i x z :
  f (m1 !! i) (Some z) = Some x →
  <[i :=x ]>(merge f m1 m2) = merge f m1 (<[i :=z ]>m2).
Proof. by intros; apply partial_alter_merge_r. Qed.
End more_merge.

Properties on the map_relation relation

Section Forall2.
Context {A B} (R : A → B → Prop) (P : A → Prop) (Q : B → Prop).
Context `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)}.

Let f (mx : option A) (my : option B) : option bool :=
  match mx, my with
  | Some x, Some y ⇒ Some (bool_decide (R x y))
  | Some x, None ⇒ Some (bool_decide (P x))
  | None, Some y ⇒ Some (bool_decide (Q y))
  | None, None ⇒ None
  end.
Lemma map_relation_alt (m1 : M A) (m2 : M B) :
  map_relation R P Q m1 m2 ↔ map_Forall (λ _, Is_true) (merge f m1 m2).
Proof.
  split.
  - intros Hm i P'; rewrite lookup_merge by done; intros.
    specialize (Hm i). destruct (m1 !! i), (m2 !! i);
      simplify_eq/=; auto using bool_decide_pack.
  - intros Hm i. specialize (Hm i). rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); simplify_eq/=; auto;
      by eapply bool_decide_unpack, Hm.
Qed.
Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x),
  ∀ y, Decision (Q y)} m1 m2 : Decision (map_relation R P Q m1 m2).
Proof.
  refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2))));
    abstract by rewrite map_relation_alt.
Defined.

Due to the finiteness of finite maps, we can extract a witness if the relation does not hold.

Lemma map_not_Forall2 (m1 : M A) (m2 : M B) :
  ¬map_relation R P Q m1 m2 ↔ ∃ i,
    (∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ ¬R x y )
    ∨ (∃ x, m1 !! i = Some x ∧ m2 !! i = None ∧ ¬P x )
    ∨ (∃ y, m1 !! i = None ∧ m2 !! i = Some y ∧ ¬Q y ).
Proof.
  split.
  - rewrite map_relation_alt, (map_not_Forall _). intros (i&?&Hm&?); ∃ i.
    rewrite lookup_merge in Hm by done.
    destruct (m1 !! i), (m2 !! i); naive_solver auto 2 using bool_decide_pack.
  - unfold map_relation, option_relation.
    by intros [i[(x&y&?&?&?)|[(x&?&?&?)|(y&?&?&?)]]] Hm;
      specialize (Hm i); simplify_option_eq.
Qed.
End Forall2.

