big_op.v

From stdpp Require Import countable fin_sets functions.
From iris.algebra Require Export big_op.
From iris.algebra Require Import list gmap.
From iris.bi Require Import derived_laws_later.
From iris.prelude Require Import options.
Import interface.bi derived_laws.bi derived_laws_later.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 :: l2 => Φ 0 x1 x2 ∗ big_sepL2 (λ n, Φ (S n)) l1 l2
  | _, _ => False
  end%I.
Global Instance: Params (@big_sepL2) 3 := {}.
Global 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_def {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.
Definition big_sepM2_aux : seal (@big_sepM2_def). Proof. by eexists. Qed.
Definition big_sepM2 := big_sepM2_aux.(unseal).
Global Arguments big_sepM2 {PROP K _ _ A B} _ _ _.
Definition big_sepM2_eq : @big_sepM2 = _ := big_sepM2_aux.(seal_eq).
Global Instance: Params (@big_sepM2) 6 := {}.
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 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] k↦y ∈ nil, Φ k y) ⊣⊢ emp.
  Proof. done. Qed.
  Lemma big_sepL_nil' P `{!Affine P} Φ : P ⊢ [∗ list] k↦y ∈ nil, Φ k y.
  Proof. apply: affine. Qed.
  Lemma big_sepL_cons Φ x l :
    ([∗ list] k↦y ∈ x :: l, Φ k y) ⊣⊢ Φ 0 x ∗ [∗ list] k↦y ∈ l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_sepL_singleton Φ x : ([∗ list] k↦y ∈ [x], Φ k y) ⊣⊢ Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_sepL_app Φ l1 l2 :
    ([∗ list] k↦y ∈ l1 ++ l2, Φ k y)
    ⊣⊢ ([∗ list] k↦y ∈ l1, Φ k y) ∗ ([∗ list] k↦y ∈ l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.
  Lemma big_sepL_snoc Φ l x :
    ([∗ list] k↦y ∈ l ++ [x], Φ k y) ⊣⊢ ([∗ list] k↦y ∈ l, Φ k y) ∗ Φ (length l) x.
  Proof. by rewrite big_opL_snoc. 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.

  (** The lemmas [big_sepL_mono], [big_sepL_ne] and [big_sepL_proper] are more
  generic than the instances as they also give [l !! k = Some y] in the premise. *)
  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_gen_proper; apply _. Qed.
  Lemma big_sepL_ne Φ Ψ l n :
    (∀ k y, l !! k = Some y → Φ k y ≡{n}≡ Ψ k y) →
    ([∗ list] k ↦ y ∈ l, Φ k y)%I ≡{n}≡ ([∗ list] k ↦ y ∈ l, Ψ k y)%I.
  Proof. apply big_opL_ne. 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.

  (** No need to declare instances for non-expansiveness and properness, we
  get both from the generic [big_opL] instances. *)
  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_sepL_mono; 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] k↦y ∈ l, emp) ⊣⊢@{PROP} emp.
  Proof. by rewrite big_opL_unit. Qed.

  Lemma big_sepL_insert_acc Φ l i x :
    l !! i = Some x →
    ([∗ list] k↦y ∈ l, Φ k y) ⊢ Φ i x ∗ (∀ y, Φ i y -∗ ([∗ list] k↦y ∈ <[i:=y]>l, Φ k y)).
  Proof.
    intros Hli. assert (i ≤ length l) by eauto using lookup_lt_Some, Nat.lt_le_incl.
    rewrite -(take_drop_middle l i x) // big_sepL_app /=.
    rewrite Nat.add_0_r take_length_le //.
    rewrite assoc -!(comm _ (Φ _ _)) -assoc. apply sep_mono_r, forall_intro=> y.
    rewrite insert_app_r_alt ?take_length_le //.
    rewrite Nat.sub_diag /=. apply wand_intro_l.
    rewrite assoc !(comm _ (Φ _ _)) -assoc big_sepL_app /=.
    by rewrite Nat.add_0_r take_length_le.
  Qed.

