fin_maps.v 81.5 KB
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(* Copyright (c) 2012-2017, Coq-std++ developers. *)
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(* This file is distributed under the terms of the BSD license. *)
(** Finite maps associate data to keys. This file defines an interface for
finite maps and collects some theory on it. Most importantly, it proves useful
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induction principles for finite maps and implements the tactic
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[simplify_map_eq] to simplify goals involving finite maps. *)
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From Coq Require Import Permutation.
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From stdpp Require Export relations orders vector fin_collections.
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(* FIXME: This file needs a 'Proof Using' hint, but the default we use
   everywhere makes for lots of extra ssumptions. *)
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(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
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course limits the space of finite map implementations, but since we are mainly
interested in finite maps with numbers as indexes, we do not consider this to
be a serious limitation. The main application of finite maps is to implement
the memory, where extensionality of Leibniz equality is very important for a
convenient use in the assertions of our axiomatic semantics. *)
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(** Finiteness is axiomatized by requiring that each map can be translated
to an association list. The translation to association lists is used to
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prove well founded recursion on finite maps. *)
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(** Finite map implementations are required to implement the [merge] function
which enables us to give a generic implementation of [union_with],
[intersection_with], and [difference_with]. *)
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Class FinMapToList K A M := map_to_list: M  list (K * A).
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Class FinMap K M `{FMap M,  A, Lookup K A (M A),  A, Empty (M A),  A,
    PartialAlter K A (M A), OMap M, Merge M,  A, FinMapToList K A (M A),
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    EqDecision K} := {
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  map_eq {A} (m1 m2 : M A) : ( i, m1 !! i = m2 !! i)  m1 = m2;
  lookup_empty {A} i : ( : M A) !! i = None;
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  lookup_partial_alter {A} f (m : M A) i :
    partial_alter f i m !! i = f (m !! i);
  lookup_partial_alter_ne {A} f (m : M A) i j :
    i  j  partial_alter f i m !! j = m !! j;
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  lookup_fmap {A B} (f : A  B) (m : M A) i : (f <$> m) !! i = f <$> m !! i;
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  NoDup_map_to_list {A} (m : M A) : NoDup (map_to_list m);
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  elem_of_map_to_list {A} (m : M A) i x :
    (i,x)  map_to_list m  m !! i = Some x;
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  lookup_omap {A B} (f : A  option B) m i : omap f m !! i = m !! i = f;
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  lookup_merge {A B C} (f: option A  option B  option C) `{!DiagNone f} m1 m2 i :
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    merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
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}.

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(** * Derived operations *)
(** All of the following functions are defined in a generic way for arbitrary
finite map implementations. These generic implementations do not cause a
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significant performance loss, which justifies including them in the finite map
interface as primitive operations. *)
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Instance map_insert `{PartialAlter K A M} : Insert K A M :=
  λ i x, partial_alter (λ _, Some x) i.
Instance map_alter `{PartialAlter K A M} : Alter K A M :=
  λ f, partial_alter (fmap f).
Instance map_delete `{PartialAlter K A M} : Delete K M :=
  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 :=
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  fold_right (λ p, <[p.1:=p.2]>) .
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Definition map_to_collection `{FinMapToList K A M,
    Singleton B C, Empty C, Union C} (f : K  A  B) (m : M) : C :=
  of_list (curry f <$> map_to_list m).
Definition map_of_collection `{Elements B C, Insert K A M, Empty M}
    (f : B  K * A) (X : C) : M :=
  map_of_list (f <$> elements X).
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Instance map_union_with `{Merge M} {A} : UnionWith A (M A) :=
  λ f, merge (union_with f).
Instance map_intersection_with `{Merge M} {A} : IntersectionWith A (M A) :=
  λ f, merge (intersection_with f).
Instance map_difference_with `{Merge M} {A} : DifferenceWith A (M A) :=
  λ f, merge (difference_with f).
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Instance map_equiv `{ A, Lookup K A (M A), Equiv A} : Equiv (M A) | 18 :=
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  λ m1 m2,  i, m1 !! i  m2 !! i.
