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(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
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From prelude Require Export base tactics orders.
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Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y,
   x, x  X  x  Y.

(** * Basic theorems *)
Section simple_collection.
  Context `{SimpleCollection A C}.
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  Implicit Types x y : A.
  Implicit Types X Y : C.
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  Lemma elem_of_empty x : x    False.
  Proof. split. apply not_elem_of_empty. done. Qed.
  Lemma elem_of_union_l x X Y : x  X  x  X  Y.
  Proof. intros. apply elem_of_union. auto. Qed.
  Lemma elem_of_union_r x X Y : x  Y  x  X  Y.
  Proof. intros. apply elem_of_union. auto. Qed.
  Global Instance: EmptySpec C.
  Proof. firstorder auto. Qed.
  Global Instance: JoinSemiLattice C.
  Proof. firstorder auto. Qed.
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  Global Instance: AntiSymm () (@collection_subseteq A C _).
  Proof. done. Qed.
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  Lemma elem_of_subseteq X Y : X  Y   x, x  X  x  Y.
  Proof. done. Qed.
  Lemma elem_of_equiv X Y : X  Y   x, x  X  x  Y.
  Proof. firstorder. Qed.
  Lemma elem_of_equiv_alt X Y :
    X  Y  ( x, x  X  x  Y)  ( x, x  Y  x  X).
  Proof. firstorder. Qed.
  Lemma elem_of_equiv_empty X : X     x, x  X.
  Proof. firstorder. Qed.
  Lemma collection_positive_l X Y : X  Y    X  .
  Proof.
    rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver.
  Qed.
  Lemma collection_positive_l_alt X Y : X    X  Y  .
  Proof. eauto using collection_positive_l. Qed.
  Lemma elem_of_singleton_1 x y : x  {[y]}  x = y.
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma elem_of_singleton_2 x y : x = y  x  {[y]}.
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma elem_of_subseteq_singleton x X : x  X  {[ x ]}  X.
  Proof.
    split.
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    - intros ??. rewrite elem_of_singleton. by intros ->.
    - intros Ex. by apply (Ex x), elem_of_singleton.
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  Qed.
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  Global Instance singleton_proper : Proper ((=) ==> ()) (singleton (B:=C)).
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  Proof. by repeat intro; subst. Qed.
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  Global Instance elem_of_proper :
    Proper ((=) ==> () ==> iff) (() : A  C  Prop) | 5.
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  Proof. intros ???; subst. firstorder. Qed.
  Lemma elem_of_union_list Xs x : x   Xs   X, X  Xs  x  X.
  Proof.
    split.
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    - induction Xs; simpl; intros HXs; [by apply elem_of_empty in HXs|].
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      setoid_rewrite elem_of_cons. apply elem_of_union in HXs. naive_solver.
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    - intros [X []]. induction 1; simpl; [by apply elem_of_union_l |].
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      intros. apply elem_of_union_r; auto.
  Qed.
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  Lemma non_empty_singleton x : ({[ x ]} : C)  .
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  Proof. intros [E _]. by apply (elem_of_empty x), E, elem_of_singleton. Qed.
  Lemma not_elem_of_singleton x y : x  {[ y ]}  x  y.
  Proof. by rewrite elem_of_singleton. Qed.
  Lemma not_elem_of_union x X Y : x  X  Y  x  X  x  Y.
  Proof. rewrite elem_of_union. tauto. Qed.

  Section leibniz.
    Context `{!LeibnizEquiv C}.
    Lemma elem_of_equiv_L X Y : X = Y   x, x  X  x  Y.
    Proof. unfold_leibniz. apply elem_of_equiv. Qed.
    Lemma elem_of_equiv_alt_L X Y :
      X = Y  ( x, x  X  x  Y)  ( x, x  Y  x  X).
    Proof. unfold_leibniz. apply elem_of_equiv_alt. Qed.
    Lemma elem_of_equiv_empty_L X : X =    x, x  X.
    Proof. unfold_leibniz. apply elem_of_equiv_empty. Qed.
    Lemma collection_positive_l_L X Y : X  Y =   X = .
    Proof. unfold_leibniz. apply collection_positive_l. Qed.
    Lemma collection_positive_l_alt_L X Y : X    X  Y  .
    Proof. unfold_leibniz. apply collection_positive_l_alt. Qed.
    Lemma non_empty_singleton_L x : {[ x ]}  .
    Proof. unfold_leibniz. apply non_empty_singleton. Qed.
  End leibniz.