Properties on the disjoint maps

Lemma map_disjoint_spec {A} (m1 m2 : M A) :
  m1 ⊥ₘ m2 ↔ ∀ i x y, m1 !! i = Some x → m2 !! i = Some y → False.
Proof.
  split; intros Hm i; specialize (Hm i);
    destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
Lemma map_disjoint_alt {A} (m1 m2 : M A) :
  m1 ⊥ₘ 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 ⊥ₘ m2 ↔ ∃ i x1 x2, m1 !! i = Some x1 ∧ m2 !! i = Some x2.
Proof.
  unfold disjoint, map_disjoint. rewrite map_not_Forall2 by solve_decision.
  split; [|naive_solver].
  intros [i[(x&y&?&?&?)|[(x&?&?&[])|(y&?&?&[])]]]; naive_solver.
Qed.
Global Instance map_disjoint_sym : Symmetric (map_disjoint : relation (M A)).
Proof. intros A m1 m2. rewrite !map_disjoint_spec. naive_solver. Qed.
Lemma map_disjoint_empty_l {A} (m : M A) : ∅ ⊥ₘ m.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_empty_r {A} (m : M A) : m ⊥ₘ ∅.
Proof. rewrite !map_disjoint_spec. intros i x y. by rewrite lookup_empty. Qed.
Lemma map_disjoint_weaken {A} (m1 m1' m2 m2' : M A) :
  m1' ⊥ₘ m2' → m1 ⊆ m1' → m2 ⊆ m2' → m1 ⊥ₘ m2.
Proof. rewrite !map_subseteq_spec, !map_disjoint_spec. eauto. Qed.
Lemma map_disjoint_weaken_l {A} (m1 m1' m2 : M A) :
  m1' ⊥ₘ m2 → m1 ⊆ m1' → m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_weaken_r {A} (m1 m2 m2' : M A) :
  m1 ⊥ₘ m2' → m2 ⊆ m2' → m1 ⊥ₘ m2.
Proof. eauto using map_disjoint_weaken. Qed.
Lemma map_disjoint_Some_l {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2 → m1 !! i = Some x → m2 !! i = None.
Proof. rewrite map_disjoint_spec, eq_None_not_Some. intros ?? [??]; eauto. Qed.
Lemma map_disjoint_Some_r {A} (m1 m2 : M A) i x:
  m1 ⊥ₘ m2 → m2 !! i = Some x → m1 !! i = None.
Proof. rewrite (symmetry_iff map_disjoint). apply map_disjoint_Some_l. Qed.
Lemma map_disjoint_singleton_l {A} (m: M A) i x : {[i :=x ]} ⊥ₘ m ↔ m !! i = None.
Proof.
  split; [|rewrite !map_disjoint_spec].
  - intro. apply (map_disjoint_Some_l {[i := x]} _ _ x);
      auto using lookup_singleton.
  - intros ? j y1 y2. destruct (decide (i = j)) as [->|].
    + rewrite lookup_singleton. intuition congruence.
    + by rewrite lookup_singleton_ne.
Qed.
Lemma map_disjoint_singleton_r {A} (m : M A) i x :
  m ⊥ₘ {[i := x ]} ↔ m !! i = None.
Proof. by rewrite (symmetry_iff map_disjoint), map_disjoint_singleton_l. Qed.
Lemma map_disjoint_singleton_l_2 {A} (m : M A) i x :
  m !! i = None → {[i := x ]} ⊥ₘ 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 ⊥ₘ {[i := x ]}.
Proof. by rewrite map_disjoint_singleton_r. Qed.
Lemma map_disjoint_delete_l {A} (m1 m2 : M A) i : m1 ⊥ₘ m2 → delete i m1 ⊥ₘ 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 ⊥ₘ m2 → m1 ⊥ₘ 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 :
  union_with f m1 m2 !! i = union_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). 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.
  destruct (m1 !! i), (m2 !! i); compute; 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_comm 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_comm _). intros i.
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
Qed.
Global Instance: Comm (=) f → Comm (@eq (M A)) (union_with f).
Proof. intros ???. apply union_with_comm. eauto. Qed.
Lemma union_with_idemp m :
  (∀ i x, m !! i = Some x → f x x = Some x ) → union_with f m m = m.
Proof.
  intros. apply (merge_idemp _). 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:?; f_equal/=; auto.
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:?; f_equal/=; auto.
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 foldr_delete_union_with (m1 m2 : M A) is :
  foldr delete (union_with f m1 m2) is =
    union_with f (foldr delete m1 is) (foldr delete m2 is).
Proof. induction is; simpl. done. by rewrite IHis, delete_union_with. Qed.
Lemma insert_union_with m1 m2 i x y z :
  f x y = Some z →
  <[i :=z ]>(union_with f m1 m2) = union_with f (<[i :=x ]>m1) (<[i :=y ]>m2).
Proof. by intros; apply (partial_alter_merge _). 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.
  by erewrite (insert_merge_l _) by (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.
  by erewrite (insert_merge_r _) by (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: Assoc (@eq (M A)) (∪).