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(** The relation [intersection_forall R] on finite maps describes that the
relation [R] holds for each pair in the intersection. *)
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Definition map_Forall `{Lookup K A M} (P : K  A  Prop) : M  Prop :=
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  λ m,  i x, m !! i = Some x  P i x.
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Definition map_relation `{ A, Lookup K A (M A)} {A B} (R : A  B  Prop)
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    (P : A  Prop) (Q : B  Prop) (m1 : M A) (m2 : M B) : Prop :=  i,
  option_relation R P Q (m1 !! i) (m2 !! i).
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Definition map_included `{ A, Lookup K A (M A)} {A}
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  (R : relation A) : relation (M A) := map_relation R (λ _, False) (λ _, True).
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Definition map_disjoint `{ A, Lookup K A (M A)} {A} : relation (M A) :=
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  map_relation (λ _ _, False) (λ _, True) (λ _, True).
Infix "⊥ₘ" := map_disjoint (at level 70) : C_scope.
Hint Extern 0 (_  _) => symmetry; eassumption.
Notation "( m ⊥ₘ.)" := (map_disjoint m) (only parsing) : C_scope.
Notation "(.⊥ₘ m )" := (λ m2, m2  m) (only parsing) : C_scope.
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Instance map_subseteq `{ A, Lookup K A (M A)} {A} : SubsetEq (M A) :=
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  map_included (=).
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(** The union of two finite maps only has a meaningful definition for maps
that are disjoint. However, as working with partial functions is inconvenient
in Coq, we define the union as a total function. In case both finite maps
have a value at the same index, we take the value of the first map. *)
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Instance map_union `{Merge M} {A} : Union (M A) := union_with (λ x _, Some x).
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Instance map_intersection `{Merge M} {A} : Intersection (M A) :=
  intersection_with (λ x _, Some x).

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

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(* The zip operation on maps combines two maps key-wise. The keys of resulting
map correspond to the keys that are in both maps. *)
Definition map_zip_with `{Merge M} {A B C} (f : A  B  C) : M A  M B  M C :=
  merge (λ mx my,
    match mx, my with Some x, Some y => Some (f x y) | _, _ => None end).
Notation map_zip := (map_zip_with pair).

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(* Folds a function [f] over a map. The order in which the function is called
is unspecified. *)
Definition map_fold `{FinMapToList K A M} {B}
  (f : K  A  B  B) (b : B) : M  B := foldr (curry f) b  map_to_list.

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

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(** ** Setoids *)
Section setoid.
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  Context `{Equiv A}.
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  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.

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  Global Instance map_equivalence :
    Equivalence (() : relation A)  Equivalence (() : relation (M A)).
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  Proof.
    split.
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    - by intros m i.
    - by intros m1 m2 ? i.
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    - by intros m1 m2 m3 ?? i; trans (m2 !! i).
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  Qed.
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  Global Instance lookup_proper (i : K) :
    Proper (() ==> ()) (lookup (M:=M A) i).
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  Proof. by intros m1 m2 Hm. Qed.
  Global Instance partial_alter_proper :
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    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (partial_alter (M:=M A)).
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  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) :
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    Proper (() ==> () ==> ()) (insert (M:=M A) i).
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  Proof. by intros ???; apply partial_alter_proper; [constructor|]. Qed.
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  Global Instance singleton_proper k :
    Proper (() ==> ()) (singletonM k : A  M A).
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  Proof.
    intros ???; apply insert_proper; [done|].
    intros ?. rewrite lookup_empty; constructor.
  Qed.
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  Global Instance delete_proper (i : K) :
    Proper (() ==> ()) (delete (M:=M A) i).
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  Proof. by apply partial_alter_proper; [constructor|]. Qed.
  Global Instance alter_proper :
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    Proper ((() ==> ()) ==> (=) ==> () ==> ()) (alter (A:=A) (M:=M A)).
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  Proof.
    intros ?? Hf; apply partial_alter_proper.
    by destruct 1; constructor; apply Hf.
  Qed.