  Section dec.
    Context `{ X Y : C, Decision (X  Y)}.
    Global Instance elem_of_dec_slow (x : A) (X : C) : Decision (x  X) | 100.
    Proof.
      refine (cast_if (decide_rel () {[ x ]} X));
        by rewrite elem_of_subseteq_singleton.
    Defined.
  End dec.
End simple_collection.

Definition of_option `{Singleton A C, Empty C} (x : option A) : C :=
  match x with None =>  | Some a => {[ a ]} end.
Fixpoint of_list `{Singleton A C, Empty C, Union C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  of_list l end.

Section of_option_list.
  Context `{SimpleCollection A C}.
  Lemma elem_of_of_option (x : A) o : x  of_option o  o = Some x.
  Proof.
    destruct o; simpl;
      rewrite ?elem_of_empty, ?elem_of_singleton; naive_solver.
  Qed.
  Lemma elem_of_of_list (x : A) l : x  of_list l  x  l.
  Proof.
    split.
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    - induction l; simpl; [by rewrite elem_of_empty|].
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      rewrite elem_of_union,elem_of_singleton; intros [->|?]; constructor; auto.
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    - induction 1; simpl; rewrite elem_of_union, elem_of_singleton; auto.
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  Qed.
End of_option_list.

Global Instance collection_guard `{CollectionMonad M} : MGuard M :=
  λ P dec A x, match dec with left H => x H | _ =>  end.

Section collection_monad_base.
  Context `{CollectionMonad M}.
  Lemma elem_of_guard `{Decision P} {A} (x : A) (X : M A) :
    x  guard P; X  P  x  X.
  Proof.
    unfold mguard, collection_guard; simpl; case_match;
      rewrite ?elem_of_empty; naive_solver.
  Qed.
  Lemma elem_of_guard_2 `{Decision P} {A} (x : A) (X : M A) :
    P  x  X  x  guard P; X.
  Proof. by rewrite elem_of_guard. Qed.
  Lemma guard_empty `{Decision P} {A} (X : M A) : guard P; X    ¬P  X  .
  Proof.
    rewrite !elem_of_equiv_empty; setoid_rewrite elem_of_guard.
    destruct (decide P); naive_solver.
  Qed.
  Lemma bind_empty {A B} (f : A  M B) X :
    X = f    X     x, x  X  f x  .
  Proof.
    setoid_rewrite elem_of_equiv_empty; setoid_rewrite elem_of_bind.
    naive_solver.
  Qed.
End collection_monad_base.

(** * Tactics *)
(** Given a hypothesis [H : _ ∈ _], the tactic [destruct_elem_of H] will
recursively split [H] for [(∪)], [(∩)], [(∖)], [map], [∅], [{[_]}]. *)
Tactic Notation "decompose_elem_of" hyp(H) :=
  let rec go H :=
  lazymatch type of H with
  | _   => apply elem_of_empty in H; destruct H
  | ?x  {[ ?y ]} =>
    apply elem_of_singleton in H; try first [subst y | subst x]
  | ?x  {[ ?y ]} =>
    apply not_elem_of_singleton in H
  | _  _  _ =>
    apply elem_of_union in H; destruct H as [H|H]; [go H|go H]
  | _  _  _ =>
    let H1 := fresh H in let H2 := fresh H in apply not_elem_of_union in H;
    destruct H as [H1 H2]; go H1; go H2
  | _  _  _ =>
    let H1 := fresh H in let H2 := fresh H in apply elem_of_intersection in H;
    destruct H as [H1 H2]; go H1; go H2
  | _  _  _ =>
    let H1 := fresh H in let H2 := fresh H in apply elem_of_difference in H;
    destruct H as [H1 H2]; go H1; go H2
  | ?x  _ <$> _ =>
    apply elem_of_fmap in H; destruct H as [? [? H]]; try (subst x); go H
  | _  _ = _ =>
    let H1 := fresh H in let H2 := fresh H in apply elem_of_bind in H;
    destruct H as [? [H1 H2]]; go H1; go H2
  | ?x  mret ?y =>
    apply elem_of_ret in H; try first [subst y | subst x]
  | _  mjoin _ = _ =>
    let H1 := fresh H in let H2 := fresh H in apply elem_of_join in H;
    destruct H as [? [H1 H2]]; go H1; go H2
  | _  guard _; _ =>
    let H1 := fresh H in let H2 := fresh H in apply elem_of_guard in H;
    destruct H as [H1 H2]; go H2
  | _  of_option _ => apply elem_of_of_option in H
  | _  of_list _ => apply elem_of_of_list in H
  | _ => idtac
  end in go H.
Tactic Notation "decompose_elem_of" :=
  repeat_on_hyps (fun H => decompose_elem_of H).

