Library iris.bi.big_op

From stdpp Require Import countable fin_sets functions.
From iris.bi Require Import derived_laws_sbi.
From iris.algebra Require Export big_op.
Set Default Proof Using "Type".
Import interface.bi derived_laws_bi.bi derived_laws_sbi.bi.

Notations for unary variants
Notation "'[∗' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_sep (λ k x, P) l) : bi_scope.
Notation "'[∗' 'list]' x ∈ l , P" :=
  (big_opL bi_sep (λ _ x, P) l) : bi_scope.
Notation "'[∗]' Ps" := (big_opL bi_sep (λ _ x, x) Ps) : bi_scope.

Notation "'[∧' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_and (λ k x, P) l) : bi_scope.
Notation "'[∧' 'list]' x ∈ l , P" :=
  (big_opL bi_and (λ _ x, P) l) : bi_scope.
Notation "'[∧]' Ps" := (big_opL bi_and (λ _ x, x) Ps) : bi_scope.

Notation "'[∨' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_or (λ k x, P) l) : bi_scope.
Notation "'[∨' 'list]' x ∈ l , P" :=
  (big_opL bi_or (λ _ x, P) l) : bi_scope.
Notation "'[∨]' Ps" := (big_opL bi_or (λ _ x, x) Ps) : bi_scope.

Notation "'[∗' 'map]' k ↦ x ∈ m , P" := (big_opM bi_sep (λ k x, P) m) : bi_scope.
Notation "'[∗' 'map]' x ∈ m , P" := (big_opM bi_sep (λ _ x, P) m) : bi_scope.

Notation "'[∗' 'set]' x ∈ X , P" := (big_opS bi_sep (λ x, P) X) : bi_scope.

Notation "'[∗' 'mset]' x ∈ X , P" := (big_opMS bi_sep (λ x, P) X) : bi_scope.

Definitions and notations for binary variants A version of the separating big operator that ranges over two lists. This version also ensures that both lists have the same length. Although this version can be defined in terms of the unary using a zip (see big_sepL2_alt), we do not define it that way to get better computational behavior (for simpl).
Fixpoint big_sepL2 {PROP : bi} {A B}
    (Φ : nat A B PROP) (l1 : list A) (l2 : list B) : PROP :=
  match l1, l2 with
  | [], []emp
  | x1 :: l1, x2 :: l2Φ 0 x1 x2 big_sepL2 (λ n, Φ (S n)) l1 l2
  | _, _False
  end%I.
Instance: Params (@big_sepL2) 3 := {}.
Arguments big_sepL2 {PROP A B} _ !_ !_ /.
Typeclasses Opaque big_sepL2.
Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P" :=
  (big_sepL2 (λ k x1 x2, P) l1 l2) : bi_scope.
Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P" :=
  (big_sepL2 (λ _ x1 x2, P) l1 l2) : bi_scope.

Definition big_sepM2 {PROP : bi} `{Countable K} {A B}
    (Φ : K A B PROP) (m1 : gmap K A) (m2 : gmap K B) : PROP :=
  ( k, is_Some (m1 !! k) is_Some (m2 !! k)
   [∗ map] k xy map_zip m1 m2, Φ k xy.1 xy.2)%I.
Instance: Params (@big_sepM2) 6 := {}.
Typeclasses Opaque big_sepM2.
Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P" :=
  (big_sepM2 (λ k x1 x2, P) m1 m2) : bi_scope.
Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P" :=
  (big_sepM2 (λ _ x1 x2, P) m1 m2) : bi_scope.

Properties

Section bi_big_op.
Context {PROP : bi}.
Implicit Types P Q : PROP.
Implicit Types Ps Qs : list PROP.
Implicit Types A : Type.

Big ops over lists

Section sep_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat A PROP.

  Lemma big_sepL_nil Φ : ([∗ list] ky nil, Φ k y) ⊣⊢ emp.
  Proof. done. Qed.
  Lemma big_sepL_nil' `{BiAffine PROP} P Φ : P [∗ list] ky nil, Φ k y.
  Proof. apply (affine _). Qed.
  Lemma big_sepL_cons Φ x l :
    ([∗ list] ky x :: l, Φ k y) ⊣⊢ Φ 0 x [∗ list] ky l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_sepL_singleton Φ x : ([∗ list] ky [x], Φ k y) ⊣⊢ Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_sepL_app Φ l1 l2 :
    ([∗ list] ky l1 ++ l2, Φ k y)
    ⊣⊢ ([∗ list] ky l1, Φ k y) ([∗ list] ky l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.

  Lemma big_sepL_mono Φ Ψ l :
    ( k y, l !! k = Some y Φ k y Ψ k y)
    ([∗ list] k y l, Φ k y) [∗ list] k y l, Ψ k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_sepL_proper Φ Ψ l :
    ( k y, l !! k = Some y Φ k y ⊣⊢ Ψ k y)
    ([∗ list] k y l, Φ k y) ⊣⊢ ([∗ list] k y l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
  Lemma big_sepL_submseteq `{BiAffine PROP} (Φ : A PROP) l1 l2 :
    l1 ⊆+ l2 ([∗ list] y l2, Φ y) [∗ list] y l1, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_sepL_app sep_elim_l.
  Qed.

  Global Instance big_sepL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢))
           (big_opL (@bi_sep PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_sepL_id_mono' :
    Proper (Forall2 (⊢) ==> (⊢)) (big_opL (@bi_sep PROP) (λ _ P, P)).
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

  Lemma big_sepL_emp l : ([∗ list] ky l, emp) ⊣⊢@{PROP} emp.
  Proof. by rewrite big_opL_unit. Qed.

  Lemma big_sepL_lookup_acc Φ l i x :
    l !! i = Some x
    ([∗ list] ky l, Φ k y) Φ i x (Φ i x -∗ ([∗ list] ky l, Φ k y)).
  Proof.
    intros Hli. rewrite -(take_drop_middle l i x) // big_sepL_app /=.
    rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl.
    rewrite assoc -!(comm _ (Φ _ _)) -assoc. by apply sep_mono_r, wand_intro_l.
  Qed.

  Lemma big_sepL_lookup Φ l i x `{!Absorbing (Φ i x)} :
    l !! i = Some x ([∗ list] ky l, Φ k y) Φ i x.
  Proof. intros. rewrite big_sepL_lookup_acc //. by rewrite sep_elim_l. Qed.

  Lemma big_sepL_elem_of (Φ : A PROP) l x `{!Absorbing (Φ x)} :
    x l ([∗ list] y l, Φ y) Φ x.
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_sepL_lookup (λ _, Φ)).
  Qed.

  Lemma big_sepL_fmap {B} (f : A B) (Φ : nat B PROP) l :
    ([∗ list] ky f <$> l, Φ k y) ⊣⊢ ([∗ list] ky l, Φ k (f y)).
  Proof. by rewrite big_opL_fmap. Qed.

  Lemma big_sepL_bind {B} (f : A list B) (Φ : B PROP) l :
    ([∗ list] y l ≫= f, Φ y) ⊣⊢ ([∗ list] x l, [∗ list] y f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

  Lemma big_sepL_sep Φ Ψ l :
    ([∗ list] kx l, Φ k x Ψ k x)
    ⊣⊢ ([∗ list] kx l, Φ k x) ([∗ list] kx l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

  Lemma big_sepL_and Φ Ψ l :
    ([∗ list] kx l, Φ k x Ψ k x)
     ([∗ list] kx l, Φ k x) ([∗ list] kx l, Ψ k x).
  Proof. auto using and_intro, big_sepL_mono, and_elim_l, and_elim_r. Qed.

  Lemma big_sepL_persistently `{BiAffine PROP} Φ l :
    <pers> ([∗ list] kx l, Φ k x) ⊣⊢ [∗ list] kx l, <pers> (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

  Lemma big_sepL_forall `{BiAffine PROP} Φ l :
    ( k x, Persistent (Φ k x))
    ([∗ list] kx l, Φ k x) ⊣⊢ ( k x, l !! k = Some x Φ k x).
  Proof.
    intros . apply (anti_symm _).
    { apply forall_introk; apply forall_introx.
      apply impl_intro_l, pure_elim_l⇒ ?; by apply: big_sepL_lookup. }
    revert Φ . induction l as [|x l IH]=> Φ ; [by auto using big_sepL_nil'|].
    rewrite big_sepL_cons. rewrite -persistent_and_sep; apply and_intro.
    - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
    - rewrite -IH. apply forall_introk; by rewrite (forall_elim (S k)).
  Qed.

