fin_maps.v 15.2 KB
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Require Export prelude.

Class FinMap K M `{ A, Empty (M A)} `{Lookup K M} `{FMap M} 
    `{PartialAlter K M} `{ A, Dom K (M A)} `{Merge M} := {
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  finmap_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;
  elem_of_dom C {A} `{Collection K C} (m : M A) i :
    i  dom C m  is_Some (m !! i);
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  merge_spec {A} f `{!PropHolds (f None None = None)} 
    (m1 m2 : M A) i : merge f m1 m2 !! i = f (m1 !! i) (m2 !! i)
}.

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Instance finmap_alter `{PartialAlter K M} : Alter K M := λ A f,
  partial_alter (fmap f).
Instance finmap_insert `{PartialAlter K M} : Insert K M := λ A k x,
  partial_alter (λ _, Some x) k.
Instance finmap_delete `{PartialAlter K M} {A} : Delete K (M A) :=
  partial_alter (λ _, None).
Instance finmap_singleton `{PartialAlter K M} {A} 
  `{Empty (M A)} : Singleton (K * A) (M A) := λ p, <[fst p:=snd p]>.
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Definition list_to_map `{Insert K M} {A} `{Empty (M A)} (l : list (K * A)) : M A :=
  insert_list l .
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Instance finmap_union `{Merge M} : UnionWith M := λ A f,
  merge (union_with f).
Instance finmap_intersection `{Merge M} : IntersectionWith M := λ A f,
  merge (intersection_with f).
Instance finmap_difference `{Merge M} : DifferenceWith M := λ A f,
  merge (difference_with f).
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Section finmap.
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Context `{FinMap K M} `{ i j : K, Decision (i = j)} {A : Type}.

Global Instance finmap_subseteq: SubsetEq (M A) := λ m n,
   i x, m !! i = Some x  n !! i = Some x.
Global Instance: BoundedPreOrder (M A).
Proof. split. firstorder. intros m i x. rewrite lookup_empty. discriminate. Qed.

Lemma lookup_subseteq_Some (m1 m2 : M A) i x :
  m1  m2  m1 !! i = Some x  m2 !! i = Some x.
Proof. auto. Qed.
Lemma lookup_subseteq_None (m1 m2 : M A) i :
  m1  m2  m2 !! i = None  m1 !! i = None.
Proof. rewrite !eq_None_not_Some. firstorder. Qed.
Lemma lookup_ne (m : M A) i j : m !! i  m !! j  i  j.
Proof. congruence. Qed.

Lemma not_elem_of_dom C `{Collection K C} (m : M A) i :
  i  dom C m  m !! i = None.
Proof. now rewrite (elem_of_dom C), eq_None_not_Some. Qed.

Lemma finmap_empty (m : M A) : ( i, m !! i = None)  m = .
Proof. intros Hm. apply finmap_eq. intros. now rewrite Hm, lookup_empty. Qed.
Lemma dom_empty C `{Collection K C} : dom C ( : M A)  .
Proof.
  split; intro.
  * rewrite (elem_of_dom C), lookup_empty. simplify_is_Some.
  * simplify_elem_of.
Qed.
Lemma dom_empty_inv C `{Collection K C} (m : M A) : dom C m    m = .
Proof.
  intros E. apply finmap_empty. intros. apply (not_elem_of_dom C).
  rewrite E. simplify_elem_of.
Qed.

Lemma lookup_empty_not i : ¬is_Some (( : M A) !! i).
Proof. rewrite lookup_empty. simplify_is_Some. Qed.
Lemma lookup_empty_Some i (x : A) : ¬ !! i = Some x.
Proof. rewrite lookup_empty. discriminate. Qed.

Lemma partial_alter_compose (m : M A) i f g :
  partial_alter (f  g) i m = partial_alter f i (partial_alter g i m).
Proof.
  intros. apply finmap_eq. intros ii. case (decide (i = ii)).
  * intros. subst. now rewrite !lookup_partial_alter.
  * intros. now rewrite !lookup_partial_alter_ne.
Qed.
Lemma partial_alter_comm (m : M A) i j f g :
  i  j 
 partial_alter f i (partial_alter g j m) = partial_alter g j (partial_alter f i m).
Proof.
  intros. apply finmap_eq. intros jj.
  destruct (decide (jj = j)).
  * subst. now rewrite lookup_partial_alter_ne,
     !lookup_partial_alter, lookup_partial_alter_ne.
  * destruct (decide (jj = i)).
    + subst. now rewrite lookup_partial_alter,
       !lookup_partial_alter_ne, lookup_partial_alter by congruence.
    + now rewrite !lookup_partial_alter_ne by congruence.
Qed.
Lemma partial_alter_self_alt (m : M A) i x :
  x = m !! i  partial_alter (λ _, x) i m = m.
Proof.
  intros. apply finmap_eq. intros ii.
  destruct (decide (i = ii)).
  * subst. now rewrite lookup_partial_alter.
  * now rewrite lookup_partial_alter_ne.
Qed.
Lemma partial_alter_self (m : M A) i : partial_alter (λ _, m !! i) i m = m.
Proof. now apply partial_alter_self_alt. Qed.