Ltac decompose_empty := repeat
  match goal with
  | H :    |- _ => clear H
  | H :  =  |- _ => clear H
  | H :   _ |- _ => symmetry in H
  | H :  = _ |- _ => symmetry in H
  | H : _  _   |- _ => apply empty_union in H; destruct H
  | H : _  _   |- _ => apply non_empty_union in H; destruct H
  | H : {[ _ ]}   |- _ => destruct (non_empty_singleton _ H)
  | H : _  _ =  |- _ => apply empty_union_L in H; destruct H
  | H : _  _   |- _ => apply non_empty_union_L in H; destruct H
  | H : {[ _ ]} =  |- _ => destruct (non_empty_singleton_L _ H)
  | H : guard _ ; _   |- _ => apply guard_empty in H; destruct H
  end.

(** The first pass of our collection tactic consists of eliminating all
occurrences of [(∪)], [(∩)], [(∖)], [(<$>)], [∅], [{[_]}], [(≡)], and [(⊆)],
by rewriting these into logically equivalent propositions. For example we
rewrite [A → x ∈ X ∪ ∅] into [A → x ∈ X ∨ False]. *)
Ltac unfold_elem_of :=
  repeat_on_hyps (fun H =>
    repeat match type of H with
    | context [ _  _ ] => setoid_rewrite elem_of_subseteq in H
    | context [ _  _ ] => setoid_rewrite subset_spec in H
    | context [ _   ] => setoid_rewrite elem_of_equiv_empty in H
    | context [ _  _ ] => setoid_rewrite elem_of_equiv_alt in H
    | context [ _ =  ] => setoid_rewrite elem_of_equiv_empty_L in H
    | context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L in H
    | context [ _   ] => setoid_rewrite elem_of_empty in H
    | context [ _  {[ _ ]} ] => setoid_rewrite elem_of_singleton in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_union in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_intersection in H
    | context [ _  _  _ ] => setoid_rewrite elem_of_difference in H
    | context [ _  _ <$> _ ] => setoid_rewrite elem_of_fmap in H
    | context [ _  mret _ ] => setoid_rewrite elem_of_ret in H
    | context [ _  _ = _ ] => setoid_rewrite elem_of_bind in H
    | context [ _  mjoin _ ] => setoid_rewrite elem_of_join in H
    | context [ _  guard _; _ ] => setoid_rewrite elem_of_guard in H
    | context [ _  of_option _ ] => setoid_rewrite elem_of_of_option in H
    | context [ _  of_list _ ] => setoid_rewrite elem_of_of_list in H
    end);
  repeat match goal with
  | |- context [ _  _ ] => setoid_rewrite elem_of_subseteq
  | |- context [ _  _ ] => setoid_rewrite subset_spec
  | |- context [ _   ] => setoid_rewrite elem_of_equiv_empty
  | |- context [ _  _ ] => setoid_rewrite elem_of_equiv_alt
  | |- context [ _ =  ] => setoid_rewrite elem_of_equiv_empty_L
  | |- context [ _ = _ ] => setoid_rewrite elem_of_equiv_alt_L
  | |- context [ _   ] => setoid_rewrite elem_of_empty
  | |- context [ _  {[ _ ]} ] => setoid_rewrite elem_of_singleton
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_union
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_intersection
  | |- context [ _  _  _ ] => setoid_rewrite elem_of_difference
  | |- context [ _  _ <$> _ ] => setoid_rewrite elem_of_fmap
  | |- context [ _  mret _ ] => setoid_rewrite elem_of_ret
  | |- context [ _  _ = _ ] => setoid_rewrite elem_of_bind
  | |- context [ _  mjoin _ ] => setoid_rewrite elem_of_join
  | |- context [ _  guard _; _ ] => setoid_rewrite elem_of_guard
  | |- context [ _  of_option _ ] => setoid_rewrite elem_of_of_option
  | |- context [ _  of_list _ ] => setoid_rewrite elem_of_of_list
  end.