Proof.
  intros A m1 m2 m3. unfold union, map_union, union_with, map_union_with.
  apply (merge_assoc _). intros i.
  by destruct (m1 !! i), (m2 !! i), (m3 !! i).
Qed.
Global Instance: IdemP (@eq (M A)) (∪).
Proof. intros A ?. by apply union_with_idemp. 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 map_positive_l {A} (m1 m2 : M A) : m1 ∪ m2 = ∅ → m1 = ∅.
Proof.
  intros Hm. apply map_empty. intros i. apply (f_equal (!! i)) in Hm.
  rewrite lookup_empty, lookup_union_None in Hm; tauto.
Qed.
Lemma map_positive_l_alt {A} (m1 m2 : M A) : m1 ≠ ∅ → m1 ∪ m2 ≠ ∅.
Proof. eauto using map_positive_l. Qed.
Lemma lookup_union_Some {A} (m1 m2 : M A) i x :
  m1 ⊥ₘ 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 ⊥ₘ m2 → m2 !! i = Some x → (m1 ∪ m2 ) !! i = Some x.
Proof. intro. rewrite lookup_union_Some; intuition. Qed.
Lemma map_union_comm {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m1 ∪ m2 = m2 ∪ m1.
Proof.
  intros Hdisjoint. apply (merge_comm (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.
  rewrite map_subseteq_spec.
  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.
  rewrite map_subseteq_spec. intros ? i x. rewrite lookup_union_Some_raw. tauto.
Qed.
Lemma map_union_subseteq_r {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m2 ⊆ m1 ∪ m2.
Proof.
  intros. rewrite map_union_comm 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. trans m2; auto using map_union_subseteq_l. Qed.
Lemma map_union_subseteq_r_alt {A} (m1 m2 m3 : M A) :
  m2 ⊥ₘ m3 → m1 ⊆ m3 → m1 ⊆ m2 ∪ m3.
Proof. intros. trans 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.
  rewrite !map_subseteq_spec. intros ???.
  rewrite !lookup_union_Some_raw. naive_solver.
Qed.
Lemma map_union_preserving_r {A} (m1 m2 m3 : M A) :
  m2 ⊥ₘ m3 → m1 ⊆ m2 → m1 ∪ m3 ⊆ m2 ∪ m3.
Proof.
  intros. rewrite !(map_union_comm _ 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 ⊥ₘ m1 → m3 ⊥ₘ m2 → m3 ∪ m1 ⊆ m3 ∪ m2 → m1 ⊆ m2.
Proof.
  rewrite !map_subseteq_spec. intros Hm31 Hm32 Hm i x ?. specialize (Hm i x).
  rewrite !lookup_union_Some in Hm by done. destruct Hm; auto.
  by rewrite map_disjoint_spec in Hm31; destruct (Hm31 i x x).
Qed.
Lemma map_union_reflecting_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 ⊆ m2 ∪ m3 → m1 ⊆ m2.
Proof.
  intros ??. rewrite !(map_union_comm _ m3) by done.
  by apply map_union_reflecting_l.
Qed.
Lemma map_union_cancel_l {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m3 ∪ m1 = m3 ∪ m2 → m1 = m2.
Proof.
  intros. apply (anti_symm (⊆));
    apply map_union_reflecting_l with m3; auto using (reflexive_eq (R:=(⊆))).
Qed.
Lemma map_union_cancel_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 = m2 ∪ m3 → m1 = m2.
Proof.
  intros. apply (anti_symm (⊆));
    apply map_union_reflecting_r with m3; auto using (reflexive_eq (R:=(⊆))).
Qed.
Lemma map_disjoint_union_l {A} (m1 m2 m3 : M A) :
  m1 ∪ m2 ⊥ₘ m3 ↔ m1 ⊥ₘ m3 ∧ m2 ⊥ₘ 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 ⊥ₘ m2 ∪ m3 ↔ m1 ⊥ₘ m2 ∧ m1 ⊥ₘ 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 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m2 ⊥ₘ m3.
Proof. by rewrite map_disjoint_union_l. Qed.
Lemma map_disjoint_union_r_2 {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m2 → m1 ⊥ₘ m3 → m1 ⊥ₘ 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_comm; [done |].
  by apply map_disjoint_singleton_l.
Qed.
Lemma map_disjoint_insert_l {A} (m1 m2 : M A) i x :
  <[i :=x ]>m1 ⊥ₘ m2 ↔ m2 !! i = None ∧ m1 ⊥ₘ 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 ⊥ₘ <[i :=x ]>m2 ↔ m1 !! i = None ∧ m1 ⊥ₘ 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 ⊥ₘ m2 → <[i :=x ]>m1 ⊥ₘ 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 ⊥ₘ m2 → m1 ⊥ₘ <[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, (assoc_L (∪)). 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, !(assoc_L (∪)).
  rewrite (map_union_comm m1); [done |].
  by apply map_disjoint_singleton_r.
Qed.
Lemma foldr_insert_union {A} (m : M A) l :
  foldr (λ p, <[p .1 :=p .2 ]>) m l = map_of_list l ∪ m.
Proof.
  induction l as [|i l IH]; simpl; [by rewrite (left_id_L _ _)|].
  by rewrite IH, 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