Lemma lookup_insert (m : M A) i x : <[i:=x]>m !! i = Some x.
Proof. unfold insert. apply lookup_partial_alter. Qed.
Lemma lookup_insert_rev (m : M A) i x y : <[i:= x ]>m !! i = Some y  x = y.
Proof. rewrite lookup_insert. congruence. Qed.
Lemma lookup_insert_ne (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_comm (m : M A) i j x y :
  i  j  <[i:=x]>(<[j:=y]>m) = <[j:=y]>(<[i:=x]>m).
Proof. apply partial_alter_comm. Qed.

Lemma lookup_insert_Some (m : M A) i j x y :
  <[i:=x]>m !! j = Some y  (i = j  x = y)  (i  j  m !! j = Some y).
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [[??]|[??]].
    + subst. apply lookup_insert.
    + now rewrite lookup_insert_ne.
Qed.
Lemma lookup_insert_None (m : M A) i j x :
  <[i:=x]>m !! j = None  m !! j = None  i  j.
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
  * intros [??]. now rewrite lookup_insert_ne.
Qed.

Lemma lookup_singleton_Some i j (x y : A) :
  {[(i, x)]} !! j = Some y  i = j  x = y.
Proof.
  unfold singleton, finmap_singleton.
  rewrite lookup_insert_Some, lookup_empty. simpl.
  intuition congruence.
Qed.
Lemma lookup_singleton_None i j (x : A) :
  {[(i, x)]} !! j = None  i  j.
Proof.
  unfold singleton, finmap_singleton.
  rewrite lookup_insert_None, lookup_empty. simpl. tauto.
Qed.

Lemma lookup_singleton i (x : A) : {[(i, x)]} !! i = Some x.
Proof. rewrite lookup_singleton_Some. tauto. Qed.
Lemma lookup_singleton_ne i j (x : A) : i  j  {[(i, x)]} !! j = None.
Proof. now rewrite lookup_singleton_None. Qed.

Lemma lookup_delete (m : M A) i : delete i m !! i = None.
Proof. apply lookup_partial_alter. Qed.
Lemma lookup_delete_ne (m : M A) i j : i  j  delete i m !! j = m !! j.
Proof. apply lookup_partial_alter_ne. Qed.

Lemma lookup_delete_Some (m : M A) i j y :
  delete i m !! j = Some y  i  j  m !! j = Some y.
Proof.
  split.
  * destruct (decide (i = j)); subst;
      rewrite ?lookup_delete, ?lookup_delete_ne; intuition congruence.
  * intros [??]. now rewrite lookup_delete_ne.
Qed.
Lemma lookup_delete_None (m : M A) i j :
  delete i m !! j = None  i = j  m !! j = None.
Proof.
  destruct (decide (i = j)).
  * subst. rewrite lookup_delete. tauto.
  * rewrite lookup_delete_ne; tauto.
Qed.

Lemma delete_empty i : delete i ( : M A) = .
Proof. rewrite <-(partial_alter_self ) at 2. now rewrite lookup_empty. Qed.
Lemma delete_singleton i (x : A) : delete i {[(i, x)]} = .
Proof. setoid_rewrite <-partial_alter_compose. apply delete_empty. Qed.
Lemma delete_comm (m : M A) i j : delete i (delete j m) = delete j (delete i m).
Proof. destruct (decide (i = j)). now subst. now apply partial_alter_comm. Qed.
Lemma delete_insert_comm (m : M A) i j x :
  i  j  delete i (<[j:=x]>m) = <[j:=x]>(delete i m).
Proof. intro. now apply partial_alter_comm. Qed.

Lemma delete_notin (m : M A) i : m !! i = None  delete i m = m.
Proof.
  intros. apply finmap_eq. intros j.
  destruct (decide (i = j)).
  * subst. now rewrite lookup_delete.
  * now apply lookup_delete_ne.
Qed.