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(** Since [firstorder] fails or loops on very small goals generated by
[solve_elem_of] already. We use the [naive_solver] tactic as a substitute.
This tactic either fails or proves the goal. *)
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Tactic Notation "solve_elem_of" tactic3(tac) :=
  setoid_subst;
  decompose_empty;
  unfold_elem_of;
  naive_solver tac.
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Tactic Notation "solve_elem_of" "-" hyp_list(Hs) "/" tactic3(tac) :=
  clear Hs; solve_elem_of tac.
Tactic Notation "solve_elem_of" "+" hyp_list(Hs) "/" tactic3(tac) :=
  revert Hs; clear; solve_elem_of tac.
Tactic Notation "solve_elem_of" := solve_elem_of eauto.
Tactic Notation "solve_elem_of" "-" hyp_list(Hs) := clear Hs; solve_elem_of.
Tactic Notation "solve_elem_of" "+" hyp_list(Hs) :=
  revert Hs; clear; solve_elem_of.
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(** * More theorems *)
Section collection.
  Context `{Collection A C}.
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  Implicit Types X Y : C.
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  Global Instance: Lattice C.
  Proof. split. apply _. firstorder auto. solve_elem_of. Qed.
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  Global Instance difference_proper :
     Proper (() ==> () ==> ()) (@difference C _).
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  Proof.
    intros X1 X2 HX Y1 Y2 HY; apply elem_of_equiv; intros x.
    by rewrite !elem_of_difference, HX, HY.
  Qed.
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  Lemma non_empty_inhabited x X : x  X  X  .
  Proof. solve_elem_of. Qed.
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  Lemma intersection_singletons x : ({[x]} : C)  {[x]}  {[x]}.
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  Proof. solve_elem_of. Qed.
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  Lemma difference_twice X Y : (X  Y)  Y  X  Y.
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  Proof. solve_elem_of. Qed.
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  Lemma subseteq_empty_difference X Y : X  Y  X  Y  .
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  Proof. solve_elem_of. Qed.
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  Lemma difference_diag X : X  X  .
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  Proof. solve_elem_of. Qed.
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  Lemma difference_union_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
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  Proof. solve_elem_of. Qed.
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  Lemma difference_union_distr_r X Y Z : Z  (X  Y)  (Z  X)  (Z  Y).
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  Proof. solve_elem_of. Qed.
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  Lemma difference_intersection_distr_l X Y Z : (X  Y)  Z  X  Z  Y  Z.
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  Proof. solve_elem_of. Qed.
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  Lemma disjoint_union_difference X Y : X  Y    (X  Y)  X  Y.
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  Proof. solve_elem_of. Qed.
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  Section leibniz.
    Context `{!LeibnizEquiv C}.
    Lemma intersection_singletons_L x : {[x]}  {[x]} = {[x]}.
    Proof. unfold_leibniz. apply intersection_singletons. Qed.
    Lemma difference_twice_L X Y : (X  Y)  Y = X  Y.
    Proof. unfold_leibniz. apply difference_twice. Qed.
    Lemma subseteq_empty_difference_L X Y : X  Y  X  Y = .
    Proof. unfold_leibniz. apply subseteq_empty_difference. Qed.
    Lemma difference_diag_L X : X  X = .
    Proof. unfold_leibniz. apply difference_diag. Qed.
    Lemma difference_union_distr_l_L X Y Z : (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_union_distr_l. Qed.
    Lemma difference_union_distr_r_L X Y Z : Z  (X  Y) = (Z  X)  (Z  Y).
    Proof. unfold_leibniz. apply difference_union_distr_r. Qed.
    Lemma difference_intersection_distr_l_L X Y Z :
      (X  Y)  Z = X  Z  Y  Z.
    Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
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    Lemma disjoint_union_difference_L X Y : X  Y =   (X  Y)  X = Y.
    Proof. unfold_leibniz. apply disjoint_union_difference. Qed.
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  End leibniz.