  Lemma big_sepL_lookup_acc Φ l i x :
    l !! i = Some x →
    ([∗ list] k↦y ∈ l, Φ k y) ⊢ Φ i x ∗ (Φ i x -∗ ([∗ list] k↦y ∈ l, Φ k y)).
  Proof. intros. by rewrite {1}big_sepL_insert_acc // (forall_elim x) list_insert_id. Qed.

  Lemma big_sepL_lookup Φ l i x `{!Absorbing (Φ i x)} :
    l !! i = Some x → ([∗ list] k↦y ∈ 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. by eapply (big_sepL_lookup (λ _, Φ)).
  Qed.

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

  Lemma big_sepL_omap {B} (f : A → option B) (Φ : B → PROP) l :
    ([∗ list] y ∈ omap f l, Φ y) ⊣⊢ ([∗ list] y ∈ l, from_option Φ emp (f y)).
  Proof. by rewrite big_opL_omap. 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] k↦x ∈ l, Φ k x ∗ Ψ k x)
    ⊣⊢ ([∗ list] k↦x ∈ l, Φ k x) ∗ ([∗ list] k↦x ∈ l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

  Lemma big_sepL_and Φ Ψ l :
    ([∗ list] k↦x ∈ l, Φ k x ∧ Ψ k x)
    ⊢ ([∗ list] k↦x ∈ l, Φ k x) ∧ ([∗ list] k↦x ∈ 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] k↦x ∈ l, Φ k x) ⊣⊢ [∗ list] k↦x ∈ l, <pers> (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

  Lemma big_sepL_intuitionistically_forall Φ l :
    □ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x) ⊢ [∗ list] k↦x ∈ l, Φ k x.
  Proof.
    revert Φ. induction l as [|x l IH]=> Φ /=; [by apply (affine _)|].
    rewrite intuitionistically_sep_dup. f_equiv.
    - rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
      by rewrite intuitionistically_elim.
    - rewrite -IH. f_equiv.
      apply forall_intro=> k; by rewrite (forall_elim (S k)).
  Qed.

  Lemma big_sepL_forall `{BiAffine PROP} Φ l :
    (∀ k x, Persistent (Φ k x)) →
    ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x).
  Proof.
    intros HΦ. apply (anti_symm _).
    { apply forall_intro=> k; apply forall_intro=> x.
      apply impl_intro_l, pure_elim_l=> ?; by apply: big_sepL_lookup. }
    rewrite -big_sepL_intuitionistically_forall. setoid_rewrite pure_impl_forall.
    by rewrite intuitionistic_intuitionistically.
  Qed.

  Lemma big_sepL_impl Φ Ψ l :
    ([∗ list] k↦x ∈ l, Φ k x) -∗
    □ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x -∗ Ψ k x) -∗
    [∗ list] k↦x ∈ l, Ψ k x.
  Proof.
    apply wand_intro_l. rewrite big_sepL_intuitionistically_forall -big_sepL_sep.
    by setoid_rewrite wand_elim_l.
  Qed.

  Lemma big_sepL_dup P `{!Affine P} l :
    □ (P -∗ P ∗ P) -∗ P -∗ [∗ list] k↦x ∈ l, P.
  Proof.
    apply wand_intro_l.
    induction l as [|x l IH]=> /=; first by apply: affine.
    rewrite intuitionistically_sep_dup {1}intuitionistically_elim.
    rewrite assoc wand_elim_r -assoc. apply sep_mono; done.
  Qed.

  Lemma big_sepL_delete Φ l i x :
    l !! i = Some x →
    ([∗ list] k↦y ∈ l, Φ k y)
    ⊣⊢ Φ i x ∗ [∗ list] k↦y ∈ 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_proper=> k y Hk. apply lookup_lt_Some in Hk.
      rewrite take_length in Hk. by rewrite decide_False; last lia.
    - apply big_sepL_proper=> k y _. by rewrite decide_False; last lia.
  Qed.