Properties of the folding the delete function

Lemma lookup_foldr_delete {A} (m : M A) is j :
  j ∈ is → foldr delete m is !! j = None.
Proof.
  induction 1 as [|i j is]; simpl; [by rewrite lookup_delete|].
  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne by done.
Qed.
Lemma lookup_foldr_delete_not_elem_of {A} (m : M A) is j :
  j ∉ is → foldr delete m is !! j = m !! j.
Proof.
  induction is; simpl; [done |]. rewrite elem_of_cons; intros.
  rewrite lookup_delete_ne; intuition.
Qed.
Lemma foldr_delete_notin {A} (m : M A) is :
  Forall (λ i, m !! i = None) is → foldr delete m is = m.
Proof. induction 1; simpl; [done |]. rewrite delete_notin; congruence. Qed.
Lemma foldr_delete_insert_ne {A} (m : M A) is j x :
  j ∉ is → foldr delete (<[j :=x ]>m) is = <[j :=x ]>(foldr delete m is).
Proof.
  induction is; simpl; [done |]. rewrite elem_of_cons. intros.
  rewrite IHis, delete_insert_ne; intuition.
Qed.
Lemma map_disjoint_foldr_delete_l {A} (m1 m2 : M A) is :
  m1 ⊥ₘ m2 → foldr delete m1 is ⊥ₘ m2.
Proof. induction is; simpl; auto using map_disjoint_delete_l. Qed.
Lemma map_disjoint_foldr_delete_r {A} (m1 m2 : M A) is :
  m1 ⊥ₘ m2 → m1 ⊥ₘ foldr delete m2 is.
Proof. induction is; simpl; auto using map_disjoint_delete_r. Qed.
Lemma foldr_delete_union {A} (m1 m2 : M A) is :
  foldr delete (m1 ∪ m2) is = foldr delete m1 is ∪ foldr delete m2 is.
Proof. apply foldr_delete_union_with. Qed.

Properties on disjointness of conversion to lists

Properties of the intersection_with operation

Lemma lookup_intersection_with {A} (f : A → A → option A) m1 m2 i :
  intersection_with f m1 m2 !! i = intersection_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-(lookup_merge _). Qed.
Lemma lookup_intersection_with_Some {A} (f : A → A → option A) m1 m2 i z :
  intersection_with f m1 m2 !! i = Some z ↔
    (∃ x y, m1 !! i = Some x ∧ m2 !! i = Some y ∧ f x y = Some z ).
Proof.
  rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
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. rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma lookup_intersection_None {A} (m1 m2 : M A) i :
  (m1 ∩ m2 ) !! i = None ↔ m1 !! i = None ∨ m2 !! i = None.
Proof.
  unfold intersection, map_intersection. rewrite lookup_intersection_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

Properties of the difference_with operation

Lemma lookup_difference_with {A} (f : A → A → option A) m1 m2 i :
  difference_with f m1 m2 !! i = difference_with f (m1 !! i) (m2 !! i).
Proof. by rewrite <-lookup_merge by done. Qed.
Lemma lookup_difference_with_Some {A} (f : A → A → option A) 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.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.