  Section dec.
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    Context `{ (x : A) (X : C), Decision (x  X)}.
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    Lemma not_elem_of_intersection x X Y : x  X  Y  x  X  x  Y.
    Proof. rewrite elem_of_intersection. destruct (decide (x  X)); tauto. Qed.
    Lemma not_elem_of_difference x X Y : x  X  Y  x  X  x  Y.
    Proof. rewrite elem_of_difference. destruct (decide (x  Y)); tauto. Qed.
    Lemma union_difference X Y : X  Y  Y  X  Y  X.
    Proof.
      split; intros x; rewrite !elem_of_union, elem_of_difference; [|intuition].
      destruct (decide (x  X)); intuition.
    Qed.
    Lemma non_empty_difference X Y : X  Y  Y  X  .
    Proof.
      intros [HXY1 HXY2] Hdiff. destruct HXY2. intros x.
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      destruct (decide (x  X)); solve_elem_of.
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    Qed.
    Lemma empty_difference_subseteq X Y : X  Y    X  Y.
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    Proof. intros ? x ?; apply dec_stable; solve_elem_of. Qed.
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    Context `{!LeibnizEquiv C}.
    Lemma union_difference_L X Y : X  Y  Y = X  Y  X.
    Proof. unfold_leibniz. apply union_difference. Qed.
    Lemma non_empty_difference_L X Y : X  Y  Y  X  .
    Proof. unfold_leibniz. apply non_empty_difference. Qed.
    Lemma empty_difference_subseteq_L X Y : X  Y =   X  Y.
    Proof. unfold_leibniz. apply empty_difference_subseteq. Qed.
  End dec.
End collection.

Section collection_ops.
  Context `{CollectionOps A C}.

  Lemma elem_of_intersection_with_list (f : A  A  option A) Xs Y x :
    x  intersection_with_list f Y Xs   xs y,
      Forall2 () xs Xs  y  Y  foldr (λ x, (= f x)) (Some y) xs = Some x.
  Proof.
    split.
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    - revert x. induction Xs; simpl; intros x HXs; [eexists [], x; intuition|].
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      rewrite elem_of_intersection_with in HXs; destruct HXs as (x1&x2&?&?&?).
      destruct (IHXs x2) as (xs & y & hy & ? & ?); trivial.
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      eexists (x1 :: xs), y. intuition (simplify_option_eq; auto).
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    - intros (xs & y & Hxs & ? & Hx). revert x Hx.
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      induction Hxs; intros; simplify_option_eq; [done |].
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      rewrite elem_of_intersection_with. naive_solver.
  Qed.

  Lemma intersection_with_list_ind (P Q : A  Prop) f Xs Y :
    ( y, y  Y  P y) 
    Forall (λ X,  x, x  X  Q x) Xs 
    ( x y z, Q x  P y  f x y = Some z  P z) 
     x, x  intersection_with_list f Y Xs  P x.
  Proof.
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    intros HY HXs Hf. induction Xs; simplify_option_eq; [done |].
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    intros x Hx. rewrite elem_of_intersection_with in Hx.
    decompose_Forall. destruct Hx as (? & ? & ? & ? & ?). eauto.
  Qed.
End collection_ops.

(** * Sets without duplicates up to an equivalence *)
Section NoDup.
  Context `{SimpleCollection A B} (R : relation A) `{!Equivalence R}.

  Definition elem_of_upto (x : A) (X : B) :=  y, y  X  R x y.
  Definition set_NoDup (X : B) :=  x y, x  X  y  X  R x y  x = y.

  Global Instance: Proper (() ==> iff) (elem_of_upto x).
  Proof. intros ??? E. unfold elem_of_upto. by setoid_rewrite E. Qed.
  Global Instance: Proper (R ==> () ==> iff) elem_of_upto.
  Proof.
    intros ?? E1 ?? E2. split; intros [z [??]]; exists z.
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    - rewrite <-E1, <-E2; intuition.
    - rewrite E1, E2; intuition.
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  Qed.
  Global Instance: Proper (() ==> iff) set_NoDup.
  Proof. firstorder. Qed.

