Library iris.algebra.big_op

From stdpp Require Export functions gmap gmultiset.
From iris.algebra Require Export monoid.
From iris.prelude Require Import options.
Local Existing Instances monoid_ne monoid_assoc monoid_comm
  monoid_left_id monoid_right_id monoid_proper
  monoid_homomorphism_rel_po monoid_homomorphism_rel_proper
  monoid_homomorphism_op_proper
  monoid_homomorphism_ne weak_monoid_homomorphism_proper.

We define the following big operators with binders build in:
  • The operator [^o list] k x l, P folds over a list l. The binder x refers to each element at index k.
  • The operator [^o map] k x m, P folds over a map m. The binder x refers to each element at index k.
  • The operator [^o set] x X, P folds over a set X. The binder x refers to each element.
Since these big operators are like quantifiers, they have the same precedence as and .

Big ops over lists

Fixpoint big_opL `{Monoid M o} {A} (f : nat A M) (xs : list A) : M :=
  match xs with
  | []monoid_unit
  | x :: xso (f 0 x) (big_opL (λ n, f (S n)) xs)
  end.
Global Instance: Params (@big_opL) 4 := {}.
Global Arguments big_opL {M} o {_ A} _ !_ /.
Typeclasses Opaque big_opL.
Notation "'[^' o 'list]' k ↦ x ∈ l , P" := (big_opL o (λ k x, P) l)
  (at level 200, o at level 1, l at level 10, k, x at level 1, right associativity,
   format "[^ o list] k ↦ x ∈ l , P") : stdpp_scope.
Notation "'[^' o 'list]' x ∈ l , P" := (big_opL o (λ _ x, P) l)
  (at level 200, o at level 1, l at level 10, x at level 1, right associativity,
   format "[^ o list] x ∈ l , P") : stdpp_scope.

Definition big_opM_def `{Monoid M o} `{Countable K} {A} (f : K A M)
  (m : gmap K A) : M := big_opL o (λ _, uncurry f) (map_to_list m).
Definition big_opM_aux : seal (@big_opM_def). Proof. by eexists. Qed.
Definition big_opM := big_opM_aux.(unseal).
Global Arguments big_opM {M} o {_ K _ _ A} _ _.
Definition big_opM_eq : @big_opM = @big_opM_def := big_opM_aux.(seal_eq).
Global Instance: Params (@big_opM) 7 := {}.
Notation "'[^' o 'map]' k ↦ x ∈ m , P" := (big_opM o (λ k x, P) m)
  (at level 200, o at level 1, m at level 10, k, x at level 1, right associativity,
   format "[^ o map] k ↦ x ∈ m , P") : stdpp_scope.
Notation "'[^' o 'map]' x ∈ m , P" := (big_opM o (λ _ x, P) m)
  (at level 200, o at level 1, m at level 10, x at level 1, right associativity,
   format "[^ o map] x ∈ m , P") : stdpp_scope.

Definition big_opS_def `{Monoid M o} `{Countable A} (f : A M)
  (X : gset A) : M := big_opL o (λ _, f) (elements X).
Definition big_opS_aux : seal (@big_opS_def). Proof. by eexists. Qed.
Definition big_opS := big_opS_aux.(unseal).
Global Arguments big_opS {M} o {_ A _ _} _ _.
Definition big_opS_eq : @big_opS = @big_opS_def := big_opS_aux.(seal_eq).
Global Instance: Params (@big_opS) 6 := {}.
Notation "'[^' o 'set]' x ∈ X , P" := (big_opS o (λ x, P) X)
  (at level 200, o at level 1, X at level 10, x at level 1, right associativity,
   format "[^ o set] x ∈ X , P") : stdpp_scope.

Definition big_opMS_def `{Monoid M o} `{Countable A} (f : A M)
  (X : gmultiset A) : M := big_opL o (λ _, f) (elements X).
Definition big_opMS_aux : seal (@big_opMS_def). Proof. by eexists. Qed.
Definition big_opMS := big_opMS_aux.(unseal).
Global Arguments big_opMS {M} o {_ A _ _} _ _.
Definition big_opMS_eq : @big_opMS = @big_opMS_def := big_opMS_aux.(seal_eq).
Global Instance: Params (@big_opMS) 6 := {}.
Notation "'[^' o 'mset]' x ∈ X , P" := (big_opMS o (λ x, P) X)
  (at level 200, o at level 1, X at level 10, x at level 1, right associativity,
   format "[^ o mset] x ∈ X , P") : stdpp_scope.

Properties about big ops

Section big_op.
Context `{Monoid M o}.
Implicit Types xs : list M.
Infix "`o`" := o (at level 50, left associativity).

Big ops over lists

Section list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types f g : nat A M.