  Lemma big_sepL_impl Φ Ψ l :
    ([∗ list] kx l, Φ k x) -∗
     ( k x, l !! k = Some x Φ k x -∗ Ψ k x) -∗
    [∗ list] kx l, Ψ k x.
  Proof.
    apply wand_intro_l. revert Φ Ψ. induction l as [|x l IH]=> Φ Ψ /=.
    { by rewrite sep_elim_r. }
    rewrite intuitionistically_sep_dup -assoc [( _ _)%I]comm -!assoc assoc.
    apply sep_mono.
    - rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
      by rewrite intuitionistically_elim wand_elim_l.
    - rewrite comm -(IH (Φ S) (Ψ S)) /=.
      apply sep_mono_l, affinely_mono, persistently_mono.
      apply forall_introk. by rewrite (forall_elim (S k)).
  Qed.

  Lemma big_sepL_delete Φ l i x :
    l !! i = Some x
    ([∗ list] ky l, Φ k y)
    ⊣⊢ Φ i x [∗ list] ky l, if decide (k = i) then emp else Φ k y.
  Proof.
    intros. rewrite -(take_drop_middle l i x) // !big_sepL_app /= Nat.add_0_r.
    rewrite take_length_le; last eauto using lookup_lt_Some, Nat.lt_le_incl.
    rewrite decide_True // left_id.
    rewrite assoc -!(comm _ (Φ _ _)) -assoc. do 2 f_equiv.
    - apply big_sepL_properk y Hk. apply lookup_lt_Some in Hk.
      rewrite take_length in Hk. by rewrite decide_False; last lia.
    - apply big_sepL_properk y _. by rewrite decide_False; last lia.
  Qed.

  Lemma big_sepL_delete' `{!BiAffine PROP} Φ l i x :
    l !! i = Some x
    ([∗ list] ky l, Φ k y) ⊣⊢ Φ i x [∗ list] ky l, k i Φ k y.
  Proof.
    intros. rewrite big_sepL_delete //. (do 2 f_equiv)=> k y.
    rewrite -decide_emp. by repeat case_decide.
  Qed.

  Lemma big_sepL_replicate l P :
    [∗] replicate (length l) P ⊣⊢ [∗ list] y l, P.
  Proof. induction l as [|x l]=> //=; by f_equiv. Qed.

  Global Instance big_sepL_nil_persistent Φ :
    Persistent ([∗ list] kx [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL_persistent Φ l :
    ( k x, Persistent (Φ k x)) Persistent ([∗ list] kx l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
  Global Instance big_sepL_persistent_id Ps :
    TCForall Persistent Ps Persistent ([∗] Ps).
  Proof. induction 1; simpl; apply _. Qed.

  Global Instance big_sepL_nil_affine Φ :
    Affine ([∗ list] kx [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL_affine Φ l :
    ( k x, Affine (Φ k x)) Affine ([∗ list] kx l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
  Global Instance big_sepL_affine_id Ps : TCForall Affine Ps Affine ([∗] Ps).
  Proof. induction 1; simpl; apply _. Qed.
End sep_list.

Section sep_list_more.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat A PROP.

  Lemma big_sepL_zip_with {B C} Φ f (l1 : list B) (l2 : list C) :
    ([∗ list] kx zip_with f l1 l2, Φ k x)
    ⊣⊢ ([∗ list] kx l1, if l2 !! k is Some y then Φ k (f x y) else emp).
  Proof.
    revert Φ l2; induction l1 as [|x l1 IH]=> Φ [|y l2] //=.
    - by rewrite big_sepL_emp left_id.
    - by rewrite IH.
  Qed.
End sep_list_more.

Lemma big_sepL2_alt {A B} (Φ : nat A B PROP) l1 l2 :
  ([∗ list] ky1;y2 l1; l2, Φ k y1 y2)
  ⊣⊢ length l1 = length l2 [∗ list] k y zip l1 l2, Φ k (y.1) (y.2).
Proof.
  apply (anti_symm _).
  - apply and_intro.
    + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=;
        auto using pure_intro, False_elim.
      rewrite IH sep_elim_r. apply pure_mono; auto.
    + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=;
        auto using pure_intro, False_elim.
      by rewrite IH.
  - apply pure_elim_l⇒ /Forall2_same_length Hl. revert Φ.
    induction Hl as [|x1 l1 x2 l2 _ _ IH]=> Φ //=. by rewrite -IH.
Qed.

Big ops over two lists

Section sep_list2.
  Context {A B : Type}.
  Implicit Types Φ Ψ : nat A B PROP.

  Lemma big_sepL2_nil Φ : ([∗ list] ky1;y2 []; [], Φ k y1 y2) ⊣⊢ emp.
  Proof. done. Qed.
  Lemma big_sepL2_nil' `{BiAffine PROP} P Φ : P [∗ list] ky1;y2 [];[], Φ k y1 y2.
  Proof. apply (affine _). Qed.

  Lemma big_sepL2_cons Φ x1 x2 l1 l2 :
    ([∗ list] ky1;y2 x1 :: l1; x2 :: l2, Φ k y1 y2)
    ⊣⊢ Φ 0 x1 x2 [∗ list] ky1;y2 l1;l2, Φ (S k) y1 y2.
  Proof. done. Qed.
  Lemma big_sepL2_cons_inv_l Φ x1 l1 l2 :
    ([∗ list] ky1;y2 x1 :: l1; l2, Φ k y1 y2) -∗
     x2 l2', l2 = x2 :: l2'
              Φ 0 x1 x2 [∗ list] ky1;y2 l1;l2', Φ (S k) y1 y2.
  Proof.
    destruct l2 as [|x2 l2]; simpl; auto using False_elim.
    by rewrite -(exist_intro x2) -(exist_intro l2) pure_True // left_id.
  Qed.
  Lemma big_sepL2_cons_inv_r Φ x2 l1 l2 :
    ([∗ list] ky1;y2 l1; x2 :: l2, Φ k y1 y2) -∗
     x1 l1', l1 = x1 :: l1'
              Φ 0 x1 x2 [∗ list] ky1;y2 l1';l2, Φ (S k) y1 y2.
  Proof.
    destruct l1 as [|x1 l1]; simpl; auto using False_elim.
    by rewrite -(exist_intro x1) -(exist_intro l1) pure_True // left_id.
  Qed.

  Lemma big_sepL2_singleton Φ x1 x2 :
    ([∗ list] ky1;y2 [x1];[x2], Φ k y1 y2) ⊣⊢ Φ 0 x1 x2.
  Proof. by rewrite /= right_id. Qed.

  Lemma big_sepL2_length Φ l1 l2 :
    ([∗ list] ky1;y2 l1; l2, Φ k y1 y2) -∗ length l1 = length l2 .
  Proof. by rewrite big_sepL2_alt and_elim_l. Qed.

  Lemma big_sepL2_app Φ l1 l2 l1' l2' :
    ([∗ list] ky1;y2 l1; l1', Φ k y1 y2) -∗
    ([∗ list] ky1;y2 l2; l2', Φ (length l1 + k) y1 y2) -∗
    ([∗ list] ky1;y2 l1 ++ l2; l1' ++ l2', Φ k y1 y2).
  Proof.
    apply wand_intro_r. revert Φ l1'. induction l1 as [|x1 l1 IH]=> Φ -[|x1' l1'] /=.
    - by rewrite left_id.
    - rewrite left_absorb. apply False_elim.
    - rewrite left_absorb. apply False_elim.
    - by rewrite -assoc IH.
  Qed.
  Lemma big_sepL2_app_inv_l Φ l1' l1'' l2 :
    ([∗ list] ky1;y2 l1' ++ l1''; l2, Φ k y1 y2) -∗
     l2' l2'', l2 = l2' ++ l2''
                ([∗ list] ky1;y2 l1';l2', Φ k y1 y2)
                ([∗ list] ky1;y2 l1'';l2'', Φ (length l1' + k) y1 y2).
  Proof.
    rewrite -(exist_intro (take (length l1') l2))
      -(exist_intro (drop (length l1') l2)) take_drop pure_True // left_id.
    revert Φ l2. induction l1' as [|x1 l1' IH]=> Φ -[|x2 l2] /=;
       [by rewrite left_id|by rewrite left_id|apply False_elim|].
    by rewrite IH -assoc.
  Qed.
  Lemma big_sepL2_app_inv_r Φ l1 l2' l2'' :
    ([∗ list] ky1;y2 l1; l2' ++ l2'', Φ k y1 y2) -∗
     l1' l1'', l1 = l1' ++ l1''
                ([∗ list] ky1;y2 l1';l2', Φ k y1 y2)
                ([∗ list] ky1;y2 l1'';l2'', Φ (length l2' + k) y1 y2).
  Proof.
    rewrite -(exist_intro (take (length l2') l1))
      -(exist_intro (drop (length l2') l1)) take_drop pure_True // left_id.
    revert Φ l1. induction l2' as [|x2 l2' IH]=> Φ -[|x1 l1] /=;
       [by rewrite left_id|by rewrite left_id|apply False_elim|].
    by rewrite IH -assoc.
  Qed.
  Lemma big_sepL2_app_inv Φ l1 l2 l1' l2' :
    length l1 = length l1'
    ([∗ list] ky1;y2 l1 ++ l2; l1' ++ l2', Φ k y1 y2) -∗
    ([∗ list] ky1;y2 l1; l1', Φ k y1 y2)
    ([∗ list] ky1;y2 l2; l2', Φ (length l1 + k)%nat y1 y2).
  Proof.
    revert Φ l1'. induction l1 as [|x1 l1 IH]=> Φ -[|x1' l1'] //= ?; simplify_eq.
    - by rewrite left_id.
    - by rewrite -assoc IH.
  Qed.