  Lemma elem_of_upto_elem_of x X : x  X  elem_of_upto x X.
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  Proof. unfold elem_of_upto. solve_elem_of. Qed.
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  Lemma elem_of_upto_empty x : ¬elem_of_upto x .
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  Proof. unfold elem_of_upto. solve_elem_of. Qed.
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  Lemma elem_of_upto_singleton x y : elem_of_upto x {[ y ]}  R x y.
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  Proof. unfold elem_of_upto. solve_elem_of. Qed.
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  Lemma elem_of_upto_union X Y x :
    elem_of_upto x (X  Y)  elem_of_upto x X  elem_of_upto x Y.
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  Proof. unfold elem_of_upto. solve_elem_of. Qed.
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  Lemma not_elem_of_upto x X : ¬elem_of_upto x X   y, y  X  ¬R x y.
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  Proof. unfold elem_of_upto. solve_elem_of. Qed.
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  Lemma set_NoDup_empty: set_NoDup .
  Proof. unfold set_NoDup. solve_elem_of. Qed.
  Lemma set_NoDup_add x X :
    ¬elem_of_upto x X  set_NoDup X  set_NoDup ({[ x ]}  X).
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  Proof. unfold set_NoDup, elem_of_upto. solve_elem_of. Qed.
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  Lemma set_NoDup_inv_add x X :
    x  X  set_NoDup ({[ x ]}  X)  ¬elem_of_upto x X.
  Proof.
    intros Hin Hnodup [y [??]].
    rewrite (Hnodup x y) in Hin; solve_elem_of.
  Qed.
  Lemma set_NoDup_inv_union_l X Y : set_NoDup (X  Y)  set_NoDup X.
  Proof. unfold set_NoDup. solve_elem_of. Qed.
  Lemma set_NoDup_inv_union_r X Y : set_NoDup (X  Y)  set_NoDup Y.
  Proof. unfold set_NoDup. solve_elem_of. Qed.
End NoDup.

(** * Quantifiers *)
Section quantifiers.
  Context `{SimpleCollection A B} (P : A  Prop).

  Definition set_Forall X :=  x, x  X  P x.
  Definition set_Exists X :=  x, x  X  P x.

  Lemma set_Forall_empty : set_Forall .
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_singleton x : set_Forall {[ x ]}  P x.
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union X Y : set_Forall X  set_Forall Y  set_Forall (X  Y).
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union_inv_1 X Y : set_Forall (X  Y)  set_Forall X.
  Proof. unfold set_Forall. solve_elem_of. Qed.
  Lemma set_Forall_union_inv_2 X Y : set_Forall (X  Y)  set_Forall Y.
  Proof. unfold set_Forall. solve_elem_of. Qed.

  Lemma set_Exists_empty : ¬set_Exists .
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  Proof. unfold set_Exists. solve_elem_of. Qed.
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  Lemma set_Exists_singleton x : set_Exists {[ x ]}  P x.
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  Proof. unfold set_Exists. solve_elem_of. Qed.
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  Lemma set_Exists_union_1 X Y : set_Exists X  set_Exists (X  Y).
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  Proof. unfold set_Exists. solve_elem_of. Qed.
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  Lemma set_Exists_union_2 X Y : set_Exists Y  set_Exists (X  Y).
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  Proof. unfold set_Exists. solve_elem_of. Qed.
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  Lemma set_Exists_union_inv X Y :
    set_Exists (X  Y)  set_Exists X  set_Exists Y.
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  Proof. unfold set_Exists. solve_elem_of. Qed.
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End quantifiers.

Section more_quantifiers.
  Context `{SimpleCollection A B}.

  Lemma set_Forall_weaken (P Q : A  Prop) (Hweaken :  x, P x  Q x) X :
    set_Forall P X  set_Forall Q X.
  Proof. unfold set_Forall. naive_solver. Qed.
  Lemma set_Exists_weaken (P Q : A  Prop) (Hweaken :  x, P x  Q x) X :
    set_Exists P X  set_Exists Q X.
  Proof. unfold set_Exists. naive_solver. Qed.
End more_quantifiers.

(** * Fresh elements *)
(** We collect some properties on the [fresh] operation. In particular we
generalize [fresh] to generate lists of fresh elements. *)
Fixpoint fresh_list `{Fresh A C, Union C, Singleton A C}
    (n : nat) (X : C) : list A :=
  match n with
  | 0 => []
  | S n => let x := fresh X in x :: fresh_list n ({[ x ]}  X)
  end.
Inductive Forall_fresh `{ElemOf A C} (X : C) : list A  Prop :=
  | Forall_fresh_nil : Forall_fresh X []
  | Forall_fresh_cons x xs :
     x  xs  x  X  Forall_fresh X xs  Forall_fresh X (x :: xs).