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; rewrite lookup_difference_with.
  destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
Lemma lookup_difference_None {A} (m1 m2 : M A) i :
  (m1 ∖ m2 ) !! i = None ↔ m1 !! i = None ∨ is_Some (m2 !! i).
Proof.
  unfold difference, map_difference; rewrite lookup_difference_with.
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
Qed.
Lemma map_disjoint_difference_l {A} (m1 m2 : M A) : m1 ⊆ m2 → m2 ∖ m1 ⊥ₘ m1.
Proof.
  intros Hm i; specialize (Hm i).
  unfold difference, map_difference; rewrite lookup_difference_with.
  by destruct (m1 !! i), (m2 !! i).
Qed.
Lemma map_disjoint_difference_r {A} (m1 m2 : M A) : m1 ⊆ m2 → m1 ⊥ₘ 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.
  rewrite map_subseteq_spec. 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 [x'|], (m2 !! i);
    try specialize (Hm1m2 x'); 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 : _ ∪ _ ⊥ₘ _ |- _ ⇒ apply map_disjoint_union_l in H; destruct H
  | H : _ ⊥ₘ _ ∪ _ |- _ ⇒ apply map_disjoint_union_r in H; destruct H
  | H : {[ _ := _ ]} ⊥ₘ _ |- _ ⇒ apply map_disjoint_singleton_l in H
  | H : _ ⊥ₘ {[ _ := _ ]} |- _ ⇒ apply map_disjoint_singleton_r in H
  | H : <[_:=_]>_ ⊥ₘ _ |- _ ⇒ apply map_disjoint_insert_l in H; destruct H
  | H : _ ⊥ₘ <[_:=_]>_ |- _ ⇒ apply map_disjoint_insert_r in H; destruct H
  | H : ⋃ _ ⊥ₘ _ |- _ ⇒ apply map_disjoint_union_list_l in H
  | H : _ ⊥ₘ ⋃ _ |- _ ⇒ apply map_disjoint_union_list_r in H
  | H : ∅ ⊥ₘ _ |- _ ⇒ clear H
  | H : _ ⊥ₘ ∅ |- _ ⇒ clear H
  | H : Forall (.⊥ₘ _) _ |- _ ⇒ rewrite Forall_vlookup in H
  | H : Forall (.⊥ₘ _) [] |- _ ⇒ clear H
  | H : Forall (.⊥ₘ _) (_ :: _) |- _ ⇒ rewrite Forall_cons in H; destruct H
  | H : Forall (.⊥ₘ _) (_ :: _) |- _ ⇒ 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 (_ ⊥ₘ _) ⇒ done : map_disjoint.
Hint Extern 2 (∅ ⊥ₘ _) ⇒ apply map_disjoint_empty_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ ∅) ⇒ apply map_disjoint_empty_r : map_disjoint.
Hint Extern 2 ({[ _ := _ ]} ⊥ₘ _) ⇒
  apply map_disjoint_singleton_l_2 : map_disjoint.
Hint Extern 2 (_ ⊥ₘ {[ _ := _ ]}) ⇒
  apply map_disjoint_singleton_r_2 : map_disjoint.
Hint Extern 2 (_ ∪ _ ⊥ₘ _) ⇒ apply map_disjoint_union_l_2 : map_disjoint.
Hint Extern 2 (_ ⊥ₘ _ ∪ _) ⇒ apply map_disjoint_union_r_2 : map_disjoint.
Hint Extern 2 (<[_:=_]>_ ⊥ₘ _) ⇒ apply map_disjoint_insert_l_2 : map_disjoint.
Hint Extern 2 (_ ⊥ₘ <[_:=_]>_) ⇒ apply map_disjoint_insert_r_2 : map_disjoint.
Hint Extern 2 (delete _ _ ⊥ₘ _) ⇒ apply map_disjoint_delete_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ delete _ _) ⇒ apply map_disjoint_delete_r : map_disjoint.
Hint Extern 2 (map_of_list _ ⊥ₘ _) ⇒
  apply map_disjoint_of_list_zip_l_2 : mem_disjoint.
Hint Extern 2 (_ ⊥ₘ map_of_list _) ⇒
  apply map_disjoint_of_list_zip_r_2 : mem_disjoint.
Hint Extern 2 (⋃ _ ⊥ₘ _) ⇒ apply map_disjoint_union_list_l_2 : mem_disjoint.
Hint Extern 2 (_ ⊥ₘ ⋃ _) ⇒ apply map_disjoint_union_list_r_2 : mem_disjoint.
Hint Extern 2 (foldr delete _ _ ⊥ₘ _) ⇒
  apply map_disjoint_foldr_delete_l : map_disjoint.
Hint Extern 2 (_ ⊥ₘ foldr delete _ _) ⇒
  apply map_disjoint_foldr_delete_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.