  Lemma big_sepL2_mono Φ Ψ l1 l2 :
    ( k y1 y2, l1 !! k = Some y1 l2 !! k = Some y2 Φ k y1 y2 Ψ k y1 y2)
    ([∗ list] k y1;y2 l1;l2, Φ k y1 y2) [∗ list] k y1;y2 l1;l2, Ψ k y1 y2.
  Proof.
    intros H. rewrite !big_sepL2_alt. f_equiv. apply big_sepL_monok [y1 y2].
    rewrite lookup_zip_with⇒ ?; simplify_option_eq; auto.
  Qed.
  Lemma big_sepL2_proper Φ Ψ l1 l2 :
    ( k y1 y2, l1 !! k = Some y1 l2 !! k = Some y2 Φ k y1 y2 ⊣⊢ Ψ k y1 y2)
    ([∗ list] k y1;y2 l1;l2, Φ k y1 y2) ⊣⊢ [∗ list] k y1;y2 l1;l2, Ψ k y1 y2.
  Proof.
    intros; apply (anti_symm _);
      apply big_sepL2_mono; auto using equiv_entails, equiv_entails_sym.
  Qed.

  Global Instance big_sepL2_ne n :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (dist n)))
      ==> (=) ==> (=) ==> (dist n))
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof.
    intros Φ1 Φ2 x1 ? <- x2 ? <-. rewrite !big_sepL2_alt. f_equiv.
    f_equivk [y1 y2]. apply .
  Qed.
  Global Instance big_sepL2_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊢)))
      ==> (=) ==> (=) ==> (⊢))
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_mono; intros; apply Hf. Qed.
  Global Instance big_sepL2_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊣⊢)))
      ==> (=) ==> (=) ==> (⊣⊢))
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_proper; intros; apply Hf. Qed.

  Lemma big_sepL2_lookup_acc Φ l1 l2 i x1 x2 :
    l1 !! i = Some x1 l2 !! i = Some x2
    ([∗ list] ky1;y2 l1;l2, Φ k y1 y2)
    Φ i x1 x2 (Φ i x1 x2 -∗ ([∗ list] ky1;y2 l1;l2, Φ k y1 y2)).
  Proof.
    intros Hl1 Hl2. rewrite big_sepL2_alt. apply pure_elim_lHl.
    rewrite {1}big_sepL_lookup_acc; last by rewrite lookup_zip_with; simplify_option_eq.
    by rewrite pure_True // left_id.
  Qed.

  Lemma big_sepL2_lookup Φ l1 l2 i x1 x2 `{!Absorbing (Φ i x1 x2)} :
    l1 !! i = Some x1 l2 !! i = Some x2
    ([∗ list] ky1;y2 l1;l2, Φ k y1 y2) Φ i x1 x2.
  Proof. intros. rewrite big_sepL2_lookup_acc //. by rewrite sep_elim_l. Qed.

  Lemma big_sepL2_fmap_l {A'} (f : A A') (Φ : nat A' B PROP) l1 l2 :
    ([∗ list] ky1;y2 f <$> l1; l2, Φ k y1 y2)
    ⊣⊢ ([∗ list] ky1;y2 l1;l2, Φ k (f y1) y2).
  Proof.
    rewrite !big_sepL2_alt fmap_length zip_with_fmap_l zip_with_zip big_sepL_fmap.
    by f_equiv; f_equivk [??].
  Qed.
  Lemma big_sepL2_fmap_r {B'} (g : B B') (Φ : nat A B' PROP) l1 l2 :
    ([∗ list] ky1;y2 l1; g <$> l2, Φ k y1 y2)
    ⊣⊢ ([∗ list] ky1;y2 l1;l2, Φ k y1 (g y2)).
  Proof.
    rewrite !big_sepL2_alt fmap_length zip_with_fmap_r zip_with_zip big_sepL_fmap.
    by f_equiv; f_equivk [??].
  Qed.

  Lemma big_sepL2_reverse_2 (Φ : A B PROP) l1 l2 :
    ([∗ list] y1;y2 l1;l2, Φ y1 y2) ([∗ list] y1;y2 reverse l1;reverse l2, Φ y1 y2).
  Proof.
    revert l2. induction l1 as [|x1 l1 IH]; intros [|x2 l2]; simpl; auto using False_elim.
    rewrite !reverse_cons (comm bi_sep) IH.
    by rewrite (big_sepL2_app _ _ [x1] _ [x2]) big_sepL2_singleton wand_elim_l.
  Qed.
  Lemma big_sepL2_reverse (Φ : A B PROP) l1 l2 :
    ([∗ list] y1;y2 reverse l1;reverse l2, Φ y1 y2) ⊣⊢ ([∗ list] y1;y2 l1;l2, Φ y1 y2).
  Proof. apply (anti_symm _); by rewrite big_sepL2_reverse_2 ?reverse_involutive. Qed.

  Lemma big_sepL2_sep Φ Ψ l1 l2 :
    ([∗ list] ky1;y2 l1;l2, Φ k y1 y2 Ψ k y1 y2)
    ⊣⊢ ([∗ list] ky1;y2 l1;l2, Φ k y1 y2) ([∗ list] ky1;y2 l1;l2, Ψ k y1 y2).
  Proof.
    rewrite !big_sepL2_alt big_sepL_sep !persistent_and_affinely_sep_l.
    rewrite -assoc (assoc _ _ (<affine> _)%I). rewrite -(comm bi_sep (<affine> _)%I).
    rewrite -assoc (assoc _ _ (<affine> _)%I) -!persistent_and_affinely_sep_l.
    by rewrite affinely_and_r persistent_and_affinely_sep_l idemp.
  Qed.

  Lemma big_sepL2_and Φ Ψ l1 l2 :
    ([∗ list] ky1;y2 l1;l2, Φ k y1 y2 Ψ k y1 y2)
     ([∗ list] ky1;y2 l1;l2, Φ k y1 y2) ([∗ list] ky1;y2 l1;l2, Ψ k y1 y2).
  Proof. auto using and_intro, big_sepL2_mono, and_elim_l, and_elim_r. Qed.

  Lemma big_sepL2_persistently `{BiAffine PROP} Φ l1 l2 :
    <pers> ([∗ list] ky1;y2 l1;l2, Φ k y1 y2)
    ⊣⊢ [∗ list] ky1;y2 l1;l2, <pers> (Φ k y1 y2).
  Proof.
    by rewrite !big_sepL2_alt persistently_and persistently_pure big_sepL_persistently.
  Qed.