  Lemma big_opL_nil f : ([^o list] ky [], f k y) = monoid_unit.
  Proof. done. Qed.
  Lemma big_opL_cons f x l :
    ([^o list] ky x :: l, f k y) = f 0 x `o` [^o list] ky l, f (S k) y.
  Proof. done. Qed.
  Lemma big_opL_singleton f x : ([^o list] ky [x], f k y) f 0 x.
  Proof. by rewrite /= right_id. Qed.
  Lemma big_opL_app f l1 l2 :
    ([^o list] ky l1 ++ l2, f k y)
     ([^o list] ky l1, f k y) `o` ([^o list] ky l2, f (length l1 + k) y).
  Proof.
    revert f. induction l1 as [|x l1 IH]=> f /=; first by rewrite left_id.
    by rewrite IH assoc.
  Qed.
  Lemma big_opL_snoc f l x :
    ([^o list] ky l ++ [x], f k y) ([^o list] ky l, f k y) `o` f (length l) x.
  Proof. rewrite big_opL_app big_opL_singleton Nat.add_0_r //. Qed.

  Lemma big_opL_unit l : ([^o list] ky l, monoid_unit) (monoid_unit : M).
  Proof. induction l; rewrite /= ?left_id //. Qed.

  Lemma big_opL_take_drop Φ l n :
    ([^o list] k x l, Φ k x)
    ([^o list] k x take n l, Φ k x) `o` ([^o list] k x drop n l, Φ (n + k) x).
  Proof.
    rewrite -{1}(take_drop n l) big_opL_app take_length.
    destruct (decide (length l n)).
    - rewrite drop_ge //=.
    - rewrite Nat.min_l //=; lia.
  Qed.

  Lemma big_opL_gen_proper_2 {B} (R : relation M) f (g : nat B M)
        l1 (l2 : list B) :
    R monoid_unit monoid_unit
    Proper (R ==> R ==> R) o
    ( k,
      match l1 !! k, l2 !! k with
      | Some y1, Some y2R (f k y1) (g k y2)
      | None, NoneTrue
      | _, _False
      end)
    R ([^o list] k y l1, f k y) ([^o list] k y l2, g k y).
  Proof.
    intros ??. revert l2 f g. induction l1 as [|x1 l1 IH]=> -[|x2 l2] //= f g Hfg.
    - by specialize (Hfg 0).
    - by specialize (Hfg 0).
    - f_equiv; [apply (Hfg 0)|]. apply IH. intros k. apply (Hfg (S k)).
  Qed.
  Lemma big_opL_gen_proper R f g l :
    Reflexive R
    Proper (R ==> R ==> R) o
    ( k y, l !! k = Some y R (f k y) (g k y))
    R ([^o list] k y l, f k y) ([^o list] k y l, g k y).
  Proof.
    intros. apply big_opL_gen_proper_2; [done..|].
    intros k. destruct (l !! k) eqn:?; auto.
  Qed.

  Lemma big_opL_ext f g l :
    ( k y, l !! k = Some y f k y = g k y)
    ([^o list] k y l, f k y) = [^o list] k y l, g k y.
  Proof. apply big_opL_gen_proper; apply _. Qed.

  Lemma big_opL_permutation (f : A M) l1 l2 :
    l1 ≡ₚ l2 ([^o list] x l1, f x) ([^o list] x l2, f x).
  Proof.
    induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto.
    - by rewrite IH.
    - by rewrite !assoc (comm _ (f x)).
    - by etrans.
  Qed.
  Global Instance big_opL_permutation' (f : A M) :
    Proper ((≡ₚ) ==> (≡)) (big_opL o (λ _, f)).
  Proof. intros xs1 xs2. apply big_opL_permutation. Qed.

The lemmas big_opL_ne and big_opL_proper are more generic than the instances as they also give l !! k = Some y in the premise.
  Lemma big_opL_ne f g l n :
    ( k y, l !! k = Some y f k y ≡{n}≡ g k y)
    ([^o list] k y l, f k y) ≡{n}≡ ([^o list] k y l, g k y).
  Proof. apply big_opL_gen_proper; apply _. Qed.
  Lemma big_opL_proper f g l :
    ( k y, l !! k = Some y f k y g k y)
    ([^o list] k y l, f k y) ([^o list] k y l, g k y).
  Proof. apply big_opL_gen_proper; apply _. Qed.

The version big_opL_proper_2 with for the list arguments can only be used if there is a setoid on A. The version for dist n can be found in algebra.list. We do not define this lemma as a Proper instance, since f_equiv will then use sometimes use this one, and other times big_opL_proper', depending on whether a setoid on A exists.
  Lemma big_opL_proper_2 `{!Equiv A} f g l1 l2 :
    l1 l2
    ( k y1 y2,
      l1 !! k = Some y1 l2 !! k = Some y2 y1 y2 f k y1 g k y2)
    ([^o list] k y l1, f k y) ([^o list] k y l2, g k y).
  Proof.
    intros Hl Hf. apply big_opL_gen_proper_2; try (apply _ || done).
    intros k. assert (l1 !! k ≡@{option A} l2 !! k) as Hlk by (by f_equiv).
    destruct (l1 !! k) eqn:?, (l2 !! k) eqn:?; inversion Hlk; naive_solver.
  Qed.