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  Lemma merge_ext f g `{!DiagNone f, !DiagNone g} :
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    (() ==> () ==> ())%signature f g 
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    (() ==> () ==> ())%signature (merge (M:=M) f) (merge g).
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  Proof.
    by intros Hf ?? Hm1 ?? Hm2 i; rewrite !lookup_merge by done; apply Hf.
  Qed.
  Global Instance union_with_proper :
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    Proper ((() ==> () ==> ()) ==> () ==> () ==>()) (union_with (M:=M A)).
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  Proof.
    intros ?? Hf ?? Hm1 ?? Hm2 i; apply (merge_ext _ _); auto.
    by do 2 destruct 1; first [apply Hf | constructor].
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  Qed.
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  Global Instance map_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (M A).
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  Proof. intros m1 m2 Hm; apply map_eq; intros i. apply leibniz_equiv, Hm. Qed.
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  Lemma map_equiv_empty (m : M A) : m    m = .
  Proof.
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    split; [intros Hm; apply map_eq; intros i|intros ->].
    - generalize (Hm i). by rewrite lookup_empty, equiv_None.
    - intros ?. rewrite lookup_empty; constructor.
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  Qed.
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  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.
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End setoid.

(** ** General properties *)
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Lemma map_eq_iff {A} (m1 m2 : M A) : m1 = m2   i, m1 !! i = m2 !! i.
Proof. split. by intros ->. apply map_eq. Qed.
Lemma map_subseteq_spec {A} (m1 m2 : M A) :
  m1  m2   i x, m1 !! i = Some x  m2 !! i = Some x.
Proof.
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  unfold subseteq, map_subseteq, map_relation. split; intros Hm i;
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    specialize (Hm i); destruct (m1 !! i), (m2 !! i); naive_solver.
Qed.
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Global Instance map_included_preorder {A} (R : relation A) :
  PreOrder R  PreOrder (map_included R).
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Proof.
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  split; [intros m i; by destruct (m !! i); simpl|].
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  intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
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  destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=;
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    done || etrans; eauto.
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Qed.
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Global Instance map_subseteq_po : PartialOrder (() : relation (M A)).
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Proof.
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  split; [apply _|].
  intros m1 m2; rewrite !map_subseteq_spec.
  intros; apply map_eq; intros i; apply option_eq; naive_solver.
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Qed.
Lemma lookup_weaken {A} (m1 m2 : M A) i x :
  m1 !! i = Some x  m1  m2  m2 !! i = Some x.
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Proof. rewrite !map_subseteq_spec. auto. Qed.
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Lemma lookup_weaken_is_Some {A} (m1 m2 : M A) i :
  is_Some (m1 !! i)  m1  m2  is_Some (m2 !! i).
Proof. inversion 1. eauto using lookup_weaken. Qed.
Lemma lookup_weaken_None {A} (m1 m2 : M A) i :
  m2 !! i = None  m1  m2  m1 !! i = None.
Proof.
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  rewrite map_subseteq_spec, !eq_None_not_Some.
  intros Hm2 Hm [??]; destruct Hm2; eauto.
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Qed.
Lemma lookup_weaken_inv {A} (m1 m2 : M A) i x y :
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  m1 !! i = Some x  m1  m2  m2 !! i = Some y  x = y.
Proof. intros Hm1 ? Hm2. eapply lookup_weaken in Hm1; eauto. congruence. Qed.
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Lemma lookup_ne {A} (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.
Lemma map_empty {A} (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply map_eq. intros. by rewrite Hm, lookup_empty. Qed.
Lemma lookup_empty_is_Some {A} i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. by inversion 1. Qed.
Lemma lookup_empty_Some {A} i (x : A) : ¬ !! i = Some x.
Proof. by rewrite lookup_empty. Qed.
Lemma map_subset_empty {A} (m : M A) : m  .
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Proof.
  intros [_ []]. rewrite map_subseteq_spec. intros ??. by rewrite lookup_empty.
Qed.
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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.
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Lemma map_subset_alt {A} (m1 m2 : M A) :
  m1  m2  m1  m2   i, m1 !! i = None  is_Some (m2 !! i).
Proof.
  rewrite strict_spec_alt. split.