  Lemma big_sepL2_impl Φ Ψ l1 l2 :
    ([∗ list] ky1;y2 l1;l2, Φ k y1 y2) -∗
     ( k x1 x2,
      l1 !! k = Some x1 l2 !! k = Some x2 Φ k x1 x2 -∗ Ψ k x1 x2) -∗
    [∗ list] ky1;y2 l1;l2, Ψ k y1 y2.
  Proof.
    apply wand_intro_l. revert Φ Ψ l2.
    induction l1 as [|x1 l1 IH]=> Φ Ψ [|x2 l2] /=; [by rewrite sep_elim_r..|].
    rewrite intuitionistically_sep_dup -assoc [( _ _)%I]comm -!assoc assoc.
    apply sep_mono.
    - rewrite (forall_elim 0) (forall_elim x1) (forall_elim x2) !pure_True // !True_impl.
      by rewrite intuitionistically_elim wand_elim_l.
    - rewrite comm -(IH (Φ S) (Ψ S)) /=.
      apply sep_mono_l, affinely_mono, persistently_mono.
      apply forall_introk. by rewrite (forall_elim (S k)).
  Qed.

  Global Instance big_sepL2_nil_persistent Φ :
    Persistent ([∗ list] ky1;y2 []; [], Φ k y1 y2).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL2_persistent Φ l1 l2 :
    ( k x1 x2, Persistent (Φ k x1 x2))
    Persistent ([∗ list] ky1;y2 l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.

  Global Instance big_sepL2_nil_affine Φ :
    Affine ([∗ list] ky1;y2 []; [], Φ k y1 y2).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL2_affine Φ l1 l2 :
    ( k x1 x2, Affine (Φ k x1 x2))
    Affine ([∗ list] ky1;y2 l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.
End sep_list2.

Section and_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat A PROP.

  Lemma big_andL_nil Φ : ([∧ list] ky nil, Φ k y) ⊣⊢ True.
  Proof. done. Qed.
  Lemma big_andL_nil' P Φ : P [∧ list] ky nil, Φ k y.
  Proof. by apply pure_intro. Qed.
  Lemma big_andL_cons Φ x l :
    ([∧ list] ky x :: l, Φ k y) ⊣⊢ Φ 0 x [∧ list] ky l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_andL_singleton Φ x : ([∧ list] ky [x], Φ k y) ⊣⊢ Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_andL_app Φ l1 l2 :
    ([∧ list] ky l1 ++ l2, Φ k y)
    ⊣⊢ ([∧ list] ky l1, Φ k y) ([∧ list] ky l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.

  Lemma big_andL_mono Φ Ψ l :
    ( k y, l !! k = Some y Φ k y Ψ k y)
    ([∧ list] k y l, Φ k y) [∧ list] k y l, Ψ k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_andL_proper Φ Ψ l :
    ( k y, l !! k = Some y Φ k y ⊣⊢ Ψ k y)
    ([∧ list] k y l, Φ k y) ⊣⊢ ([∧ list] k y l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
  Lemma big_andL_submseteq (Φ : A PROP) l1 l2 :
    l1 ⊆+ l2 ([∧ list] y l2, Φ y) [∧ list] y l1, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_andL_app and_elim_l.
  Qed.

  Global Instance big_andL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢))
           (big_opL (@bi_and PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_andL_id_mono' :
    Proper (Forall2 (⊢) ==> (⊢)) (big_opL (@bi_and PROP) (λ _ P, P)).
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

  Lemma big_andL_lookup Φ l i x :
    l !! i = Some x ([∧ list] ky l, Φ k y) Φ i x.
  Proof.
    intros. rewrite -(take_drop_middle l i x) // big_andL_app /=.
    rewrite Nat.add_0_r take_length_le;
      eauto using lookup_lt_Some, Nat.lt_le_incl, and_elim_l', and_elim_r'.
  Qed.

  Lemma big_andL_elem_of (Φ : A PROP) l x :
    x l ([∧ list] y l, Φ y) Φ x.
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_andL_lookup (λ _, Φ)).
  Qed.

  Lemma big_andL_fmap {B} (f : A B) (Φ : nat B PROP) l :
    ([∧ list] ky f <$> l, Φ k y) ⊣⊢ ([∧ list] ky l, Φ k (f y)).
  Proof. by rewrite big_opL_fmap. Qed.

  Lemma big_andL_bind {B} (f : A list B) (Φ : B PROP) l :
    ([∧ list] y l ≫= f, Φ y) ⊣⊢ ([∧ list] x l, [∧ list] y f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

  Lemma big_andL_and Φ Ψ l :
    ([∧ list] kx l, Φ k x Ψ k x)
    ⊣⊢ ([∧ list] kx l, Φ k x) ([∧ list] kx l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

  Lemma big_andL_persistently Φ l :
    <pers> ([∧ list] kx l, Φ k x) ⊣⊢ [∧ list] kx l, <pers> (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

  Lemma big_andL_forall Φ l :
    ([∧ list] kx l, Φ k x) ⊣⊢ ( k x, l !! k = Some x Φ k x).
  Proof.
    apply (anti_symm _).
    { apply forall_introk; apply forall_introx.
      apply impl_intro_l, pure_elim_l⇒ ?; by apply: big_andL_lookup. }
    revert Φ. induction l as [|x l IH]=> Φ; [by auto using big_andL_nil'|].
    rewrite big_andL_cons. apply and_intro.
    - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
    - rewrite -IH. apply forall_introk; by rewrite (forall_elim (S k)).
  Qed.

  Global Instance big_andL_nil_persistent Φ :
    Persistent ([∧ list] kx [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_andL_persistent Φ l :
    ( k x, Persistent (Φ k x)) Persistent ([∧ list] kx l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
End and_list.

Section or_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat A PROP.

  Lemma big_orL_nil Φ : ([∨ list] ky nil, Φ k y) ⊣⊢ False.
  Proof. done. Qed.
  Lemma big_orL_cons Φ x l :
    ([∨ list] ky x :: l, Φ k y) ⊣⊢ Φ 0 x [∨ list] ky l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_orL_singleton Φ x : ([∨ list] ky [x], Φ k y) ⊣⊢ Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_orL_app Φ l1 l2 :
    ([∨ list] ky l1 ++ l2, Φ k y)
    ⊣⊢ ([∨ list] ky l1, Φ k y) ([∨ list] ky l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.

  Lemma big_orL_mono Φ Ψ l :
    ( k y, l !! k = Some y Φ k y Ψ k y)
    ([∨ list] k y l, Φ k y) [∨ list] k y l, Ψ k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_orL_proper Φ Ψ l :
    ( k y, l !! k = Some y Φ k y ⊣⊢ Ψ k y)
    ([∨ list] k y l, Φ k y) ⊣⊢ ([∨ list] k y l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
  Lemma big_orL_submseteq (Φ : A PROP) l1 l2 :
    l1 ⊆+ l2 ([∨ list] y l1, Φ y) [∨ list] y l2, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_orL_app -or_intro_l.
  Qed.

  Global Instance big_orL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢))
           (big_opL (@bi_or PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_orL_id_mono' :
    Proper (Forall2 (⊢) ==> (⊢)) (big_opL (@bi_or PROP) (λ _ P, P)).
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

  Lemma big_orL_lookup Φ l i x :
    l !! i = Some x Φ i x ([∨ list] ky l, Φ k y).
  Proof.
    intros. rewrite -(take_drop_middle l i x) // big_orL_app /=.
    rewrite Nat.add_0_r take_length_le;
      eauto using lookup_lt_Some, Nat.lt_le_incl, or_intro_l', or_intro_r'.
  Qed.

  Lemma big_orL_elem_of (Φ : A PROP) l x :
    x l Φ x ([∨ list] y l, Φ y).
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_orL_lookup (λ _, Φ)).
  Qed.

  Lemma big_orL_fmap {B} (f : A B) (Φ : nat B PROP) l :
    ([∨ list] ky f <$> l, Φ k y) ⊣⊢ ([∨ list] ky l, Φ k (f y)).
  Proof. by rewrite big_opL_fmap. Qed.

  Lemma big_orL_bind {B} (f : A list B) (Φ : B PROP) l :
    ([∨ list] y l ≫= f, Φ y) ⊣⊢ ([∨ list] x l, [∨ list] y f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

  Lemma big_orL_or Φ Ψ l :
    ([∨ list] kx l, Φ k x Ψ k x)
    ⊣⊢ ([∨ list] kx l, Φ k x) ([∨ list] kx l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

  Lemma big_orL_persistently Φ l :
    <pers> ([∨ list] kx l, Φ k x) ⊣⊢ [∨ list] kx l, <pers> (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

  Lemma big_orL_exist Φ l :
    ([∨ list] kx l, Φ k x) ⊣⊢ ( k x, l !! k = Some x Φ k x).
  Proof.
    apply (anti_symm _).
    { revert Φ. induction l as [|x l IH]=> Φ.
      { rewrite big_orL_nil. apply False_elim. }
      rewrite big_orL_cons. apply or_elim.
      - by rewrite -(exist_intro 0) -(exist_intro x) pure_True // left_id.
      - rewrite IH. apply exist_elimk. by rewrite -(exist_intro (S k)). }
    apply exist_elimk; apply exist_elimx. apply pure_elim_l⇒ ?.
    by apply: big_orL_lookup.
  Qed.