  Lemma big_sepL_delete' `{!BiAffine PROP} Φ l i x :
    l !! i = Some x →
    ([∗ list] k↦y ∈ l, Φ k y) ⊣⊢ Φ i x ∗ [∗ list] k↦y ∈ 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.

  Lemma big_sepL_later `{BiAffine PROP} Φ l :
    ▷ ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷ Φ k x).
  Proof. apply (big_opL_commute _). Qed.
  Lemma big_sepL_later_2 Φ l :
    ([∗ list] k↦x ∈ l, ▷ Φ k x) ⊢ ▷ [∗ list] k↦x ∈ l, Φ k x.
  Proof. by rewrite (big_opL_commute _). Qed.

  Lemma big_sepL_laterN `{BiAffine PROP} Φ n l :
    ▷^n ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷^n Φ k x).
  Proof. apply (big_opL_commute _). Qed.
  Lemma big_sepL_laterN_2 Φ n l :
    ([∗ list] k↦x ∈ l, ▷^n Φ k x) ⊢ ▷^n [∗ list] k↦x ∈ l, Φ k x.
  Proof. by rewrite (big_opL_commute _). Qed.

  Global Instance big_sepL_nil_persistent Φ :
    Persistent ([∗ list] k↦x ∈ [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL_persistent Φ l :
    (∀ k x, Persistent (Φ k x)) → Persistent ([∗ list] k↦x ∈ 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] k↦x ∈ [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL_affine Φ l :
    (∀ k x, Affine (Φ k x)) → Affine ([∗ list] k↦x ∈ 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.

  Global Instance big_sepL_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
    Timeless ([∗ list] k↦x ∈ [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL_timeless `{!Timeless (emp%I : PROP)} Φ l :
    (∀ k x, Timeless (Φ k x)) → Timeless ([∗ list] k↦x ∈ l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
  Global Instance big_sepL_timeless_id `{!Timeless (emp%I : PROP)} Ps :
    TCForall Timeless Ps → Timeless ([∗] Ps).
  Proof. induction 1; simpl; apply _. Qed.
End sep_list.

(* Some lemmas depend on the generalized versions of the above ones. *)

Lemma big_sepL_sep_zip {A B} (Φ : nat → A → PROP) (Ψ : nat → B → PROP) l1 l2 :
  length l1 = length l2 →
  ([∗ list] k↦x ∈ l1, Φ k x) ∗ ([∗ list] k↦x ∈ l2, Ψ k x) ⊣⊢
  ([∗ list] k↦xy ∈ zip l1 l2, Φ k xy.1 ∗ Ψ k xy.2).
Proof.
  intros Hlen. rewrite big_sepL_sep.
  rewrite -(big_sepL_fmap fst) -(big_sepL_fmap snd).
  rewrite fst_zip; last lia. by rewrite snd_zip; last lia.
Qed.

Lemma big_sepL_zip_with {A B C} (Φ : nat → A → PROP) f (l1 : list B) (l2 : list C) :
  ([∗ list] k↦x ∈ zip_with f l1 l2, Φ k x)
    ⊣⊢ ([∗ list] k↦x ∈ 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.

Lemma big_sepL2_alt {A B} (Φ : nat → A → B → PROP) l1 l2 :
  ([∗ list] k↦y1;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] k↦y1;y2 ∈ []; [], Φ k y1 y2) ⊣⊢ emp.
  Proof. done. Qed.
  Lemma big_sepL2_nil' P `{!Affine P} Φ : P ⊢ [∗ list] k↦y1;y2 ∈ [];[], Φ k y1 y2.
  Proof. apply: affine. Qed.
  Lemma big_sepL2_nil_inv_l Φ l2 :
    ([∗ list] k↦y1;y2 ∈ []; l2, Φ k y1 y2) -∗ ⌜l2 = []⌝.
  Proof. destruct l2; simpl; auto using False_elim, pure_intro. Qed.
  Lemma big_sepL2_nil_inv_r Φ l1 :
    ([∗ list] k↦y1;y2 ∈ l1; [], Φ k y1 y2) -∗ ⌜l1 = []⌝.
  Proof. destruct l1; simpl; auto using False_elim, pure_intro. Qed.