  - intros [? Heq]; split; [done|].
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    destruct (decide (Exists (λ ix, m1 !! ix.1 = None) (map_to_list m2)))
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      as [[[i x] [?%elem_of_map_to_list ?]]%Exists_exists
         |Hm%(not_Exists_Forall _)]; [eauto|].
    destruct Heq; apply (anti_symm _), map_subseteq_spec; [done|intros i x Hi].
    assert (is_Some (m1 !! i)) as [x' ?].
    { by apply not_eq_None_Some,
        (proj1 (Forall_forall _ _) Hm (i,x)), elem_of_map_to_list. }
    by rewrite <-(lookup_weaken_inv m1 m2 i x' x).
  - intros [? (i&?&x&?)]; split; [done|]. congruence.
Qed.

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(** ** Properties of the [partial_alter] operation *)
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Lemma partial_alter_ext {A} (f g : option A  option A) (m : M A) i :
  ( x, m !! i = x  f x = g x)  partial_alter f i m = partial_alter g i m.
Proof.
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  intros. apply map_eq; intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne; auto.
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Qed.
Lemma partial_alter_compose {A} f g (m : M A) i:
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  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
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  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|?];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
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Qed.
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Lemma partial_alter_commute {A} f g (m : M A) i j :
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  i  j  partial_alter f i (partial_alter g j m) =
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    partial_alter g j (partial_alter f i m).
Proof.
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  intros. apply map_eq; intros jj. destruct (decide (jj = j)) as [->|?].
  { by rewrite lookup_partial_alter_ne,
      !lookup_partial_alter, lookup_partial_alter_ne. }
  destruct (decide (jj = i)) as [->|?].
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  - by rewrite lookup_partial_alter,
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     !lookup_partial_alter_ne, lookup_partial_alter by congruence.
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  - by rewrite !lookup_partial_alter_ne by congruence.
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Qed.
Lemma partial_alter_self_alt {A} (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
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  intros. apply map_eq. intros ii. by destruct (decide (i = ii)) as [->|];
    rewrite ?lookup_partial_alter, ?lookup_partial_alter_ne.
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Qed.
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Lemma partial_alter_self {A} (m : M A) i : partial_alter (λ _, m !! i) i m = m.
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Proof. by apply partial_alter_self_alt. Qed.
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Lemma partial_alter_subseteq {A} f (m : M A) i :
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  m !! i = None  m  partial_alter f i m.
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Proof.
  rewrite map_subseteq_spec. intros Hi j x Hj.
  rewrite lookup_partial_alter_ne; congruence.
Qed.
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Lemma partial_alter_subset {A} f (m : M A) i :
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  m !! i = None  is_Some (f (m !! i))  m  partial_alter f i m.
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Proof.
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  intros Hi Hfi. apply map_subset_alt; split; [by apply partial_alter_subseteq|].
  exists i. by rewrite lookup_partial_alter.
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Qed.

(** ** Properties of the [alter] operation *)
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Lemma lookup_alter {A} (f : A  A) m i : alter f i m !! i = f <$> m !! i.
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Proof. unfold alter. apply lookup_partial_alter. Qed.
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Lemma lookup_alter_ne {A} (f : A  A) m i j : i  j  alter f i m !! j = m !! j.
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Proof. unfold alter. apply lookup_partial_alter_ne. Qed.
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Lemma alter_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.
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Lemma alter_compose {A} (f g : A  A) (m : M A) i:
  alter (f  g) i m = alter f i (alter g i m).
Proof.
  unfold alter, map_alter. rewrite <-partial_alter_compose.
  apply partial_alter_ext. by intros [?|].
Qed.
Lemma alter_commute {A} (f g : A  A) (m : M A) i j :
  i  j  alter f i (alter g j m) = alter g j (alter f i m).
Proof. apply partial_alter_commute. Qed.
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Lemma lookup_alter_Some {A} (f : A  A) m i j y :
  alter f i m !! j = Some y 
    (i = j   x, m !! j = Some x  y = f x)  (i  j  m !! j = Some y).
Proof.
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  destruct (decide (i = j)) as [->|?].