  Lemma big_orL_sep_l P Φ l :
    P ([∨ list] kx l, Φ k x) ⊣⊢ ([∨ list] kx l, P Φ k x).
  Proof.
    rewrite !big_orL_exist sep_exist_l.
    f_equivk. rewrite sep_exist_l. f_equivx.
    by rewrite !persistent_and_affinely_sep_l !assoc (comm _ P).
 Qed.
  Lemma big_orL_sep_r Q Φ l :
    ([∨ list] kx l, Φ k x) Q ⊣⊢ ([∨ list] kx l, Φ k x Q).
  Proof. setoid_rewrite (comm bi_sep). apply big_orL_sep_l. Qed.

  Global Instance big_orL_nil_persistent Φ :
    Persistent ([∨ list] kx [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_orL_persistent Φ l :
    ( k x, Persistent (Φ k x)) Persistent ([∨ list] kx l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
End or_list.

Big ops over finite maps

Section map.
  Context `{Countable K} {A : Type}.
  Implicit Types m : gmap K A.
  Implicit Types Φ Ψ : K A PROP.

  Lemma big_sepM_mono Φ Ψ m :
    ( k x, m !! k = Some x Φ k x Ψ k x)
    ([∗ map] k x m, Φ k x) [∗ map] k x m, Ψ k x.
  Proof. apply big_opM_forall; apply _ || auto. Qed.
  Lemma big_sepM_proper Φ Ψ m :
    ( k x, m !! k = Some x Φ k x ⊣⊢ Ψ k x)
    ([∗ map] k x m, Φ k x) ⊣⊢ ([∗ map] k x m, Ψ k x).
  Proof. apply big_opM_proper. Qed.
  Lemma big_sepM_subseteq `{BiAffine PROP} Φ m1 m2 :
    m2 m1 ([∗ map] k x m1, Φ k x) [∗ map] k x m2, Φ k x.
  Proof. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed.

  Global Instance big_sepM_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (=) ==> (⊢))
           (big_opM (@bi_sep PROP) (K:=K) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_sepM_mono⇒ ???; apply Hf. Qed.

  Lemma big_sepM_empty Φ : ([∗ map] kx , Φ k x) ⊣⊢ emp.
  Proof. by rewrite big_opM_empty. Qed.
  Lemma big_sepM_empty' `{BiAffine PROP} P Φ : P [∗ map] kx , Φ k x.
  Proof. rewrite big_sepM_empty. apply: affine. Qed.

  Lemma big_sepM_insert Φ m i x :
    m !! i = None
    ([∗ map] ky <[i:=x]> m, Φ k y) ⊣⊢ Φ i x [∗ map] ky m, Φ k y.
  Proof. apply big_opM_insert. Qed.

  Lemma big_sepM_delete Φ m i x :
    m !! i = Some x
    ([∗ map] ky m, Φ k y) ⊣⊢ Φ i x [∗ map] ky delete i m, Φ k y.
  Proof. apply big_opM_delete. Qed.

  Lemma big_sepM_insert_2 Φ m i x :
    TCOr ( x, Affine (Φ i x)) (Absorbing (Φ i x))
    Φ i x -∗ ([∗ map] ky m, Φ k y) -∗ [∗ map] ky <[i:=x]> m, Φ k y.
  Proof.
    intros Ha. apply wand_intro_r. destruct (m !! i) as [y|] eqn:Hi; last first.
    { by rewrite -big_sepM_insert. }
    assert (TCOr (Affine (Φ i y)) (Absorbing (Φ i x))).
    { destruct Ha; try apply _. }
    rewrite big_sepM_delete // assoc.
    rewrite (sep_elim_l (Φ i x)) -big_sepM_insert ?lookup_delete //.
    by rewrite insert_delete.
  Qed.

  Lemma big_sepM_lookup_acc Φ m i x :
    m !! i = Some x
    ([∗ map] ky m, Φ k y) Φ i x (Φ i x -∗ ([∗ map] ky m, Φ k y)).
  Proof.
    intros. rewrite big_sepM_delete //. by apply sep_mono_r, wand_intro_l.
  Qed.

  Lemma big_sepM_lookup Φ m i x `{!Absorbing (Φ i x)} :
    m !! i = Some x ([∗ map] ky m, Φ k y) Φ i x.
  Proof. intros. rewrite big_sepM_lookup_acc //. by rewrite sep_elim_l. Qed.

  Lemma big_sepM_lookup_dom (Φ : K PROP) m i `{!Absorbing (Φ i)} :
    is_Some (m !! i) ([∗ map] k_ m, Φ k) Φ i.
  Proof. intros [x ?]. by eapply (big_sepM_lookup (λ i x, Φ i)). Qed.

  Lemma big_sepM_singleton Φ i x : ([∗ map] ky {[i:=x]}, Φ k y) ⊣⊢ Φ i x.
  Proof. by rewrite big_opM_singleton. Qed.

  Lemma big_sepM_fmap {B} (f : A B) (Φ : K B PROP) m :
    ([∗ map] ky f <$> m, Φ k y) ⊣⊢ ([∗ map] ky m, Φ k (f y)).
  Proof. by rewrite big_opM_fmap. Qed.

  Lemma big_sepM_insert_override Φ m i x x' :
    m !! i = Some x (Φ i x ⊣⊢ Φ i x')
    ([∗ map] ky <[i:=x']> m, Φ k y) ⊣⊢ ([∗ map] ky m, Φ k y).
  Proof. apply big_opM_insert_override. Qed.

  Lemma big_sepM_insert_override_1 Φ m i x x' :
    m !! i = Some x
    ([∗ map] ky <[i:=x']> m, Φ k y)
      (Φ i x' -∗ Φ i x) -∗ ([∗ map] ky m, Φ k y).
  Proof.
    intros ?. apply wand_intro_l.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
    by rewrite assoc wand_elim_l -big_sepM_delete.
  Qed.

  Lemma big_sepM_insert_override_2 Φ m i x x' :
    m !! i = Some x
    ([∗ map] ky m, Φ k y)
      (Φ i x -∗ Φ i x') -∗ ([∗ map] ky <[i:=x']> m, Φ k y).
  Proof.
    intros ?. apply wand_intro_l.
    rewrite {1}big_sepM_delete //; rewrite assoc wand_elim_l.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
  Qed.

  Lemma big_sepM_insert_acc Φ m i x :
    m !! i = Some x
    ([∗ map] ky m, Φ k y)
      Φ i x ( x', Φ i x' -∗ ([∗ map] ky <[i:=x']> m, Φ k y)).
  Proof.
    intros ?. rewrite {1}big_sepM_delete //. apply sep_mono; [done|].
    apply forall_introx'.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
    by apply wand_intro_l.
  Qed.

  Lemma big_sepM_fn_insert {B} (Ψ : K A B PROP) (f : K B) m i x b :
    m !! i = None
       ([∗ map] ky <[i:=x]> m, Ψ k y (<[i:=b]> f k))
    ⊣⊢ (Ψ i x b [∗ map] ky m, Ψ k y (f k)).
  Proof. apply big_opM_fn_insert. Qed.

  Lemma big_sepM_fn_insert' (Φ : K PROP) m i x P :
    m !! i = None
    ([∗ map] ky <[i:=x]> m, <[i:=P]> Φ k) ⊣⊢ (P [∗ map] ky m, Φ k).
  Proof. apply big_opM_fn_insert'. Qed.

  Lemma big_sepM_union Φ m1 m2 :
    m1 ##ₘ m2
    ([∗ map] ky m1 m2, Φ k y)
    ⊣⊢ ([∗ map] ky m1, Φ k y) ([∗ map] ky m2, Φ k y).
  Proof. apply big_opM_union. Qed.

  Lemma big_sepM_sep Φ Ψ m :
    ([∗ map] kx m, Φ k x Ψ k x)
    ⊣⊢ ([∗ map] kx m, Φ k x) ([∗ map] kx m, Ψ k x).
  Proof. apply big_opM_op. Qed.