  Global Instance big_opL_ne' n :
    Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
           (big_opL o (A:=A)).
  Proof. intros f f' Hf l ? <-. apply big_opL_ne; intros; apply Hf. Qed.
  Global Instance big_opL_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (=) ==> (≡))
           (big_opL o (A:=A)).
  Proof. intros f f' Hf l ? <-. apply big_opL_proper; intros; apply Hf. Qed.

  Lemma big_opL_consZ_l (f : Z A M) x l :
    ([^o list] ky x :: l, f k y) = f 0 x `o` [^o list] ky l, f (1 + k)%Z y.
  Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed.
  Lemma big_opL_consZ_r (f : Z A M) x l :
    ([^o list] ky x :: l, f k y) = f 0 x `o` [^o list] ky l, f (k + 1)%Z y.
  Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed.

  Lemma big_opL_fmap {B} (h : A B) (f : nat B M) l :
    ([^o list] ky h <$> l, f k y) ([^o list] ky l, f k (h y)).
  Proof. revert f. induction l as [|x l IH]=> f; csimpl⇒ //. by rewrite IH. Qed.

  Lemma big_opL_omap {B} (h : A option B) (f : B M) l :
    ([^o list] y omap h l, f y) ([^o list] y l, from_option f monoid_unit (h y)).
  Proof.
    revert f. induction l as [|x l IH]=> f //; csimpl.
    case_match; csimpl; by rewrite IH // left_id.
  Qed.

  Lemma big_opL_op f g l :
    ([^o list] kx l, f k x `o` g k x)
     ([^o list] kx l, f k x) `o` ([^o list] kx l, g k x).
  Proof.
    revert f g; induction l as [|x l IH]=> f g /=; first by rewrite left_id.
    by rewrite IH -!assoc (assoc _ (g _ _)) [(g _ _ `o` _)]comm -!assoc.
  Qed.
End list.

Lemma big_opL_bind {A B} (h : A list B) (f : B M) l :
  ([^o list] y l ≫= h, f y) ([^o list] x l, [^o list] y h x, f y).
Proof.
  revert f. induction l as [|x l IH]=> f; csimpl⇒ //. by rewrite big_opL_app IH.
Qed.

Lemma big_opL_sep_zip_with {A B C} (f : A B C) (g1 : C A) (g2 : C B)
    (h1 : nat A M) (h2 : nat B M) l1 l2 :
  ( x y, g1 (f x y) = x)
  ( x y, g2 (f x y) = y)
  length l1 = length l2
  ([^o list] kxy zip_with f l1 l2, h1 k (g1 xy) `o` h2 k (g2 xy))
  ([^o list] kx l1, h1 k x) `o` ([^o list] ky l2, h2 k y).
Proof.
  intros Hlen Hg1 Hg2. rewrite big_opL_op.
  rewrite -(big_opL_fmap g1) -(big_opL_fmap g2).
  rewrite fmap_zip_with_r; [|auto with lia..].
  by rewrite fmap_zip_with_l; [|auto with lia..].
Qed.

Lemma big_opL_sep_zip {A B} (h1 : nat A M) (h2 : nat B M) l1 l2 :
  length l1 = length l2
  ([^o list] kxy zip l1 l2, h1 k xy.1 `o` h2 k xy.2)
  ([^o list] kx l1, h1 k x) `o` ([^o list] ky l2, h2 k y).
Proof. by apply big_opL_sep_zip_with. Qed.

Big ops over finite maps


Lemma big_opM_empty `{Countable K} {B} (f : K B M) :
  ([^o map] kx , f k x) = monoid_unit.
Proof. by rewrite big_opM_eq /big_opM_def map_to_list_empty. Qed.

Lemma big_opM_insert `{Countable K} {B} (f : K B M) (m : gmap K B) i x :
  m !! i = None
  ([^o map] ky <[i:=x]> m, f k y) f i x `o` [^o map] ky m, f k y.
Proof. intros ?. by rewrite big_opM_eq /big_opM_def map_to_list_insert. Qed.

Lemma big_opM_delete `{Countable K} {B} (f : K B M) (m : gmap K B) i x :
  m !! i = Some x
  ([^o map] ky m, f k y) f i x `o` [^o map] ky delete i m, f k y.
Proof.
  intros. rewrite -big_opM_insert ?lookup_delete //.
  by rewrite insert_delete.
Qed.

Section gmap.
  Context `{Countable K} {A : Type}.
  Implicit Types m : gmap K A.
  Implicit Types f g : K A M.