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  - rewrite lookup_alter. naive_solver (simplify_option_eq; eauto).
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  - rewrite lookup_alter_ne by done. naive_solver.
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Qed.
Lemma lookup_alter_None {A} (f : A  A) m i j :
  alter f i m !! j = None  m !! j = None.
Proof.
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  by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_alter, ?fmap_None, ?lookup_alter_ne.
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Qed.
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Lemma lookup_alter_is_Some {A} (f : A  A) m i j :
  is_Some (alter f i m !! j)  is_Some (m !! j).
Proof. by rewrite <-!not_eq_None_Some, lookup_alter_None. Qed.
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Lemma alter_id {A} (f : A  A) m i :
  ( x, m !! i = Some x  f x = x)  alter f i m = m.
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Proof.
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  intros Hi; apply map_eq; intros j; destruct (decide (i = j)) as [->|?].
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  { rewrite lookup_alter; destruct (m !! j); f_equal/=; auto. }
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  by rewrite lookup_alter_ne by done.
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Qed.
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Lemma alter_mono {A} f (m1 m2 : M A) i : m1  m2  alter f i m1  alter f i m2.
Proof.
  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_alter_Some. naive_solver.
Qed.
Lemma alter_strict_mono {A} f (m1 m2 : M A) i :
  m1  m2  alter f i m1  alter f i m2.
Proof.
  rewrite !map_subset_alt.
  intros [? (j&?&?)]; split; auto using alter_mono.
  exists j. by rewrite lookup_alter_None, lookup_alter_is_Some.
Qed.
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(** ** Properties of the [delete] operation *)
Lemma lookup_delete {A} (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne {A} (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.
Lemma lookup_delete_Some {A} (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
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  - destruct (decide (i = j)) as [->|?];
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      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
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  - intros [??]. by rewrite lookup_delete_ne.
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Qed.
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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.
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Lemma lookup_delete_None {A} (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
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  destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne; tauto.
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Qed.
Lemma delete_empty {A} i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. by rewrite lookup_empty. Qed.
Lemma delete_commute {A} (m : M A) i j :
  delete i (delete j m) = delete j (delete i m).
Proof. destruct (decide (i = j)). by subst. by apply partial_alter_commute. Qed.
Lemma delete_insert_ne {A} (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. by apply partial_alter_commute. Qed.
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Lemma delete_notin {A} (m : M A) i : m !! i = None  delete i m = m.
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Proof.
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  intros. apply map_eq. intros j. by destruct (decide (i = j)) as [->|?];
    rewrite ?lookup_delete, ?lookup_delete_ne.
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Qed.
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Lemma delete_idemp {A} (m : M A) i :
  delete i (delete i m) = delete i m.
Proof. by setoid_rewrite <-partial_alter_compose. Qed.
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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.
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Lemma delete_insert_delete {A} (m : M A) i x :
  delete i (<[i:=x]>m) = delete i m.
Proof. by setoid_rewrite <-partial_alter_compose. Qed.
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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.
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Lemma delete_subseteq {A} (m : M A) i : delete i m  m.
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Proof.
  rewrite !map_subseteq_spec. intros j x. rewrite lookup_delete_Some. tauto.
Qed.
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Lemma delete_subset {A} (m : M A) i : is_Some (m !! i)  delete i m  m.
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Proof.
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  intros [x ?]; apply map_subset_alt; split; [apply delete_subseteq|].
  exists i. rewrite lookup_delete; eauto.
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Qed.
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Lemma delete_mono {A} (m1 m2 : M A) i : m1  m2  delete i m1  delete i m2.
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Proof.
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  rewrite !map_subseteq_spec. intros ? j x.
  rewrite !lookup_delete_Some. intuition eauto.
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Qed.

(** ** Properties of the [insert] operation *)
Lemma lookup_insert {A} (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
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Lemma lookup_insert_rev {A}  (m : M A) i x y : <[i:=x]>m !! i = Some y  x = y.
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Proof. rewrite lookup_insert. congruence. Qed.
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Lemma lookup_insert_ne {A} (m : M A) i j x : i  j  <[i:=x]>m !! j = m !! j.