  Lemma big_sepM_and Φ Ψ m :
    ([∗ map] kx m, Φ k x Ψ k x)
     ([∗ map] kx m, Φ k x) ([∗ map] kx m, Ψ k x).
  Proof. auto using and_intro, big_sepM_mono, and_elim_l, and_elim_r. Qed.

  Lemma big_sepM_persistently `{BiAffine PROP} Φ m :
    (<pers> ([∗ map] kx m, Φ k x)) ⊣⊢ ([∗ map] kx m, <pers> (Φ k x)).
  Proof. apply (big_opM_commute _). Qed.

  Lemma big_sepM_forall `{BiAffine PROP} Φ m :
    ( k x, Persistent (Φ k x))
    ([∗ map] kx m, Φ k x) ⊣⊢ ( k x, m !! k = Some x Φ k x).
  Proof.
    intros. apply (anti_symm _).
    { apply forall_introk; apply forall_introx.
      apply impl_intro_l, pure_elim_l⇒ ?; by apply: big_sepM_lookup. }
    induction m as [|i x m ? IH] using map_ind; auto using big_sepM_empty'.
    rewrite big_sepM_insert // -persistent_and_sep. apply and_intro.
    - rewrite (forall_elim i) (forall_elim x) lookup_insert.
      by rewrite pure_True // True_impl.
    - rewrite -IH. apply forall_monok; apply forall_monoy.
      apply impl_intro_l, pure_elim_l⇒ ?.
      rewrite lookup_insert_ne; last by intros ?; simplify_map_eq.
      by rewrite pure_True // True_impl.
  Qed.

  Lemma big_sepM_impl Φ Ψ m :
    ([∗ map] kx m, Φ k x) -∗
     ( k x, m !! k = Some x Φ k x -∗ Ψ k x) -∗
    [∗ map] kx m, Ψ k x.
  Proof.
    apply wand_intro_l. induction m as [|i x m ? IH] using map_ind.
    { by rewrite sep_elim_r. }
    rewrite !big_sepM_insert // intuitionistically_sep_dup.
    rewrite -assoc [( _ _)%I]comm -!assoc assoc. apply sep_mono.
    - rewrite (forall_elim i) (forall_elim x) pure_True ?lookup_insert //.
      by rewrite True_impl intuitionistically_elim wand_elim_l.
    - rewrite comm -IH /=.
      apply sep_mono_l, affinely_mono, persistently_mono, forall_monok.
      apply forall_monoy. apply impl_intro_l, pure_elim_l⇒ ?.
      rewrite lookup_insert_ne; last by intros ?; simplify_map_eq.
      by rewrite pure_True // True_impl.
  Qed.

  Global Instance big_sepM_empty_persistent Φ :
    Persistent ([∗ map] kx , Φ k x).
  Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
  Global Instance big_sepM_persistent Φ m :
    ( k x, Persistent (Φ k x)) Persistent ([∗ map] kx m, Φ k x).
  Proof. intros. apply big_sepL_persistent_ [??]; apply _. Qed.

  Global Instance big_sepM_empty_affine Φ :
    Affine ([∗ map] kx , Φ k x).
  Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
  Global Instance big_sepM_affine Φ m :
    ( k x, Affine (Φ k x)) Affine ([∗ map] kx m, Φ k x).
  Proof. intros. apply big_sepL_affine_ [??]; apply _. Qed.
End map.

Big ops over two maps

Section map2.
  Context `{Countable K} {A B : Type}.
  Implicit Types Φ Ψ : K A B PROP.

  Lemma big_sepM2_dom Φ m1 m2 :
    ([∗ map] ky1;y2 m1; m2, Φ k y1 y2) -∗ dom (gset K) m1 = dom (gset K) m2 .
  Proof.
    rewrite /big_sepM2 and_elim_l. apply pure_monoHm.
    set_unfoldk. by rewrite !elem_of_dom.
  Qed.

  Lemma big_sepM2_mono Φ Ψ m1 m2 :
    ( k y1 y2, m1 !! k = Some y1 m2 !! k = Some y2 Φ k y1 y2 Ψ k y1 y2)
    ([∗ map] k y1;y2 m1;m2, Φ k y1 y2) [∗ map] k y1;y2 m1;m2, Ψ k y1 y2.
  Proof.
    intros Hm1m2. rewrite /big_sepM2. apply and_mono_r, big_sepM_mono.
    intros k [x1 x2]. rewrite map_lookup_zip_with.
    specialize (Hm1m2 k x1 x2).
    destruct (m1 !! k) as [y1|]; last done.
    destruct (m2 !! k) as [y2|]; simpl; last done.
    intros ?; simplify_eq. by apply Hm1m2.
  Qed.
  Lemma big_sepM2_proper Φ Ψ m1 m2 :
    ( k y1 y2, m1 !! k = Some y1 m2 !! k = Some y2 Φ k y1 y2 ⊣⊢ Ψ k y1 y2)
    ([∗ map] k y1;y2 m1;m2, Φ k y1 y2) ⊣⊢ [∗ map] k y1;y2 m1;m2, Ψ k y1 y2.
  Proof.
    intros; apply (anti_symm _);
      apply big_sepM2_mono; auto using equiv_entails, equiv_entails_sym.
  Qed.

  Global Instance big_sepM2_ne n :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (dist n)))
      ==> (=) ==> (=) ==> (dist n))
           (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)).
  Proof.
    intros Φ1 Φ2 x1 ? <- x2 ? <-. rewrite /big_sepM2. f_equiv.
    f_equivk [y1 y2]. apply .
  Qed.
  Global Instance big_sepM2_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊢)))
      ==> (=) ==> (=) ==> (⊢))
           (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)).
  Proof. intros f g Hf m1 ? <- m2 ? <-. apply big_sepM2_mono; intros; apply Hf. Qed.
  Global Instance big_sepM2_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (⊣⊢)))
      ==> (=) ==> (=) ==> (⊣⊢))
           (big_sepM2 (PROP:=PROP) (K:=K) (A:=A) (B:=B)).
  Proof. intros f g Hf m1 ? <- m2 ? <-. apply big_sepM2_proper; intros; apply Hf. Qed.

  Lemma big_sepM2_empty Φ : ([∗ map] ky1;y2 ; , Φ k y1 y2) ⊣⊢ emp.
  Proof.
    rewrite /big_sepM2 pure_True ?left_id //.
    intros k. rewrite !lookup_empty; split; by inversion 1.
  Qed.
  Lemma big_sepM2_empty' `{BiAffine PROP} P Φ : P [∗ map] ky1;y2 ;, Φ k y1 y2.
  Proof. rewrite big_sepM2_empty. apply (affine _). Qed.

  Lemma big_sepM2_empty_l m1 Φ :
    ([∗ map] ky1;y2 m1; , Φ k y1 y2) m1 = .
  Proof.
    rewrite big_sepM2_dom dom_empty_L.
    apply pure_mono, dom_empty_inv_L.
  Qed.

  Lemma big_sepM2_empty_r m2 Φ :
    ([∗ map] ky1;y2 ; m2, Φ k y1 y2) m2 = .
  Proof.
    rewrite big_sepM2_dom dom_empty_L.
    apply pure_mono=>?. eapply (dom_empty_inv_L (D:=gset K)). eauto.
  Qed.

  Lemma big_sepM2_insert Φ m1 m2 i x1 x2 :
    m1 !! i = None m2 !! i = None
    ([∗ map] ky1;y2 <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2)
    ⊣⊢ Φ i x1 x2 [∗ map] ky1;y2 m1;m2, Φ k y1 y2.
  Proof.
    intros Hm1 Hm2. rewrite /big_sepM2 -map_insert_zip_with.
    rewrite big_sepM_insert;
      last by rewrite map_lookup_zip_with Hm1.
    rewrite !persistent_and_affinely_sep_l /=.
    rewrite sep_assoc (sep_comm _ (Φ _ _ _)) -sep_assoc.
    repeat apply sep_proper=>//.
    apply affinely_proper, pure_proper.
    split.
    - intros H1 k. destruct (decide (i = k)) as [->|?].
      + rewrite Hm1 Hm2 //. by split; intros ?; exfalso; eapply is_Some_None.
      + specialize (H1 k). revert H1. rewrite !lookup_insert_ne //.
    - intros H1 k. destruct (decide (i = k)) as [->|?].
      + rewrite !lookup_insert. split; by econstructor.
      + rewrite !lookup_insert_ne //.
  Qed.