  Lemma big_sepL2_cons Φ x1 x2 l1 l2 :
    ([∗ list] k↦y1;y2 ∈ x1 :: l1; x2 :: l2, Φ k y1 y2)
    ⊣⊢ Φ 0 x1 x2 ∗ [∗ list] k↦y1;y2 ∈ l1;l2, Φ (S k) y1 y2.
  Proof. done. Qed.
  Lemma big_sepL2_cons_inv_l Φ x1 l1 l2 :
    ([∗ list] k↦y1;y2 ∈ x1 :: l1; l2, Φ k y1 y2) -∗
    ∃ x2 l2', ⌜ l2 = x2 :: l2' ⌝ ∧
              Φ 0 x1 x2 ∗ [∗ list] k↦y1;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] k↦y1;y2 ∈ l1; x2 :: l2, Φ k y1 y2) -∗
    ∃ x1 l1', ⌜ l1 = x1 :: l1' ⌝ ∧
              Φ 0 x1 x2 ∗ [∗ list] k↦y1;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] k↦y1;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] k↦y1;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] k↦y1;y2 ∈ l1; l1', Φ k y1 y2) -∗
    ([∗ list] k↦y1;y2 ∈ l2; l2', Φ (length l1 + k) y1 y2) -∗
    ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1' ++ l1''; l2, Φ k y1 y2) -∗
    ∃ l2' l2'', ⌜ l2 = l2' ++ l2'' ⌝ ∧
                ([∗ list] k↦y1;y2 ∈ l1';l2', Φ k y1 y2) ∗
                ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1; l2' ++ l2'', Φ k y1 y2) -∗
    ∃ l1' l1'', ⌜ l1 = l1' ++ l1'' ⌝ ∧
                ([∗ list] k↦y1;y2 ∈ l1';l2', Φ k y1 y2) ∗
                ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1 ++ l2; l1' ++ l2', Φ k y1 y2) -∗
    ([∗ list] k↦y1;y2 ∈ l1; l1', Φ k y1 y2) ∗
    ([∗ list] k↦y1;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_snoc Φ x1 x2 l1 l2 :
    ([∗ list] k↦y1;y2 ∈ (l1 ++ [x1]);(l2 ++ [x2]), Φ k y1 y2) ⊣⊢
    ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ∗ Φ (length l1) x1 x2.
  Proof.
    apply (anti_symm (⊢)); last first.
    - apply wand_elim_l'. rewrite big_sepL2_app. apply wand_mono; last done.
      rewrite big_sepL2_singleton Nat.add_0_r. done.
    - rewrite big_sepL2_app_inv_l. apply exist_elim=>l2l. apply exist_elim=>l2r.
      apply pure_elim_l=>Hl2.
      apply (pure_elim (length [x1] = length l2r)).
      { rewrite !big_sepL2_length sep_elim_r. done. }
      simpl. destruct l2r as [? l2r|]; first done.
      destruct l2r as [|]; last done. intros _.
      apply app_inj_tail in Hl2 as [-> ->].
      apply sep_mono_r.
      rewrite big_sepL2_singleton Nat.add_0_r. done.
  Qed.