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Proof. unfold insert. apply lookup_partial_alter_ne. Qed.
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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.
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Lemma insert_commute {A} (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_commute. Qed.
Lemma lookup_insert_Some {A} (m : M A) i j x y :
  <[i:=x]>m !! j = Some y  (i = j  x = y)  (i  j  m !! j = Some y).
Proof.
  split.
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  - destruct (decide (i = j)) as [->|?];
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      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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  - intros [[-> ->]|[??]]; [apply lookup_insert|]. by rewrite lookup_insert_ne.
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Qed.
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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.
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Lemma lookup_insert_is_Some' {A} (m : M A) i j x :
  is_Some (<[i:=x]>m !! j)  i = j  is_Some (m !! j).
Proof. rewrite lookup_insert_is_Some. destruct (decide (i=j)); naive_solver. Qed.
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Lemma lookup_insert_None {A} (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
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  split; [|by intros [??]; rewrite lookup_insert_ne].
  destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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Qed.
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Lemma insert_id {A} (m : M A) i x : m !! i = Some x  <[i:=x]>m = m.
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Proof.
  intros; apply map_eq; intros j; destruct (decide (i = j)) as [->|];
    by rewrite ?lookup_insert, ?lookup_insert_ne by done.
Qed.
Lemma insert_included {A} R `{!Reflexive R} (m : M A) i x :
  ( y, m !! i = Some y  R y x)  map_included R m (<[i:=x]>m).
Proof.
  intros ? j; destruct (decide (i = j)) as [->|].
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  - rewrite lookup_insert. destruct (m !! j); simpl; eauto.
  - rewrite lookup_insert_ne by done. by destruct (m !! j); simpl.
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Qed.
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Lemma insert_empty {A} i (x : A) : <[i:=x]> = {[i := x]}.
Proof. done. Qed.
Lemma insert_non_empty {A} (m : M A) i x : <[i:=x]>m  .
Proof.
  intros Hi%(f_equal (!! i)). by rewrite lookup_insert, lookup_empty in Hi.
Qed.

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Lemma insert_subseteq {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
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Proof. apply partial_alter_subseteq. Qed.
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Lemma insert_subset {A} (m : M A) i x : m !! i = None  m  <[i:=x]>m.
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Proof. intro. apply partial_alter_subset; eauto. Qed.
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Lemma insert_mono {A} (m1 m2 : M A) i x : m1  m2  <[i:=x]> m1  <[i:=x]>m2.
Proof.
  rewrite !map_subseteq_spec.
  intros Hm j y. rewrite !lookup_insert_Some. naive_solver.
Qed.
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Lemma insert_subseteq_r {A} (m1 m2 : M A) i x :
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  m1 !! i = None  m1  m2  m1  <[i:=x]>m2.
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Proof.
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  intros. trans (<[i:=x]> m1); eauto using insert_subseteq, insert_mono.
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Qed.
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Lemma insert_delete_subseteq {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
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Proof.
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  rewrite !map_subseteq_spec. intros Hi Hix j y Hj.
  destruct (decide (i = j)) as [->|]; [congruence|].
  rewrite lookup_delete_ne by done.
  apply Hix; by rewrite lookup_insert_ne by done.
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Qed.
Lemma delete_insert_subseteq {A} (m1 m2 : M A) i x :
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  m1 !! i = Some x  delete i m1  m2  m1  <[i:=x]> m2.
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Proof.
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  rewrite !map_subseteq_spec.
  intros Hix Hi j y Hj. destruct (decide (i = j)) as [->|?].
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  - rewrite lookup_insert. congruence.
  - rewrite lookup_insert_ne by done. apply Hi. by rewrite lookup_delete_ne.
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Qed.
Lemma insert_delete_subset {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2  m1  delete i m2.
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Proof.
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  intros ? [Hm12 Hm21]; split; [eauto using insert_delete_subseteq|].
  contradict Hm21. apply delete_insert_subseteq; auto.
  eapply lookup_weaken, Hm12. by rewrite lookup_insert.
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Qed.