  Lemma big_sepM2_delete Φ m1 m2 i x1 x2 :
    m1 !! i = Some x1 m2 !! i = Some x2
    ([∗ map] kx;y m1;m2, Φ k x y) ⊣⊢
      Φ i x1 x2 [∗ map] kx;y delete i m1;delete i m2, Φ k x y.
  Proof.
    rewrite /big_sepM2Hx1 Hx2.
    rewrite !persistent_and_affinely_sep_l /=.
    rewrite sep_assoc (sep_comm (Φ _ _ _)) -sep_assoc.
    apply sep_proper.
    - apply affinely_proper, pure_proper. split.
      + intros Hm k. destruct (decide (i = k)) as [->|?].
        { rewrite !lookup_delete. split; intros []%is_Some_None. }
        rewrite !lookup_delete_ne //.
      + intros Hm k. specialize (Hm k). revert Hm. destruct (decide (i = k)) as [->|?].
        { intros _. rewrite Hx1 Hx2. split; eauto. }
        rewrite !lookup_delete_ne //.
    - rewrite -map_delete_zip_with.
      apply (big_sepM_delete (λ i xx, Φ i xx.1 xx.2) (map_zip m1 m2) i (x1,x2)).
      by rewrite map_lookup_zip_with Hx1 Hx2.
  Qed.

  Lemma big_sepM2_insert_acc Φ m1 m2 i x1 x2 :
    m1 !! i = Some x1 m2 !! i = Some x2
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)
    Φ i x1 x2 ( x1' x2', Φ i x1' x2' -∗
        ([∗ map] ky1;y2 <[i:=x1']>m1;<[i:=x2']>m2, Φ k y1 y2)).
  Proof.
    intros ??. rewrite {1}big_sepM2_delete //. apply sep_mono; [done|].
    apply forall_introx1'. apply forall_introx2'.
    rewrite -(insert_delete m1) -(insert_delete m2) big_sepM2_insert ?lookup_delete //.
    by apply wand_intro_l.
  Qed.

  Lemma big_sepM2_insert_2 Φ m1 m2 i x1 x2 :
    TCOr ( x y, Affine (Φ i x y)) (Absorbing (Φ i x1 x2))
    Φ i x1 x2 -∗
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2) -∗
    ([∗ map] ky1;y2 <[i:=x1]>m1; <[i:=x2]>m2, Φ k y1 y2).
  Proof.
    intros Ha. rewrite /big_sepM2.
    assert (TCOr ( x, Affine (Φ i x.1 x.2)) (Absorbing (Φ i x1 x2))).
    { destruct Ha; try apply _. }
    apply wand_intro_r.
    rewrite !persistent_and_affinely_sep_l /=.
    rewrite (sep_comm (Φ _ _ _)) -sep_assoc. apply sep_mono.
    { apply affinely_mono, pure_mono. intros Hm k.
      destruct (decide (i = k)) as [->|].
      - rewrite !lookup_insert. split; eauto.
      - rewrite !lookup_insert_ne //. }
    rewrite (big_sepM_insert_2 (λ k xx, Φ k xx.1 xx.2) (map_zip m1 m2) i (x1, x2)).
    rewrite map_insert_zip_with. apply wand_elim_r.
  Qed.

  Lemma big_sepM2_lookup_acc Φ m1 m2 i x1 x2 :
    m1 !! i = Some x1 m2 !! i = Some x2
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)
    Φ i x1 x2 (Φ i x1 x2 -∗ ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)).
  Proof.
    intros Hm1 Hm2. etrans; first apply big_sepM2_insert_acc=>//.
    apply sep_mono_r. rewrite (forall_elim x1) (forall_elim x2).
    rewrite !insert_id //.
 Qed.

  Lemma big_sepM2_lookup Φ m1 m2 i x1 x2 `{!Absorbing (Φ i x1 x2)} :
    m1 !! i = Some x1 m2 !! i = Some x2
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2) Φ i x1 x2.
  Proof. intros. rewrite big_sepM2_lookup_acc //. by rewrite sep_elim_l. Qed.

  Lemma big_sepM2_lookup_1 Φ m1 m2 i x1 `{! x2, Absorbing (Φ i x1 x2)} :
    m1 !! i = Some x1
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)
     x2, m2 !! i = Some x2 Φ i x1 x2.
  Proof.
    intros Hm1. rewrite /big_sepM2.
    rewrite persistent_and_sep_1.
    apply wand_elim_l'. apply pure_elim'Hm.
    assert (is_Some (m2 !! i)) as [x2 Hm2].
    { apply Hm. by econstructor. }
    apply wand_intro_r. rewrite -(exist_intro x2).
    rewrite /big_sepM2 (big_sepM_lookup _ _ i (x1,x2));
      last by rewrite map_lookup_zip_with Hm1 Hm2.
    rewrite pure_True// sep_elim_r.
    apply and_intro=>//. by apply pure_intro.
  Qed.

  Lemma big_sepM2_lookup_2 Φ m1 m2 i x2 `{! x1, Absorbing (Φ i x1 x2)} :
    m2 !! i = Some x2
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)
     x1, m1 !! i = Some x1 Φ i x1 x2.
  Proof.
    intros Hm2. rewrite /big_sepM2.
    rewrite persistent_and_sep_1.
    apply wand_elim_l'. apply pure_elim'Hm.
    assert (is_Some (m1 !! i)) as [x1 Hm1].
    { apply Hm. by econstructor. }
    apply wand_intro_r. rewrite -(exist_intro x1).
    rewrite /big_sepM2 (big_sepM_lookup _ _ i (x1,x2));
      last by rewrite map_lookup_zip_with Hm1 Hm2.
    rewrite pure_True// sep_elim_r.
    apply and_intro=>//. by apply pure_intro.
  Qed.

  Lemma big_sepM2_singleton Φ i x1 x2 :
    ([∗ map] ky1;y2 {[ i := x1 ]}; {[ i := x2 ]}, Φ k y1 y2) ⊣⊢ Φ i x1 x2.
  Proof.
    rewrite /big_sepM2 map_zip_with_singleton big_sepM_singleton.
    apply (anti_symm _).
    - apply and_elim_r.
    - rewrite <- (left_id True%I (∧)%I (Φ i x1 x2)).
      apply and_mono=>//. apply pure_mono_ k.
      rewrite !lookup_insert_is_Some' !lookup_empty -!not_eq_None_Some.
      naive_solver.
  Qed.

  Lemma big_sepM2_fmap {A' B'} (f : A A') (g : B B') (Φ : K A' B' PROP) m1 m2 :
    ([∗ map] ky1;y2 f <$> m1; g <$> m2, Φ k y1 y2)
    ⊣⊢ ([∗ map] ky1;y2 m1;m2, Φ k (f y1) (g y2)).
  Proof.
    rewrite /big_sepM2. rewrite map_fmap_zip.
    apply and_proper.
    - apply pure_proper. split.
      + intros Hm k. specialize (Hm k). revert Hm.
        by rewrite !lookup_fmap !fmap_is_Some.
      + intros Hm k. specialize (Hm k). revert Hm.
        by rewrite !lookup_fmap !fmap_is_Some.
    - by rewrite big_sepM_fmap.
  Qed.

  Lemma big_sepM2_fmap_l {A'} (f : A A') (Φ : K A' B PROP) m1 m2 :
    ([∗ map] ky1;y2 f <$> m1; m2, Φ k y1 y2)
    ⊣⊢ ([∗ map] ky1;y2 m1;m2, Φ k (f y1) y2).
  Proof. rewrite -{1}(map_fmap_id m2). apply big_sepM2_fmap. Qed.

  Lemma big_sepM2_fmap_r {B'} (g : B B') (Φ : K A B' PROP) m1 m2 :
    ([∗ map] ky1;y2 m1; g <$> m2, Φ k y1 y2)
    ⊣⊢ ([∗ map] ky1;y2 m1;m2, Φ k y1 (g y2)).
  Proof. rewrite -{1}(map_fmap_id m1). apply big_sepM2_fmap. Qed.