  (** The lemmas [big_sepL2_mono], [big_sepL2_ne] and [big_sepL2_proper] are more
  generic than the instances as they also give [li !! k = Some yi] in the premise. *)
  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_mono=> k [y1 y2].
    rewrite lookup_zip_with=> ?; simplify_option_eq; auto.
  Qed.
  Lemma big_sepL2_ne Φ Ψ l1 l2 n :
    (∀ k y1 y2, l1 !! k = Some y1 → l2 !! k = Some y2 → Φ k y1 y2 ≡{n}≡ Ψ k y1 y2) →
    ([∗ list] k ↦ y1;y2 ∈ l1;l2, Φ k y1 y2)%I ≡{n}≡ ([∗ list] k ↦ y1;y2 ∈ l1;l2, Ψ k y1 y2)%I.
  Proof.
    intros H. rewrite !big_sepL2_alt. f_equiv. apply big_sepL_ne=> k [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.
  Lemma big_sepL2_proper_2 `{!Equiv A, !Equiv B} Φ Ψ l1 l2 l1' l2' :
    l1 ≡ l1' → l2 ≡ l2' →
    (∀ k y1 y1' y2 y2',
      l1 !! k = Some y1 → l1' !! k = Some y1' → y1 ≡ y1' →
      l2 !! k = Some y2 → l2' !! k = Some y2' → y2 ≡ 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 Hl1 Hl2 Hf. rewrite !big_sepL2_alt. f_equiv.
    { do 2 f_equiv; by apply length_proper. }
    apply big_opL_proper_2; [by f_equiv|].
    intros k [x1 y1] [x2 y2] (?&?&[=<- <-]&?&?)%lookup_zip_with_Some
      (?&?&[=<- <-]&?&?)%lookup_zip_with_Some [??]; naive_solver.
  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 f g Hf l1 ? <- l2 ? <-. apply big_sepL2_ne; intros; apply Hf. 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_insert_acc Φ l1 l2 i x1 x2 :
    l1 !! i = Some x1 → l2 !! i = Some x2 →
    ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢
    Φ i x1 x2 ∗ (∀ y1 y2, Φ i y1 y2 -∗ ([∗ list] k↦y1;y2 ∈ <[i:=y1]>l1;<[i:=y2]>l2, Φ k y1 y2)).
  Proof.
    intros Hl1 Hl2. rewrite big_sepL2_alt. apply pure_elim_l=> Hl.
    rewrite {1}big_sepL_insert_acc; last by rewrite lookup_zip_with; simplify_option_eq.
    apply sep_mono_r. apply forall_intro => y1. apply forall_intro => y2.
    rewrite big_sepL2_alt !insert_length pure_True // left_id -insert_zip_with.
    by rewrite (forall_elim (y1, y2)).
  Qed.

  Lemma big_sepL2_lookup_acc Φ l1 l2 i x1 x2 :
    l1 !! i = Some x1 → l2 !! i = Some x2 →
    ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢
    Φ i x1 x2 ∗ (Φ i x1 x2 -∗ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2)).
  Proof.
    intros. rewrite {1}big_sepL2_insert_acc // (forall_elim x1) (forall_elim x2).
    by rewrite !list_insert_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] k↦y1;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] k↦y1;y2 ∈ f <$> l1; l2, Φ k y1 y2)
    ⊣⊢ ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k (f y1) y2).
  Proof.
    rewrite !big_sepL2_alt fmap_length zip_with_fmap_l zip_with_zip big_sepL_fmap.
    by f_equiv; f_equiv=> k [??].
  Qed.
  Lemma big_sepL2_fmap_r {B'} (g : B → B') (Φ : nat → A → B' → PROP) l1 l2 :
    ([∗ list] k↦y1;y2 ∈ l1; g <$> l2, Φ k y1 y2)
    ⊣⊢ ([∗ list] k↦y1;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_equiv=> k [??].
  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_replicate_l l x Φ n :
    length l = n →
    ([∗ list] k↦x1;x2 ∈ replicate n x; l, Φ k x1 x2) ⊣⊢ [∗ list] k↦x2 ∈ l, Φ k x x2.
  Proof. intros <-. revert Φ. induction l as [|y l IH]=> //= Φ. by rewrite IH. Qed.
  Lemma big_sepL2_replicate_r l x Φ n :
    length l = n →
    ([∗ list] k↦x1;x2 ∈ l;replicate n x, Φ k x1 x2) ⊣⊢ [∗ list] k↦x1 ∈ l, Φ k x1 x.
  Proof. intros <-. revert Φ. induction l as [|y l IH]=> //= Φ. by rewrite IH. Qed.