Lemma insert_subset_inv {A} (m1 m2 : M A) i x :
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  m1 !! i = None  <[i:=x]> m1  m2 
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   m2', m2 = <[i:=x]>m2'  m1  m2'  m2' !! i = None.
Proof.
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  intros Hi Hm1m2. exists (delete i m2). split_and?.
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  - rewrite insert_delete, insert_id. done.
    eapply lookup_weaken, strict_include; eauto. by rewrite lookup_insert.
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  - eauto using insert_delete_subset.
  - by rewrite lookup_delete.
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Qed.

(** ** Properties of the singleton maps *)
Lemma lookup_singleton_Some {A} i j (x y : A) :
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  {[i := x]} !! j = Some y  i = j  x = y.
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Proof.
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  rewrite <-insert_empty,lookup_insert_Some, lookup_empty; intuition congruence.
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Qed.
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Lemma lookup_singleton_None {A} i j (x : A) : {[i := x]} !! j = None  i  j.
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Proof. rewrite <-insert_empty,lookup_insert_None, lookup_empty; tauto. Qed.
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Lemma lookup_singleton {A} i (x : A) : {[i := x]} !! i = Some x.
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Proof. by rewrite lookup_singleton_Some. Qed.
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Lemma lookup_singleton_ne {A} i j (x : A) : i  j  {[i := x]} !! j = None.
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Proof. by rewrite lookup_singleton_None. Qed.
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Lemma map_non_empty_singleton {A} i (x : A) : {[i := x]}  .
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Proof.
  intros Hix. apply (f_equal (!! i)) in Hix.
  by rewrite lookup_empty, lookup_singleton in Hix.
Qed.
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Lemma insert_singleton {A} i (x y : A) : <[i:=y]>{[i := x]} = {[i := y]}.
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Proof.
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  unfold singletonM, map_singleton, insert, map_insert.
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  by rewrite <-partial_alter_compose.
Qed.
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Lemma alter_singleton {A} (f : A  A) i x : alter f i {[i := x]} = {[i := f x]}.
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Proof.
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  intros. apply map_eq. intros i'. destruct (decide (i = i')) as [->|?].
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  - by rewrite lookup_alter, !lookup_singleton.
  - by rewrite lookup_alter_ne, !lookup_singleton_ne.
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Qed.
Lemma alter_singleton_ne {A} (f : A  A) i j x :
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  i  j  alter f i {[j := x]} = {[j := x]}.
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Proof.
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  intros. apply map_eq; intros i'. by destruct (decide (i = i')) as [->|?];
    rewrite ?lookup_alter, ?lookup_singleton_ne, ?lookup_alter_ne by done.
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Qed.
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Lemma singleton_non_empty {A} i (x : A) : {[i:=x]}  .
Proof. apply insert_non_empty. Qed.
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Lemma delete_singleton {A} i (x : A) : delete i {[i := x]} = .
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
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Lemma delete_singleton_ne {A} i j (x : A) :
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  i  j  delete i {[j := x]} = {[j := x]}.
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Proof. intro. apply delete_notin. by apply lookup_singleton_ne. Qed.
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(** ** Properties of the map operations *)
Lemma fmap_empty {A B} (f : A  B) : f <$>  = .
Proof. apply map_empty; intros i. by rewrite lookup_fmap, lookup_empty. Qed.
Lemma omap_empty {A B} (f : A  option B) : omap f  = .
Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
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Lemma fmap_insert {A B} (f: A  B) m i x: f <$> <[i:=x]>m = <[i:=f x]>(f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
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  - by rewrite lookup_fmap, !lookup_insert.
  - by rewrite lookup_fmap, !lookup_insert_ne, lookup_fmap by done.
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Qed.
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Lemma fmap_delete {A B} (f: A  B) m i: f <$> delete i m = delete i (f <$> m).
Proof.
  apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
  - by rewrite lookup_fmap, !lookup_delete.
  - by rewrite lookup_fmap, !lookup_delete_ne, lookup_fmap by done.
Qed.
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Lemma omap_insert {A B} (f : A  option B) m i x y :
  f x = Some y  omap f (<[i:=x]>m) = <[i:=y]>(omap f m).