  Lemma big_opM_gen_proper_2 {B} (R : relation M) f (g : K B M)
        m1 (m2 : gmap K B) :
    subrelation (≡) R Equivalence R
    Proper (R ==> R ==> R) o
    ( k,
      match m1 !! k, m2 !! k with
      | Some y1, Some y2R (f k y1) (g k y2)
      | None, NoneTrue
      | _, _False
      end)
    R ([^o map] k x m1, f k x) ([^o map] k x m2, g k x).
  Proof.
    intros HR ??. revert m2 f g.
    induction m1 as [|k x1 m1 Hm1k IH] using map_indm2 f g Hfg.
    { destruct m2 as [|k x2 m2 _ _] using map_ind.
      { rewrite !big_opM_empty. by apply HR. }
      generalize (Hfg k). by rewrite lookup_empty lookup_insert. }
    generalize (Hfg k). rewrite lookup_insert.
    destruct (m2 !! k) as [x2|] eqn:Hm2k; [intros Hk|done].
    etrans; [by apply HR, big_opM_insert|].
    etrans; [|by symmetry; apply HR, big_opM_delete].
    f_equiv; [done|]. apply IHk'. destruct (decide (k = k')) as [->|?].
    - by rewrite lookup_delete Hm1k.
    - generalize (Hfg k'). rewrite lookup_insert_ne // lookup_delete_ne //.
  Qed.

  Lemma big_opM_gen_proper R f g m :
    Reflexive R
    Proper (R ==> R ==> R) o
    ( k x, m !! k = Some x R (f k x) (g k x))
    R ([^o map] k x m, f k x) ([^o map] k x m, g k x).
  Proof.
    intros ?? Hf. rewrite big_opM_eq. apply (big_opL_gen_proper R); auto.
    intros k [i x] ?%elem_of_list_lookup_2. by apply Hf, elem_of_map_to_list.
  Qed.

  Lemma big_opM_ext f g m :
    ( k x, m !! k = Some x f k x = g k x)
    ([^o map] k x m, f k x) = ([^o map] k x m, g k x).
  Proof. apply big_opM_gen_proper; apply _. Qed.

The lemmas big_opM_ne and big_opM_proper are more generic than the instances as they also give m !! k = Some y in the premise.
  Lemma big_opM_ne f g m n :
    ( k x, m !! k = Some x f k x ≡{n}≡ g k x)
    ([^o map] k x m, f k x) ≡{n}≡ ([^o map] k x m, g k x).
  Proof. apply big_opM_gen_proper; apply _. Qed.
  Lemma big_opM_proper f g m :
    ( k x, m !! k = Some x f k x g k x)
    ([^o map] k x m, f k x) ([^o map] k x m, g k x).
  Proof. apply big_opM_gen_proper; apply _. Qed.
The version big_opM_proper_2 with for the map arguments can only be used if there is a setoid on A. The version for dist n can be found in algebra.gmap. We do not define this lemma as a Proper instance, since f_equiv will then use sometimes use this one, and other times big_opM_proper', depending on whether a setoid on A exists.
  Lemma big_opM_proper_2 `{!Equiv A} f g m1 m2 :
    m1 m2
    ( k y1 y2,
      m1 !! k = Some y1 m2 !! k = Some y2 y1 y2 f k y1 g k y2)
    ([^o map] k y m1, f k y) ([^o map] k y m2, g k y).
  Proof.
    intros Hl Hf. apply big_opM_gen_proper_2; try (apply _ || done).
    intros k. assert (m1 !! k ≡@{option A} m2 !! k) as Hlk by (by f_equiv).
    destruct (m1 !! k) eqn:?, (m2 !! k) eqn:?; inversion Hlk; naive_solver.
  Qed.

  Global Instance big_opM_ne' n :
    Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
           (big_opM o (K:=K) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opM_ne; intros; apply Hf. Qed.
  Global Instance big_opM_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (=) ==> (≡))
           (big_opM o (K:=K) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opM_proper; intros; apply Hf. Qed.

  Lemma big_opM_map_to_list f m :
    ([^o map] kx m, f k x) [^o list] xk map_to_list m, f (xk.1) (xk.2).
  Proof. rewrite big_opM_eq. apply big_opL_proper'; [|done]. by intros ? [??]. Qed.

  Lemma big_opM_singleton f i x : ([^o map] ky {[i:=x]}, f k y) f i x.
  Proof.
    rewrite -insert_empty big_opM_insert/=; last eauto using lookup_empty.
    by rewrite big_opM_empty right_id.
  Qed.

  Lemma big_opM_unit m : ([^o map] ky m, monoid_unit) (monoid_unit : M).
  Proof.
    by induction m using map_ind; rewrite /= ?big_opM_insert ?left_id // big_opM_eq.
  Qed.

  Lemma big_opM_fmap {B} (h : A B) (f : K B M) m :
    ([^o map] ky h <$> m, f k y) ([^o map] ky m, f k (h y)).
  Proof.
    rewrite big_opM_eq /big_opM_def map_to_list_fmap big_opL_fmap.
    by apply big_opL_proper⇒ ? [??].
  Qed.

  Lemma big_opM_omap {B} (h : A option B) (f : K B M) m :
    ([^o map] ky omap h m, f k y)
     [^o map] ky m, from_option (f k) monoid_unit (h y).
  Proof.
    revert f. induction m as [|i x m Hmi IH] using map_indf.
    { by rewrite omap_empty !big_opM_empty. }
    assert (omap h m !! i = None) by (by rewrite lookup_omap Hmi).
    destruct (h x) as [y|] eqn:Hhx.
    - by rewrite omap_insert Hhx //= !big_opM_insert // IH Hhx.
    - rewrite omap_insert_None // delete_notin // big_opM_insert //.
      by rewrite Hhx /= left_id.
  Qed.