Lemma delete_partial_alter (m : M A) i f :
  m !! i = None  delete i (partial_alter f i m) = m.
Proof.
  intros. unfold delete, finmap_delete. rewrite <-partial_alter_compose. 
  rapply partial_alter_self_alt. congruence.
Qed.
Lemma delete_partial_alter_dom C `{Collection K C} (m : M A) i f : 
  i  dom C m  delete i (partial_alter f i m) = m.
Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
Lemma delete_insert (m : M A) i x : m !! i = None  delete i (<[i:=x]>m) = m.
Proof. apply delete_partial_alter. Qed.
Lemma delete_insert_dom C `{Collection K C} (m : M A) i x :
  i  dom C m  delete i (<[i:=x]>m) = m.
Proof. rewrite (not_elem_of_dom C). apply delete_partial_alter. Qed.
Lemma insert_delete (m : M A) i x : m !! i = Some x  <[i:=x]>(delete i m) = m.
Proof.
  intros Hmi. unfold delete, finmap_delete, insert, finmap_insert.
  rewrite <-partial_alter_compose. unfold compose. rewrite <-Hmi.
  now apply partial_alter_self_alt.
Qed.

Lemma elem_of_dom_delete C `{Collection K C} (m : M A) i j :
  i  dom C (delete j m)  i  j  i  dom C m.
Proof.
  rewrite !(elem_of_dom C). unfold is_Some.
  setoid_rewrite lookup_delete_Some. firstorder auto.
Qed.
Lemma not_elem_of_dom_delete C `{Collection K C} (m : M A) i :
  i  dom C (delete i m).
Proof. apply (not_elem_of_dom C), lookup_delete. Qed.
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Lemma lookup_delete_list (m : M A) is j :
  In j is  delete_list is m !! j = None.
Proof.
  induction is as [|i is]; simpl; [easy |].
  intros [?|?].
  * subst. now rewrite lookup_delete.
  * destruct (decide (i = j)).
    + subst. now rewrite lookup_delete.
    + rewrite lookup_delete_ne; auto.
Qed.
Lemma lookup_delete_list_notin (m : M A) is j :
  ¬In j is  delete_list is m !! j = m !! j.
Proof.
  induction is; simpl; [easy |].
  intros. rewrite lookup_delete_ne; tauto.
Qed.

Lemma delete_list_notin (m : M A) is :
  Forall (λ i, m !! i = None) is  delete_list is m = m.
Proof.
  induction 1; simpl; [easy |].
  rewrite delete_notin; congruence.
Qed.
Lemma delete_list_insert_comm (m : M A) is j x :
  ¬In j is  delete_list is (<[j:=x]>m) = <[j:=x]>(delete_list is m).
Proof.
  induction is; simpl; [easy |].
  intros. rewrite IHis, delete_insert_comm; tauto.
Qed.

Lemma finmap_ind C (P : M A  Prop) `{FinCollection K C} :
  P   ( i x m, i  dom C m  P m  P (<[i:=x]>m))   m, P m.
Proof.
  intros Hemp Hinsert.
  intros m. apply (collection_ind (λ X,  m, dom C m  X  P m)) with (dom C m).
  * solve_proper.
  * clear m. intros m Hm. rewrite finmap_empty.
    + easy.
    + intros. rewrite <-(not_elem_of_dom C), Hm.
      now simplify_elem_of.
  * clear m. intros i X Hi IH m Hdom.
    assert (is_Some (m !! i)) as [x Hx].
    { apply (elem_of_dom C).
      rewrite Hdom. clear Hdom.
      now simplify_elem_of. }
    rewrite <-(insert_delete m i x) by easy.
    apply Hinsert.
    { now apply (not_elem_of_dom_delete C). }
    apply IH. apply elem_of_equiv. intros.
    rewrite (elem_of_dom_delete C). rewrite Hdom.
    clear Hdom. simplify_elem_of.
  * easy.
Qed.

Section merge.
  Context (f : option A  option A  option A).

  Global Instance: LeftId (=) None f  LeftId (=)  (merge f).
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  Proof.
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    intros ??. apply finmap_eq. intros.
    now rewrite !(merge_spec f), lookup_empty, (left_id None f).
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  Qed.
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  Global Instance: RightId (=) None f  RightId (=)  (merge f).
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  Proof.
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    intros ??. apply finmap_eq. intros.
    now rewrite !(merge_spec f), lookup_empty, (right_id None f).
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  Qed.
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  Global Instance: Idempotent (=) f  Idempotent (=) (merge f).
  Proof. intros ??. apply finmap_eq. intros. now rewrite !(merge_spec f). Qed.