  Lemma big_sepM2_sep Φ Ψ m1 m2 :
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2 Ψ k y1 y2)
    ⊣⊢ ([∗ map] ky1;y2 m1;m2, Φ k y1 y2) ([∗ map] ky1;y2 m1;m2, Ψ k y1 y2).
  Proof.
    rewrite /big_sepM2.
    rewrite -{1}(and_idem k : K, is_Some (m1 !! k) is_Some (m2 !! k)%I).
    rewrite -and_assoc.
    rewrite !persistent_and_affinely_sep_l /=.
    rewrite -sep_assoc. apply sep_proper=>//.
    rewrite sep_assoc (sep_comm _ (<affine> _)%I) -sep_assoc.
    apply sep_proper=>//. apply big_sepM_sep.
  Qed.

  Lemma big_sepM2_and Φ Ψ m1 m2 :
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2 Ψ k y1 y2)
     ([∗ map] ky1;y2 m1;m2, Φ k y1 y2) ([∗ map] ky1;y2 m1;m2, Ψ k y1 y2).
  Proof. auto using and_intro, big_sepM2_mono, and_elim_l, and_elim_r. Qed.

  Lemma big_sepM2_persistently `{BiAffine PROP} Φ m1 m2 :
    <pers> ([∗ map] ky1;y2 m1;m2, Φ k y1 y2)
    ⊣⊢ [∗ map] ky1;y2 m1;m2, <pers> (Φ k y1 y2).
  Proof.
    by rewrite /big_sepM2 persistently_and
         persistently_pure big_sepM_persistently.
  Qed.

 Lemma big_sepM2_forall `{BiAffine PROP} Φ m1 m2 :
   ( k x1 x2, Persistent (Φ k x1 x2))
   ([∗ map] kx1;x2 m1;m2, Φ k x1 x2) ⊣⊢
      k : K, is_Some (m1 !! k) is_Some (m2 !! k)
      ( k x1 x2, m1 !! k = Some x1 m2 !! k = Some x2 Φ k x1 x2).
 Proof.
   rewrite /big_sepM2⇒ ?. apply and_proper=>//.
   rewrite big_sepM_forall. apply forall_properk.
   apply (anti_symm _).
   - apply forall_introx1. apply forall_introx2.
     rewrite (forall_elim (x1,x2)).
     rewrite impl_curry -pure_and. apply impl_mono=>//.
     apply pure_mono. rewrite map_lookup_zip_with. by intros [-> ->].
   - apply forall_intro⇒ [[x1 x2]].
     rewrite (forall_elim x1) (forall_elim x2).
     rewrite impl_curry -pure_and. apply impl_mono=>//.
     apply pure_mono. rewrite map_lookup_zip_with.
     by destruct (m1 !! k), (m2 !! k)=>/= // ?; simplify_eq.
 Qed.

 Lemma big_sepM2_impl Φ Ψ m1 m2 :
    ([∗ map] ky1;y2 m1;m2, Φ k y1 y2) -∗
     ( k x1 x2,
      m1 !! k = Some x1 m2 !! k = Some x2 Φ k x1 x2 -∗ Ψ k x1 x2) -∗
    [∗ map] ky1;y2 m1;m2, Ψ k y1 y2.
  Proof.
    apply wand_intro_l.
    rewrite /big_sepM2.
    rewrite !persistent_and_affinely_sep_l /=.
    rewrite sep_assoc. rewrite (sep_comm ( _)%I) -sep_assoc.
    apply sep_mono_r. apply wand_elim_r'.
    rewrite (big_sepM_impl _ (λ k xx, Ψ k xx.1 xx.2)). apply wand_mono=>//.
    apply intuitionistically_mono'.
    apply forall_monok. apply forall_intro⇒ [[x y]].
    rewrite (forall_elim x) (forall_elim y).
    rewrite impl_curry -pure_and. apply impl_mono=>//.
    apply pure_mono. rewrite map_lookup_zip_with.
    destruct (m1 !! k) as [x1|], (m2 !! k) as [x2|]; naive_solver.
  Qed.

  Global Instance big_sepM2_empty_persistent Φ :
    Persistent ([∗ map] ky1;y2 ; , Φ k y1 y2).
  Proof. rewrite big_sepM2_empty. apply _. Qed.
  Global Instance big_sepM2_persistent Φ m1 m2 :
    ( k x1 x2, Persistent (Φ k x1 x2))
    Persistent ([∗ map] ky1;y2 m1;m2, Φ k y1 y2).
  Proof. rewrite /big_sepM2. apply _. Qed.

  Global Instance big_sepM2_empty_affine Φ :
    Affine ([∗ map] ky1;y2 ; , Φ k y1 y2).
  Proof. rewrite big_sepM2_empty. apply _. Qed.
  Global Instance big_sepM2_affine Φ m1 m2 :
    ( k x1 x2, Affine (Φ k x1 x2))
    Affine ([∗ map] ky1;y2 m1;m2, Φ k y1 y2).
  Proof. rewrite /big_sepM2. apply _. Qed.
End map2.

Big ops over finite sets

Section gset.
  Context `{Countable A}.
  Implicit Types X : gset A.
  Implicit Types Φ : A PROP.

  Lemma big_sepS_mono Φ Ψ X :
    ( x, x X Φ x Ψ x)
    ([∗ set] x X, Φ x) [∗ set] x X, Ψ x.
  Proof. intros. apply big_opS_forall; apply _ || auto. Qed.
  Lemma big_sepS_proper Φ Ψ X :
    ( x, x X Φ x ⊣⊢ Ψ x)
    ([∗ set] x X, Φ x) ⊣⊢ ([∗ set] x X, Ψ x).
  Proof. apply big_opS_proper. Qed.
  Lemma big_sepS_subseteq `{BiAffine PROP} Φ X Y :
    Y X ([∗ set] x X, Φ x) [∗ set] x Y, Φ x.
  Proof. intros. by apply big_sepL_submseteq, elements_submseteq. Qed.

  Global Instance big_sepS_mono' :
     Proper (pointwise_relation _ (⊢) ==> (=) ==> (⊢)) (big_opS (@bi_sep PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. by apply big_sepS_mono. Qed.

  Lemma big_sepS_empty Φ : ([∗ set] x , Φ x) ⊣⊢ emp.
  Proof. by rewrite big_opS_empty. Qed.
  Lemma big_sepS_empty' `{!BiAffine PROP} P Φ : P [∗ set] x , Φ x.
  Proof. rewrite big_sepS_empty. apply: affine. Qed.

  Lemma big_sepS_insert Φ X x :
    x X ([∗ set] y {[ x ]} X, Φ y) ⊣⊢ (Φ x [∗ set] y X, Φ y).
  Proof. apply big_opS_insert. Qed.

  Lemma big_sepS_fn_insert {B} (Ψ : A B PROP) f X x b :
    x X
       ([∗ set] y {[ x ]} X, Ψ y (<[x:=b]> f y))
    ⊣⊢ (Ψ x b [∗ set] y X, Ψ y (f y)).
  Proof. apply big_opS_fn_insert. Qed.

  Lemma big_sepS_fn_insert' Φ X x P :
    x X ([∗ set] y {[ x ]} X, <[x:=P]> Φ y) ⊣⊢ (P [∗ set] y X, Φ y).
  Proof. apply big_opS_fn_insert'. Qed.

  Lemma big_sepS_union Φ X Y :
    X ## Y
    ([∗ set] y X Y, Φ y) ⊣⊢ ([∗ set] y X, Φ y) ([∗ set] y Y, Φ y).
  Proof. apply big_opS_union. Qed.

  Lemma big_sepS_delete Φ X x :
    x X ([∗ set] y X, Φ y) ⊣⊢ Φ x [∗ set] y X {[ x ]}, Φ y.
  Proof. apply big_opS_delete. Qed.

  Lemma big_sepS_elem_of Φ X x `{!Absorbing (Φ x)} :
    x X ([∗ set] y X, Φ y) Φ x.
  Proof. intros. rewrite big_sepS_delete //. by rewrite sep_elim_l. Qed.

  Lemma big_sepS_elem_of_acc Φ X x :
    x X
    ([∗ set] y X, Φ y) Φ x (Φ x -∗ ([∗ set] y X, Φ y)).
  Proof.
    intros. rewrite big_sepS_delete //. by apply sep_mono_r, wand_intro_l.
  Qed.

  Lemma big_sepS_singleton Φ x : ([∗ set] y {[ x ]}, Φ y) ⊣⊢ Φ x.
  Proof. apply big_opS_singleton. Qed.

  Lemma big_sepS_filter' (P : A Prop) `{ x, Decision (P x)} Φ X :
    ([∗ set] y filter P X, Φ y)
    ⊣⊢ ([∗ set] y X, if decide (