  Lemma big_sepL2_sep Φ Ψ l1 l2 :
    ([∗ list] k↦y1;y2 ∈ l1;l2, 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.
    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] k↦y1;y2 ∈ l1;l2, 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. 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] k↦y1;y2 ∈ l1;l2, Φ k y1 y2)
    ⊣⊢ [∗ list] k↦y1;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_intuitionistically_forall Φ l1 l2 :
    length l1 = length l2 →
    □ (∀ k x1 x2, ⌜l1 !! k = Some x1⌝ → ⌜l2 !! k = Some x2⌝ → Φ k x1 x2) ⊢
    [∗ list] k↦x1;x2 ∈ l1;l2, Φ k x1 x2.
  Proof.
    revert l2 Φ. induction l1 as [|x1 l1 IH]=> -[|x2 l2] Φ ?; simplify_eq/=.
    { by apply (affine _). }
    rewrite intuitionistically_sep_dup. f_equiv.
    - rewrite (forall_elim 0) (forall_elim x1) (forall_elim x2).
      by rewrite !pure_True // !True_impl intuitionistically_elim.
    - rewrite -IH //. f_equiv.
      by apply forall_intro=> k; by rewrite (forall_elim (S k)).
  Qed.

  Lemma big_sepL2_forall `{BiAffine PROP} Φ l1 l2 :
    (∀ k x1 x2, Persistent (Φ k x1 x2)) →
    ([∗ list] k↦x1;x2 ∈ l1;l2, Φ k x1 x2) ⊣⊢
      ⌜length l1 = length l2⌝
      ∧ (∀ k x1 x2, ⌜l1 !! k = Some x1⌝ → ⌜l2 !! k = Some x2⌝ → Φ k x1 x2).
  Proof.
    intros. apply (anti_symm _).
    - apply and_intro; [apply big_sepL2_length|].
      apply forall_intro=> k. apply forall_intro=> x1. apply forall_intro=> x2.
      do 2 (apply impl_intro_l; apply pure_elim_l=> ?). by apply :big_sepL2_lookup.
    - apply pure_elim_l=> ?. rewrite -big_sepL2_intuitionistically_forall //.
      repeat setoid_rewrite pure_impl_forall.
      by rewrite intuitionistic_intuitionistically.
  Qed.

  Lemma big_sepL2_impl Φ Ψ l1 l2 :
    ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1;l2, Ψ k y1 y2.
  Proof.
    rewrite -(idemp bi_and (big_sepL2 _ _ _)) {1}big_sepL2_length.
    apply pure_elim_l=> ?. rewrite big_sepL2_intuitionistically_forall //.
    apply bi.wand_intro_l. rewrite -big_sepL2_sep. by setoid_rewrite wand_elim_l.
  Qed.

  Lemma big_sepL2_later_1 `{BiAffine PROP} Φ l1 l2 :
    (▷ [∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2) ⊢ ◇ [∗ list] k↦y1;y2 ∈ l1;l2, ▷ Φ k y1 y2.
  Proof.
    rewrite !big_sepL2_alt later_and big_sepL_later (timeless ⌜ _ ⌝%I).
    rewrite except_0_and. auto using and_mono, except_0_intro.
  Qed.

  Lemma big_sepL2_later_2 Φ l1 l2 :
    ([∗ list] k↦y1;y2 ∈ l1;l2, ▷ Φ k y1 y2) ⊢ ▷ [∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2.
  Proof.
    rewrite !big_sepL2_alt later_and big_sepL_later_2.
    auto using and_mono, later_intro.
  Qed.