  Context `{!PropHolds (f None None = None)}.

  Lemma merge_spec_alt m1 m2 m :
    ( i, m !! i = f (m1 !! i) (m2 !! i))  merge f m1 m2 = m.
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  Proof.
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    split; [| intro; subst; apply (merge_spec _) ].
    intros Hlookup. apply finmap_eq. intros. rewrite Hlookup.
    apply (merge_spec _).
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  Qed.
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  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 finmap_eq. intros. now rewrite !(merge_spec f). Qed.
  Global Instance: Commutative (=) f  Commutative (=) (merge f).
  Proof. intros ???. apply merge_comm. intros. now apply (commutative 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 finmap_eq. intros. now rewrite !(merge_spec f). Qed.
  Global Instance: Associative (=) f  Associative (=) (merge f).
  Proof. intros ????. apply merge_assoc. intros. now apply (associative f). Qed.
End merge.

Section union_intersection.
  Context (f : A  A  A).

  Lemma finmap_union_merge m1 m2 i x y :
    m1 !! i = Some x  m2 !! i = Some y  union_with f m1 m2 !! i = Some (f x y).
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  Proof.
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    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
  Qed.   
  Lemma finmap_union_l m1 m2 i x :
    m1 !! i = Some x  m2 !! i = None  union_with f m1 m2 !! i = Some x.
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  Proof.
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    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
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  Qed.
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  Lemma finmap_union_r m1 m2 i y :
    m1 !! i = None  m2 !! i = Some y  union_with f m1 m2 !! i = Some y.
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  Proof.
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    intros Hx Hy. unfold union_with, finmap_union.
    now rewrite (merge_spec _), Hx, Hy.
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  Qed.
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  Lemma finmap_union_None m1 m2 i :
    union_with f m1 m2 !! i = None  m1 !! i = None  m2 !! i = None.
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  Proof.
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    unfold union_with, finmap_union. rewrite (merge_spec _).
    destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
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  Qed.

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  Global Instance: LeftId (=)  (union_with f : M A  M A  M A) := _.
  Global Instance: RightId (=)  (union_with f : M A  M A  M A) := _.
  Global Instance:
    Commutative (=) f  Commutative (=) (union_with f : M A  M A  M A) := _.
  Global Instance:
    Associative (=) f  Associative (=) (union_with f : M A  M A  M A) := _.
  Global Instance:
    Idempotent (=) f  Idempotent (=) (union_with f : M A  M A  M A) := _.
End union_intersection.
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Lemma lookup_insert_list (m : M A) l1 l2 i x :
  (y, ¬In (i,y) l1)  insert_list (l1 ++ (i,x) :: l2) m !! i = Some x.
Proof.
  induction l1 as [|[j y] l1 IH]; simpl.
   intros. now rewrite lookup_insert.
  intros Hy. rewrite lookup_insert_ne. apply IH.
   firstorder.
  intro. apply (Hy y). left. congruence.
Qed.
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Lemma lookup_insert_list_not_in (m : M A) l i :
  (y, ¬In (i,y) l)  insert_list l m !! i = m !! i.
Proof.
  induction l as [|[j y] l IH]; simpl.
   easy.
  intros Hy. rewrite lookup_insert_ne.
   firstorder.
  intro. apply (Hy y). left. congruence.
Qed.
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End finmap.
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Tactic Notation "simplify_map" "by" tactic(T) := repeat
  match goal with
  | _ => progress simplify_eqs
  | H : context[  !! _ ] |- _ => rewrite lookup_empty in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert in H
  | H : context[ (<[_:=_]>_) !! _ ] |- _ => rewrite lookup_insert_ne in H by T
  | H : context[ (delete _ _) !! _ ] |- _ => rewrite lookup_delete in H
  | H : context[ (delete _ _) !! _ ] |- _ => rewrite lookup_delete_ne in H by T
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton in H
  | H : context[ {[ _ ]} !! _ ] |- _ => rewrite lookup_singleton_ne in H by T
  | |- context[  !! _ ] => rewrite lookup_empty
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert
  | |- context[ (<[_:=_]>_) !! _ ] => rewrite lookup_insert_ne by T
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete
  | |- context[ (delete _ _) !! _ ] => rewrite lookup_delete_ne by T
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton
  | |- context[ {[ _ ]} !! _ ] => rewrite lookup_singleton_ne by T
  end.
Tactic Notation "simplify_map" := simplify_map by auto.