  Lemma big_opM_insert_delete `{Countable K} {B} (f : K B M) (m : gmap K B) i x :
    ([^o map] ky <[i:=x]> m, f k y) f i x `o` [^o map] ky delete i m, f k y.
  Proof.
    rewrite -insert_delete_insert big_opM_insert; first done. by rewrite lookup_delete.
  Qed.

  Lemma big_opM_insert_override (f : K A M) m i x x' :
    m !! i = Some x f i x f i x'
    ([^o map] ky <[i:=x']> m, f k y) ([^o map] ky m, f k y).
  Proof.
    intros ? Hx. rewrite -insert_delete_insert big_opM_insert ?lookup_delete //.
    by rewrite -Hx -big_opM_delete.
  Qed.

  Lemma big_opM_fn_insert {B} (g : K A B M) (f : K B) m i (x : A) b :
    m !! i = None
    ([^o map] ky <[i:=x]> m, g k y (<[i:=b]> f k))
     g i x b `o` [^o map] ky m, g k y (f k).
  Proof.
    intros. rewrite big_opM_insert // fn_lookup_insert.
    f_equiv; apply big_opM_proper; autok y ?.
    by rewrite fn_lookup_insert_ne; last set_solver.
  Qed.
  Lemma big_opM_fn_insert' (f : K M) m i x P :
    m !! i = None
    ([^o map] ky <[i:=x]> m, <[i:=P]> f k) (P `o` [^o map] ky m, f k).
  Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed.

  Lemma big_opM_filter' (φ : K × A Prop) `{ kx, Decision kx)} f m :
    ([^o map] k x filter φ m, f k x)
     ([^o map] k x m, if decide (φ (k, x)) then f k x else monoid_unit).
  Proof.
    induction m as [|k v m ? IH] using map_ind.
    { by rewrite map_filter_empty !big_opM_empty. }
    destruct (decide(k, v))).
    - rewrite map_filter_insert_True //.
      assert (filter φ m !! k = None) by (apply map_filter_lookup_None; eauto).
      by rewrite !big_opM_insert // decide_True // IH.
    - rewrite map_filter_insert_not' //; last by congruence.
      rewrite !big_opM_insert // decide_False // IH. by rewrite left_id.
  Qed.

  Lemma big_opM_union f m1 m2 :
    m1 ##ₘ m2
    ([^o map] ky m1 m2, f k y)
     ([^o map] ky m1, f k y) `o` ([^o map] ky m2, f k y).
  Proof.
    intros. induction m1 as [|i x m ? IH] using map_ind.
    { by rewrite big_opM_empty !left_id. }
    decompose_map_disjoint.
    rewrite -insert_union_l !big_opM_insert //;
      last by apply lookup_union_None.
    rewrite -assoc IH //.
  Qed.

  Lemma big_opM_op f g m :
    ([^o map] kx m, f k x `o` g k x)
     ([^o map] kx m, f k x) `o` ([^o map] kx m, g k x).
  Proof.
    rewrite big_opM_eq /big_opM_def -big_opL_op. by apply big_opL_proper⇒ ? [??].
  Qed.
End gmap.

Lemma big_opM_sep_zip_with `{Countable K} {A B C}
    (f : A B C) (g1 : C A) (g2 : C B)
    (h1 : K A M) (h2 : K B M) m1 m2 :
  ( x y, g1 (f x y) = x)
  ( x y, g2 (f x y) = y)
  ( k, is_Some (m1 !! k) is_Some (m2 !! k))
  ([^o map] kxy map_zip_with f m1 m2, h1 k (g1 xy) `o` h2 k (g2 xy))
  ([^o map] kx m1, h1 k x) `o` ([^o map] ky m2, h2 k y).
Proof.
  intros Hdom Hg1 Hg2. rewrite big_opM_op.
  rewrite -(big_opM_fmap g1) -(big_opM_fmap g2).
  rewrite map_fmap_zip_with_r; [|naive_solver..].
  by rewrite map_fmap_zip_with_l; [|naive_solver..].
Qed.

Lemma big_opM_sep_zip `{Countable K} {A B}
    (h1 : K A M) (h2 : K B M) m1 m2 :
  ( k, is_Some (m1 !! k) is_Some (m2 !! k))
  ([^o map] kxy map_zip m1 m2, h1 k xy.1 `o` h2 k xy.2)
  ([^o map] kx m1, h1 k x) `o` ([^o map] ky m2, h2 k y).
Proof. intros. by apply big_opM_sep_zip_with. Qed.

Big ops over finite sets

Section gset.
  Context `{Countable A}.
  Implicit Types X : gset A.
  Implicit Types f : A M.