  Lemma big_sepL2_laterN_2 Φ n l1 l2 :
    ([∗ list] k↦y1;y2 ∈ l1;l2, ▷^n Φ k y1 y2) ⊢ ▷^n [∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2.
  Proof.
    rewrite !big_sepL2_alt laterN_and big_sepL_laterN_2.
    auto using and_mono, laterN_intro.
  Qed.

  Lemma big_sepL2_flip Φ l1 l2 :
    ([∗ list] k↦y1;y2 ∈ l2; l1, Φ k y2 y1) ⊣⊢ ([∗ list] k↦y1;y2 ∈ l1; l2, Φ k y1 y2).
  Proof.
    revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2]//=; simplify_eq.
    by rewrite IH.
  Qed.

  Lemma big_sepL2_merge (Φ : nat → A → PROP) (Ψ : nat → B → PROP)
      (l1 : list A) (l2 : list B) :
    length l1 = length l2 →
    ([∗ list] k↦y1 ∈ l1, Φ k y1) -∗
    ([∗ list] k↦y2 ∈ l2, Ψ k y2) -∗
    ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 ∗ Ψ k y2).
  Proof.
    intros Hlen. apply wand_intro_r.
    rewrite big_sepL_sep_zip // big_sepL2_alt pure_True // left_id //.
  Qed.

  Global Instance big_sepL2_nil_persistent Φ :
    Persistent ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.

  Global Instance big_sepL2_nil_affine Φ :
    Affine ([∗ list] k↦y1;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] k↦y1;y2 ∈ l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.

  Global Instance big_sepL2_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
    Timeless ([∗ list] k↦y1;y2 ∈ []; [], Φ k y1 y2).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL2_timeless `{!Timeless (emp%I : PROP)} Φ l1 l2 :
    (∀ k x1 x2, Timeless (Φ k x1 x2)) →
    Timeless ([∗ list] k↦y1;y2 ∈ l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.
End sep_list2.

Lemma big_sepL2_ne_2 {A B : ofe}
    (Φ Ψ : nat → A → B → PROP) l1 l2 l1' l2' n :
  l1 ≡{n}≡ l1' → l2 ≡{n}≡ l2' →
  (∀ k y1 y1' y2 y2',
    l1 !! k = Some y1 → l1' !! k = Some y1' → y1 ≡{n}≡ y1' →
    l2 !! k = Some y2 → l2' !! k = Some y2' → y2 ≡{n}≡ y2' →
    Φ k y1 y2 ≡{n}≡ Ψ k y1' y2') →
  ([∗ list] k ↦ y1;y2 ∈ l1;l2, Φ k y1 y2)%I ≡{n}≡ ([∗ list] k ↦ y1;y2 ∈ l1';l2', Ψ k y1 y2)%I.
Proof.
  intros Hl1 Hl2 Hf. rewrite !big_sepL2_alt. f_equiv.
  { do 2 f_equiv; by apply: length_ne. }
  apply big_opL_ne_2; [by f_equiv|].
  intros k [x1 y1] [x2 y2] (?&?&[=<- <-]&?&?)%lookup_zip_with_Some
    (?&?&[=<- <-]&?&?)%lookup_zip_with_Some [??]; naive_solver.
Qed.

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

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

  (** The lemmas [big_andL_mono], [big_andL_ne] and [big_andL_proper] are more
  generic than the instances as they also give [l !! k = Some y] in the premise. *)
  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_gen_proper; apply _. Qed.
  Lemma big_andL_ne Φ Ψ l n :
    (∀ k y, l !! k = Some y → Φ k y ≡{n}≡ Ψ k y) →
    ([∧ list] k ↦ y ∈ l, Φ k y)%I ≡{n}≡ ([∧ list] k ↦ y ∈ l, Ψ k y)%I.
  Proof. apply big_opL_ne. 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.

  (** No need to declare instances for non-expansiveness and properness, we
  get both from the generic [big_opL] instances. *)
  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_andL_mono; 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] k↦y ∈ 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. by eapply (big_andL_lookup (λ _, Φ)).
  Qed.