Section fresh.
  Context `{FreshSpec A C}.
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  Implicit Types X Y : C.
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  Global Instance fresh_proper: Proper (() ==> (=)) (fresh (C:=C)).
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  Proof. intros ???. by apply fresh_proper_alt, elem_of_equiv. Qed.
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  Global Instance fresh_list_proper:
    Proper ((=) ==> () ==> (=)) (fresh_list (C:=C)).
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  Proof.
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    intros ? n ->. induction n as [|n IH]; intros ?? E; f_equal/=; [by rewrite E|].
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    apply IH. by rewrite E.
  Qed.

  Lemma Forall_fresh_NoDup X xs : Forall_fresh X xs  NoDup xs.
  Proof. induction 1; by constructor. Qed.
  Lemma Forall_fresh_elem_of X xs x : Forall_fresh X xs  x  xs  x  X.
  Proof.
    intros HX; revert x; rewrite <-Forall_forall.
    by induction HX; constructor.
  Qed.
  Lemma Forall_fresh_alt X xs :
    Forall_fresh X xs  NoDup xs   x, x  xs  x  X.
  Proof.
    split; eauto using Forall_fresh_NoDup, Forall_fresh_elem_of.
    rewrite <-Forall_forall.
    intros [Hxs Hxs']. induction Hxs; decompose_Forall_hyps; constructor; auto.
  Qed.
  Lemma Forall_fresh_subseteq X Y xs :
    Forall_fresh X xs  Y  X  Forall_fresh Y xs.
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  Proof. rewrite !Forall_fresh_alt; solve_elem_of. Qed.
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  Lemma fresh_list_length n X : length (fresh_list n X) = n.
  Proof. revert X. induction n; simpl; auto. Qed.
  Lemma fresh_list_is_fresh n X x : x  fresh_list n X  x  X.
  Proof.
    revert X. induction n as [|n IH]; intros X; simpl;[by rewrite elem_of_nil|].
    rewrite elem_of_cons; intros [->| Hin]; [apply is_fresh|].
    apply IH in Hin; solve_elem_of.
  Qed.
  Lemma NoDup_fresh_list n X : NoDup (fresh_list n X).
  Proof.
    revert X. induction n; simpl; constructor; auto.
    intros Hin; apply fresh_list_is_fresh in Hin; solve_elem_of.
  Qed.
  Lemma Forall_fresh_list X n : Forall_fresh X (fresh_list n X).
  Proof.
    rewrite Forall_fresh_alt; eauto using NoDup_fresh_list, fresh_list_is_fresh.
  Qed.
End fresh.

(** * Properties of implementations of collections that form a monad *)
Section collection_monad.
  Context `{CollectionMonad M}.