  Lemma big_opS_gen_proper R f g X :
    Reflexive R Proper (R ==> R ==> R) o
    ( x, x X R (f x) (g x))
    R ([^o set] x X, f x) ([^o set] x X, g x).
  Proof.
    rewrite big_opS_eq. intros ?? Hf. apply (big_opL_gen_proper R); auto.
    intros k x ?%elem_of_list_lookup_2. by apply Hf, elem_of_elements.
  Qed.

  Lemma big_opS_ext f g X :
    ( x, x X f x = g x)
    ([^o set] x X, f x) = ([^o set] x X, g x).
  Proof. apply big_opS_gen_proper; apply _. Qed.

The lemmas big_opS_ne and big_opS_proper are more generic than the instances as they also give x X in the premise.
  Lemma big_opS_ne f g X n :
    ( x, x X f x ≡{n}≡ g x)
    ([^o set] x X, f x) ≡{n}≡ ([^o set] x X, g x).
  Proof. apply big_opS_gen_proper; apply _. Qed.
  Lemma big_opS_proper f g X :
    ( x, x X f x g x)
    ([^o set] x X, f x) ([^o set] x X, g x).
  Proof. apply big_opS_gen_proper; apply _. Qed.

  Global Instance big_opS_ne' n :
    Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opS_ne; intros; apply Hf. Qed.
  Global Instance big_opS_proper' :
    Proper (pointwise_relation _ (≡) ==> (=) ==> (≡)) (big_opS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opS_proper; intros; apply Hf. Qed.

  Lemma big_opS_elements f X :
    ([^o set] x X, f x) [^o list] x elements X, f x.
  Proof. by rewrite big_opS_eq. Qed.

  Lemma big_opS_empty f : ([^o set] x , f x) = monoid_unit.
  Proof. by rewrite big_opS_eq /big_opS_def elements_empty. Qed.

  Lemma big_opS_insert f X x :
    x X ([^o set] y {[ x ]} X, f y) (f x `o` [^o set] y X, f y).
  Proof. intros. by rewrite !big_opS_elements elements_union_singleton. Qed.
  Lemma big_opS_fn_insert {B} (f : A B M) h X x b :
    x X
    ([^o set] y {[ x ]} X, f y (<[x:=b]> h y))
     f x b `o` [^o set] y X, f y (h y).
  Proof.
    intros. rewrite big_opS_insert // fn_lookup_insert.
    f_equiv; apply big_opS_proper; autoy ?.
    by rewrite fn_lookup_insert_ne; last set_solver.
  Qed.
  Lemma big_opS_fn_insert' f X x P :
    x X ([^o set] y {[ x ]} X, <[x:=P]> f y) (P `o` [^o set] y X, f y).
  Proof. apply (big_opS_fn_insert (λ y, id)). Qed.

  Lemma big_opS_union f X Y :
    X ## Y
    ([^o set] y X Y, f y) ([^o set] y X, f y) `o` ([^o set] y Y, f y).
  Proof.
    intros. induction X as [|x X ? IH] using set_ind_L.
    { by rewrite left_id_L big_opS_empty left_id. }
    rewrite -assoc_L !big_opS_insert; [|set_solver..].
    by rewrite -assoc IH; last set_solver.
  Qed.

  Lemma big_opS_delete f X x :
    x X ([^o set] y X, f y) f x `o` [^o set] y X {[ x ]}, f y.
  Proof.
    intros. rewrite -big_opS_insert; last set_solver.
    by rewrite -union_difference_L; last set_solver.
  Qed.

  Lemma big_opS_singleton f x : ([^o set] y {[ x ]}, f y) f x.
  Proof. intros. by rewrite big_opS_elements elements_singleton /= right_id. Qed.

  Lemma big_opS_unit X : ([^o set] y X, monoid_unit) (monoid_unit : M).
  Proof.
    by induction X using set_ind_L; rewrite /= ?big_opS_insert ?left_id // big_opS_eq.
  Qed.

  Lemma big_opS_filter' (φ : A Prop) `{ x, Decision x)} f X :
    ([^o set] y filter φ X, f y)
     ([^o set] y X, if decide (φ y) then f y else monoid_unit).
  Proof.
    induction X as [|x X ? IH] using set_ind_L.
    { by rewrite filter_empty_L !big_opS_empty. }
    destruct (decidex)).
    - rewrite filter_union_L filter_singleton_L //.
      rewrite !big_opS_insert //; last set_solver.
      by rewrite decide_True // IH.
    - rewrite filter_union_L filter_singleton_not_L // left_id_L.
      by rewrite !big_opS_insert // decide_False // IH left_id.
  Qed.

  Lemma big_opS_op f g X :
    ([^o set] y X, f y `o` g y) ([^o set] y X, f y) `o` ([^o set] y X, g y).
  Proof. by rewrite !big_opS_elements -big_opL_op. Qed.