  Lemma big_andL_fmap {B} (f : A → B) (Φ : nat → B → PROP) l :
    ([∧ list] k↦y ∈ f <$> l, Φ k y) ⊣⊢ ([∧ list] k↦y ∈ 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] k↦x ∈ l, Φ k x ∧ Ψ k x)
    ⊣⊢ ([∧ list] k↦x ∈ l, Φ k x) ∧ ([∧ list] k↦x ∈ l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

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

  Lemma big_andL_forall Φ l :
    ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x).
  Proof.
    apply (anti_symm _).
    { apply forall_intro=> k; apply forall_intro=> x.
      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_intro=> k; by rewrite (forall_elim (S k)).
  Qed.

  Global Instance big_andL_nil_persistent Φ :
    Persistent ([∧ list] k↦x ∈ [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_andL_persistent Φ l :
    (∀ k x, Persistent (Φ k x)) → Persistent ([∧ list] k↦x ∈ 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] k↦y ∈ nil, Φ k y) ⊣⊢ False.
  Proof. done. Qed.
  Lemma big_orL_cons Φ x l :
    ([∨ list] k↦y ∈ x :: l, Φ k y) ⊣⊢ Φ 0 x ∨ [∨ list] k↦y ∈ l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_orL_singleton Φ x : ([∨ list] k↦y ∈ [x], Φ k y) ⊣⊢ Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_orL_app Φ l1 l2 :
    ([∨ list] k↦y ∈ l1 ++ l2, Φ k y)
    ⊣⊢ ([∨ list] k↦y ∈ l1, Φ k y) ∨ ([∨ list] k↦y ∈ l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. 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.

  (** The lemmas [big_orL_mono], [big_orL_ne] and [big_orL_proper] are more
  generic than the instances as they also give [l !! k = Some y] in the premise. *)
  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_gen_proper; apply _. Qed.
  Lemma big_orL_ne Φ Ψ l n :
    (∀ k y, l !! k = Some y → Φ k y ≡{n}≡ Ψ k y) →
    ([∨ list] k ↦ y ∈ l, Φ k y)%I ≡{n}≡ ([∨ list] k ↦ y ∈ l, Ψ k y)%I.
  Proof. apply big_opL_ne. 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.

  (** No need to declare instances for non-expansiveness and properness, we
  get both from the generic [big_opL] instances. *)
  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_orL_mono; 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] k↦y ∈ 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; by eapply (big_orL_lookup (λ _, Φ)).
  Qed.

  Lemma big_orL_fmap {B} (f : A → B) (Φ : nat → B → PROP) l :
    ([∨ list] k↦y ∈ f <$> l, Φ k y) ⊣⊢ ([∨ list] k↦y ∈ 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] k↦x ∈ l, Φ k x ∨ Ψ k x)
    ⊣⊢ ([∨ list] k↦x ∈ l, Φ k x) ∨ ([∨ list] k↦x ∈ l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

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

  Lemma big_orL_exist Φ l :
    ([∨ list] k↦x ∈ 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_elim=> k. by rewrite -(exist_intro (S k)). }
    apply exist_elim=> k; apply exist_elim=> x. apply pure_elim_l=> ?.
    by apply: big_orL_lookup.
  Qed.

  Lemma big_orL_sep_l P Φ l :
    P ∗ ([∨ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∨ list] k↦x ∈ l, P ∗ Φ k x).
  Proof.
    rewrite !big_orL_exist sep_exist_l.
    f_equiv=> k. rewrite sep_exist_l. f_equiv=> x.
    by rewrite !persistent_and_affinely_sep_l !assoc (comm _ P).
 Qed.
  Lemma big_orL_sep_r Q Φ l :
    ([∨ list] k↦x ∈ l, Φ k x) ∗ Q ⊣⊢ ([∨ list] k↦x ∈ 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] k↦x ∈ [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_orL_persistent Φ l :
    (∀ k x, Persistent (Φ k x)) → Persistent ([∨ list] k↦x ∈ 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}