Tactic Notation "simpl_map" "by" tactic3(tac) := repeat
  match goal with
  | H : context[ ∅ !! _ ] |- _ ⇒ rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ ⇒
    rewrite lookup_insert in H || rewrite lookup_insert_ne in H by tac
  | H : context[ (alter _ _ _) !! _] |- _ ⇒
    rewrite lookup_alter in H || rewrite lookup_alter_ne in H by tac
  | H : context[ (delete _ _) !! _] |- _ ⇒
    rewrite lookup_delete in H || rewrite lookup_delete_ne in H by tac
  | H : context[ {[ _ := _ ]} !! _ ] |- _ ⇒
    rewrite lookup_singleton in H || rewrite lookup_singleton_ne in H by tac
  | H : context[ (_ <$> _) !! _ ] |- _ ⇒ rewrite lookup_fmap in H
  | H : context[ (omap _ _) !! _ ] |- _ ⇒ rewrite lookup_omap in H
  | H : context[ lookup (A:=?A) ?i (?m1 ∪ ?m2) ] |- _ ⇒
    let x := fresh in evar (x:A);
    let x' := eval unfold x in x in clear x;
    let E := fresh in
    assert ((m1 ∪ m2) !! i = Some x') as E by (clear H; by tac);
    rewrite E in H; clear E
  | |- context[ ∅ !! _ ] ⇒ rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] ⇒
    rewrite lookup_insert || rewrite lookup_insert_ne by tac
  | |- context[ (alter _ _ _) !! _ ] ⇒
    rewrite lookup_alter || rewrite lookup_alter_ne by tac
  | |- context[ (delete _ _) !! _ ] ⇒
    rewrite lookup_delete || rewrite lookup_delete_ne by tac
  | |- context[ {[ _ := _ ]} !! _ ] ⇒
    rewrite lookup_singleton || rewrite lookup_singleton_ne by tac
  | |- context[ (_ <$> _) !! _ ] ⇒ rewrite lookup_fmap
  | |- context[ (omap _ _) !! _ ] ⇒ rewrite lookup_omap
  | |- context [ lookup (A:=?A) ?i ?m ] ⇒
    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
  end.

Create HintDb simpl_map.
Tactic Notation "simpl_map" := simpl_map by eauto with simpl_map map_disjoint.

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.

Tactic Notation "simplify_map_eq" "by" tactic3(tac) :=
  decompose_map_disjoint;
  repeat match goal with
  | _ ⇒ progress simpl_map by tac
  | _ ⇒ progress simplify_eq/=
  | _ ⇒ progress simpl_option by tac
  | H : {[ _ := _ ]} !! _ = None |- _ ⇒ rewrite lookup_singleton_None in H
  | H : {[ _ := _ ]} !! _ = Some _ |- _ ⇒
    rewrite lookup_singleton_Some in H; destruct H
  | 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
  | H1 : ?m1 !! ?i = Some ?x, H2 : ?m2 !! ?i = None |- _ ⇒
    let H3 := fresh in
    apply (lookup_weaken _ m2) in H1; [congruence|by tac]
  | H : ?m ∪ _ = ?m ∪ _ |- _ ⇒
    apply map_union_cancel_l in H; [|by tac|by tac]
  | H : _ ∪ ?m = _ ∪ ?m |- _ ⇒
    apply map_union_cancel_r in H; [|by tac|by tac]
  | H : {[?i := ?x]} = ∅ |- _ ⇒ by destruct (map_non_empty_singleton i x)
  | H : ∅ = {[?i := ?x]} |- _ ⇒ by destruct (map_non_empty_singleton i x)
  | H : ?m !! ?i = Some _, H2 : ?m !! ?j = None |- _ ⇒
     unless (i ≠ j) by done;
     assert (i ≠ j) by (by intros ?; simplify_eq)
  end.
Tactic Notation "simplify_map_eq" "/=" "by" tactic3(tac) :=
  repeat (progress csimpl in × || simplify_map_eq by tac).
Tactic Notation "simplify_map_eq" :=
  simplify_map_eq by eauto with simpl_map map_disjoint.
Tactic Notation "simplify_map_eq" "/=" :=
  simplify_map_eq/= by eauto with simpl_map map_disjoint.