Proof.
  intros; apply map_eq; intros i'; destruct (decide (i' = i)) as [->|].
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  - by rewrite lookup_omap, !lookup_insert.
  - by rewrite lookup_omap, !lookup_insert_ne, lookup_omap by done.
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Qed.
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Lemma map_fmap_singleton {A B} (f : A  B) i x : f <$> {[i := x]} = {[i := f x]}.
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Proof.
  by unfold singletonM, map_singleton; rewrite fmap_insert, map_fmap_empty.
Qed.
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Lemma omap_singleton {A B} (f : A  option B) i x y :
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  f x = Some y  omap f {[ i := x ]} = {[ i := y ]}.
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Proof.
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  intros. unfold singletonM, map_singleton.
  by erewrite omap_insert, omap_empty by eauto.
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Qed.
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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.
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Lemma map_fmap_equiv_ext `{Equiv A, Equiv B} (f1 f2 : A  B) m :
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  ( 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.
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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.
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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.
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Lemma map_fmap_mono {A B} (f : A  B) (m1 m2 : M A) :
  m1  m2  f <$> m1  f <$> m2.
Proof.
  rewrite !map_subseteq_spec; intros Hm i x.
  rewrite !lookup_fmap, !fmap_Some. naive_solver.
Qed.
Lemma map_fmap_strict_mono {A B} (f : A  B) (m1 m2 : M A) :
  m1  m2  f <$> m1  f <$> m2.
Proof.
  rewrite !map_subset_alt.
  intros [? (j&?&?)]; split; auto using map_fmap_mono.
  exists j. by rewrite !lookup_fmap, fmap_None, fmap_is_Some.
Qed.
Lemma map_omap_mono {A B} (f : A  option B) (m1 m2 : M A) :
  m1  m2  omap f m1  omap f m2.
Proof.
  rewrite !map_subseteq_spec; intros Hm i x.
  rewrite !lookup_omap, !bind_Some. naive_solver.
Qed.

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(** ** Properties of conversion to lists *)
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Lemma elem_of_map_to_list' {A} (m : M A) ix :
  ix  map_to_list m  m !! ix.1 = Some (ix.2).
Proof. destruct ix as [i x]. apply elem_of_map_to_list. Qed.
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Lemma map_to_list_unique {A} (m : M A) i x y :
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  (i,x)  map_to_list m  (i,y)  map_to_list m  x = y.
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Proof. rewrite !elem_of_map_to_list. congruence. Qed.
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Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
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Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
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Lemma elem_of_map_of_list_1' {A} (l : list (K * A)) i x :
  ( y, (i,y)  l  x = y)  (i,x)  l  map_of_list l !! i = Some x.
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Proof.
  induction l as [|[j y] l IH]; csimpl; [by rewrite elem_of_nil|].
  setoid_rewrite elem_of_cons.
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  intros Hdup [?|?]; simplify_eq; [by rewrite lookup_insert|].
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  destruct (decide (i = j)) as [->|].
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  - rewrite lookup_insert; f_equal; eauto using eq_sym.
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  - rewrite lookup_insert_ne by done; eauto.
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Qed.
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Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
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  NoDup (l.*1)  (i,x)  l  map_of_list l !! i = Some x.
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Proof.
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  intros ? Hx; apply elem_of_map_of_list_1'; eauto using NoDup_fmap_fst.
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  intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
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  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
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    by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
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Qed.
Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
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  map_of_list l !! i = Some x  (i,x)  l.
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Proof.
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  induction l as [|[j y] l IH]; simpl; [by rewrite lookup_empty|].
  rewrite elem_of_cons. destruct (decide (i = j)) as [->|];
    rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
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Qed.
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Lemma elem_of_map_of_list' {A} (l : list (K * A)) i x :
  ( x', (i,x)  l  (i,x')  l  x = x') 
  (i,x)  l  map_of_list l !! i = Some x.
Proof. split; auto using elem_of_map_of_list_1', elem_of_map_of_list_2. Qed.
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Lemma elem_of_map_of_list {A} (l :