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  Global Instance collection_fmap_mono {A B} :
    Proper (pointwise_relation _ (=) ==> () ==> ()) (@fmap M _ A B).
  Proof. intros f g ? X Y ?; solve_elem_of. Qed.
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  Global Instance collection_fmap_proper {A B} :
    Proper (pointwise_relation _ (=) ==> () ==> ()) (@fmap M _ A B).
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  Proof. intros f g ? X Y [??]; split; solve_elem_of. Qed.
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  Global Instance collection_bind_mono {A B} :
    Proper (((=) ==> ()) ==> () ==> ()) (@mbind M _ A B).
  Proof. unfold respectful; intros f g Hfg X Y ?; solve_elem_of. Qed.
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  Global Instance collection_bind_proper {A B} :
    Proper (((=) ==> ()) ==> () ==> ()) (@mbind M _ A B).
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  Proof. unfold respectful; intros f g Hfg X Y [??]; split; solve_elem_of. Qed.
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  Global Instance collection_join_mono {A} :
    Proper (() ==> ()) (@mjoin M _ A).
  Proof. intros X Y ?; solve_elem_of. Qed.
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  Global Instance collection_join_proper {A} :
    Proper (() ==> ()) (@mjoin M _ A).
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  Proof. intros X Y [??]; split; solve_elem_of. Qed.
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  Lemma collection_bind_singleton {A B} (f : A  M B) x : {[ x ]} = f  f x.
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  Proof. solve_elem_of. Qed.
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  Lemma collection_guard_True {A} `{Decision P} (X : M A) : P  guard P; X  X.
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  Proof. solve_elem_of. Qed.
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  Lemma collection_fmap_compose {A B C} (f : A  B) (g : B  C) (X : M A) :
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    g  f <$> X  g <$> (f <$> X).
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  Proof. solve_elem_of. Qed.
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  Lemma elem_of_fmap_1 {A B} (f : A  B) (X : M A) (y : B) :
    y  f <$> X   x, y = f x  x  X.
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  Proof. solve_elem_of. Qed.
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  Lemma elem_of_fmap_2 {A B} (f : A  B) (X : M A) (x : A) :
    x  X  f x  f <$> X.
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  Proof. solve_elem_of. Qed.
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  Lemma elem_of_fmap_2_alt {A B} (f : A  B) (X : M A) (x : A) (y : B) :
    x  X  y = f x  y  f <$> X.
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  Proof. solve_elem_of. Qed.
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  Lemma elem_of_mapM {A B} (f : A  M B) l k :
    l  mapM f k  Forall2 (λ x y, x  f y) l k.
  Proof.
    split.
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    - revert l. induction k; solve_elem_of.
    - induction 1; solve_elem_of.
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  Qed.
  Lemma collection_mapM_length {A B} (f : A  M B) l k :
    l  mapM f k  length l = length k.
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  Proof. revert l; induction k; solve_elem_of. Qed.
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  Lemma elem_of_mapM_fmap {A B} (f : A  B) (g : B  M A) l k :
    Forall (λ x,  y, y  g x  f y = x) l  k  mapM g l  fmap f k = l.
  Proof.
    intros Hl. revert k. induction Hl; simpl; intros;
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      decompose_elem_of; f_equal/=; auto.
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  Qed.
  Lemma elem_of_mapM_Forall {A B} (f : A  M B) (P : B  Prop) l k :
    l  mapM f k  Forall (λ x,  y, y  f x  P y) k  Forall P l.
  Proof. rewrite elem_of_mapM. apply Forall2_Forall_l. Qed.
  Lemma elem_of_mapM_Forall2_l {A B C} (f : A  M B) (P: B  C  Prop) l1 l2 k :
    l1  mapM f k  Forall2 (λ x y,  z, z  f x  P z y) k l2 
    Forall2 P l1 l2.
  Proof.
    rewrite elem_of_mapM. intros Hl1. revert l2.
    induction Hl1; inversion_clear 1; constructor; auto.
  Qed.
End collection_monad.
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(** Finite collections *)
Definition set_finite `{ElemOf A B} (X : B) :=  l : list A,  x, x  X  x  l.

Section finite.
  Context `{SimpleCollection A B}.
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  Global Instance set_finite_subseteq :
     Proper (flip () ==> impl) (@set_finite A B _).
  Proof. intros X Y HX [l Hl]; exists l; solve_elem_of. Qed.
  Global Instance set_finite_proper : Proper (() ==> iff) (@set_finite A B _).
  Proof. by intros X Y [??]; split; apply set_finite_subseteq. Qed.
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  Lemma empty_finite : set_finite .
  Proof. by exists []; intros ?; rewrite elem_of_empty. Qed.
  Lemma singleton_finite (x : A) : set_finite {[ x ]}.
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  Proof. exists [x]; intros y ->%elem_of_singleton; left. Qed.
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  Lemma union_finite X Y : set_finite X  set_finite Y  set_finite (X  Y).
  Proof.
    intros [lX ?] [lY ?]; exists (lX ++ lY); intros x.
    rewrite elem_of_union, elem_of_app; naive_solver.
  Qed.
  Lemma union_finite_inv_l X Y : set_finite (X  Y)  set_finite X.
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  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
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  Lemma union_finite_inv_r X Y : set_finite (X  Y)  set_finite Y.
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  Proof. intros [l ?]; exists l; solve_elem_of. Qed.
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End finite.

Section more_finite.
  Context `{Collection A B}.
  Lemma intersection_finite_l X Y : set_finite X  set_finite (X  Y).
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  Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed.
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  Lemma intersection_finite_r X Y : set_finite Y  set_finite (X  Y).
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  Proof. intros [l ?]; exists l; intros x [??]%elem_of_intersection; auto. Qed.
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  Lemma difference_finite X Y : set_finite X  set_finite (X  Y).
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  Proof. intros [l ?]; exists l; intros x [??]%elem_of_difference; auto. Qed.
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  Lemma difference_finite_inv X Y `{ x, Decision (x  Y)} :
    set_finite Y  set_finite (X  Y)  set_finite X.
  Proof.
    intros [l ?] [k ?]; exists (l ++ k).
    intros x ?; destruct (decide (x  Y)); rewrite elem_of_app; solve_elem_of.
  Qed.
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End more_finite.