  Lemma big_opS_list_to_set f (l : list A) :
    NoDup l
    ([^o set] x list_to_set l, f x) [^o list] x l, f x.
  Proof.
    induction 1 as [|x l ?? IHl].
    - rewrite big_opS_empty //.
    - rewrite /= big_opS_union; last set_solver.
      by rewrite big_opS_singleton IHl.
  Qed.
End gset.

Lemma big_opS_set_map `{Countable A, Countable B} (h : A B) (X : gset A) (f : B M) :
  Inj (=) (=) h
  ([^o set] x set_map h X, f x) ([^o set] x X, f (h x)).
Proof.
  intros Hinj.
  induction X as [|x X ? IH] using set_ind_L.
  { by rewrite set_map_empty !big_opS_empty. }
  rewrite set_map_union_L set_map_singleton_L.
  rewrite !big_opS_union; [|set_solver..].
  rewrite !big_opS_singleton IH //.
Qed.

Lemma big_opM_dom `{Countable K} {A} (f : K M) (m : gmap K A) :
  ([^o map] k_ m, f k) ([^o set] k dom _ m, f k).
Proof.
  induction m as [|i x ?? IH] using map_ind.
  { by rewrite big_opM_eq big_opS_eq dom_empty_L. }
  by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom.
Qed.

Big ops over finite msets

Section gmultiset.
  Context `{Countable A}.
  Implicit Types X : gmultiset A.
  Implicit Types f : A M.

  Lemma big_opMS_gen_proper R f g X :
    Reflexive R Proper (R ==> R ==> R) o
    ( x, x X R (f x) (g x))
    R ([^o mset] x X, f x) ([^o mset] x X, g x).
  Proof.
    rewrite big_opMS_eq. intros ?? Hf. apply (big_opL_gen_proper R); auto.
    intros k x ?%elem_of_list_lookup_2. by apply Hf, gmultiset_elem_of_elements.
  Qed.

  Lemma big_opMS_ext f g X :
    ( x, x X f x = g x)
    ([^o mset] x X, f x) = ([^o mset] x X, g x).
  Proof. apply big_opMS_gen_proper; apply _. Qed.

The lemmas big_opMS_ne and big_opMS_proper are more generic than the instances as they also give x X in the premise.
  Lemma big_opMS_ne f g X n :
    ( x, x X f x ≡{n}≡ g x)
    ([^o mset] x X, f x) ≡{n}≡ ([^o mset] x X, g x).
  Proof. apply big_opMS_gen_proper; apply _. Qed.
  Lemma big_opMS_proper f g X :
    ( x, x X f x g x)
    ([^o mset] x X, f x) ([^o mset] x X, g x).
  Proof. apply big_opMS_gen_proper; apply _. Qed.

  Global Instance big_opMS_ne' n :
    Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opMS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opMS_ne; intros; apply Hf. Qed.
  Global Instance big_opMS_proper' :
    Proper (pointwise_relation _ (≡) ==> (=) ==> (≡)) (big_opMS o (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opMS_proper; intros; apply Hf. Qed.

  Lemma big_opMS_elements f X :
    ([^o mset] x X, f x) [^o list] x elements X, f x.
  Proof. by rewrite big_opMS_eq. Qed.

  Lemma big_opMS_empty f : ([^o mset] x , f x) = monoid_unit.
  Proof. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_empty. Qed.

  Lemma big_opMS_disj_union f X Y :
    ([^o mset] y X Y, f y) ([^o mset] y X, f y) `o` [^o mset] y Y, f y.
  Proof. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_disj_union big_opL_app. Qed.

  Lemma big_opMS_singleton f x : ([^o mset] y {[+ x +]}, f y) f x.
  Proof.
    intros. by rewrite big_opMS_eq /big_opMS_def gmultiset_elements_singleton /= right_id.
  Qed.

  Lemma big_opMS_insert f X x :
    ([^o mset] y {[+ x +]} X, f y) (f x `o` [^o mset] y X, f y).
  Proof. intros. rewrite big_opMS_disj_union big_opMS_singleton //. Qed.

  Lemma big_opMS_delete f X x :
    x X ([^o mset] y X, f y) f x `o` [^o mset] y X {[+ x +]}, f y.
  Proof.
    intros. rewrite -big_opMS_singleton -big_opMS_disj_union.
    by rewrite -gmultiset_disj_union_difference'.
  Qed.

  Lemma big_opMS_unit X : ([^o mset] y X, monoid_unit) (monoid_unit : M).
  Proof.
    by induction X using gmultiset_ind;
      rewrite /= ?big_opMS_disj_union ?big_opMS_singleton ?left_id // big_opMS_eq.
  Qed.

  Lemma big_opMS_op f g X :
    ([^o mset] y X, f y `o` g y) ([^o mset] y X, f y) `o` ([^o mset] y X, g y).
  Proof. by rewrite big_opMS_eq /big_opMS_def -big_opL_op. Qed.
End gmultiset.

Commuting lemmas
Lemma big_opL_opL {A B} (f : nat A nat B M) (l1 : list A) (l2 : list B) :
  ([^o list] k1x1 l1, [^o list] k2x2 l2, f k1 x1 k2 x2)
  ([^o list] k2x2 l2, [^o list] k1x1 l1, f k1 x1 k2 x2).
Proof.
  revert f l2. induction l1 as [|x1 l1 IH]; simpl; intros Φ l2.
  { by rewrite big_opL_unit. }
  by rewrite IH big_opL_op.
Qed.
Lemma big_opL_opM {A} `{Countable K} {B}
    (f : nat A K B M) (l1 : list A) (m2 : gmap K B) :
  ([^o list] k1x1 l1, [^o map] k2x2 m2, f k1 x1 k2 x2)
  ([^o map] k2x2 m2, [^o list] k1x1 l1, f k1 x1 k2 x2).
Proof. repeat setoid_rewrite big_opM_map_to_list. by rewrite big_opL_opL. Qed.
Lemma big_opL_opS {A} `{Countable B}
    (f : nat A B M) (l1 : list A) (X2 : gset B) :
  ([^o list] k1x1 l1, [^o set] x2 X2, f k1 x1 x2)
  ([^o set] x2 X2, [^o list] k1x1 l1, f k1 x1 x2).
Proof. repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL. Qed.
Lemma big_opL_opMS {A} `{Countable B}
    (f : nat A B M) (l1 : list A) (X2 : gmultiset B) :
  ([^o list] k1x1 l1, [^o mset] x2 X2, f k1 x1 x2)
  ([^o mset] x2 X2, [^o list] k1x1 l1, f k1 x1 x2).
Proof. repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL. Qed.

Lemma big_opM_opL {A} `{Countable K} {B}
    (f : K A nat B M) (m1 : gmap K A) (l2 : list B) :
  ([^o map] k1x1 m1, [^o list] k2x2 l2, f k1 x1 k2 x2)
  ([^o list] k2x2 l2, [^o map] k1x1 m1, f k1 x1 k2 x2).
Proof. symmetry. apply big_opL_opM. Qed.
Lemma big_opM_opM `{Countable K1} {A} `{Countable K2} {B}
    (f : K1 A K2 B M) (m1 : gmap K1 A) (m2 : gmap K2 B) :
  ([^o map] k1x1 m1, [^o map] k2x2 m2, f k1 x1 k2 x2)
  ([^o map] k2x2 m2, [^o map] k1x1 m1, f k1 x1 k2 x2).
Proof. repeat setoid_rewrite big_opM_map_to_list. by rewrite big_opL_opL. Qed.
Lemma big_opM_opS `{Countable K} {A} `{Countable B}
    (f : K A B M) (m1 : gmap K A) (X2 : gset B) :
  ([^o map] k1x1 m1, [^o set] x2 X2, f k1 x1 x2)
  ([^o set] x2 X2, [^o map] k1x1 m1, f k1 x1 x2).
Proof.
  repeat setoid_rewrite big_opM_map_to_list.
  repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL.
Qed.
Lemma big_opM_opMS `{Countable K} {A} `{Countable B} (f : K A B M)
    (m1 : gmap K A) (X2 : gmultiset B) :
  ([^o map] k1x1 m1, [^o mset] x2 X2, f k1 x1 x2)
  ([^o mset] x2 X2, [^o map] k1x1 m1, f k1 x1 x2).
Proof.
  repeat setoid_rewrite big_opM_map_to_list.
  repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL.
Qed.

Lemma big_opS_opL `{Countable A} {B}
    (f : A nat B M) (X1 : gset A) (l2 : list B) :
  ([^o set] x1 X1, [^o list] k2x2 l2, f x1 k2 x2)
  ([^o list] k2x2 l2, [^o set] x1 X1, f x1 k2 x2).
Proof. symmetry. apply big_opL_opS. Qed.
Lemma big_opS_opM `{Countable A} `{Countable K} {B}
    (f : A K B M) (X1 : gset A) (m2 : gmap K B) :
  ([^o set] x1 X1, [^o map] k2x2 m2, f x1 k2 x2)
  ([^o map] k2x2 m2, [^o set] x1 X1, f x1 k2 x2).
Proof. symmetry. apply big_opM_opS. Qed.
Lemma big_opS_opS `{Countable A, Countable B}
    (X : gset A) (Y : gset B) (f : A B M) :
  ([^o set] x X, [^o set] y Y, f x y) ([^o set] y Y, [^o set] x X, f x y).
Proof. repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL. Qed.
Lemma big_opS_opMS `{Countable A, Countable B}
    (X : gset A) (Y : gmultiset B) (f : A B M) :
  ([^o set] x X, [^o mset] y Y, f x y) ([^o mset] y Y, [^o set] x X, f x y).
Proof.
  repeat setoid_rewrite big_opS_elements.
  repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL.
Qed.

Lemma big_opMS_opL `{Countable A} {B}
    (f : A nat B M) (X1 : gmultiset A) (l2 : list B) :
  ([^o mset] x1 X1, [^o list] k2x2 l2, f x1 k2 x2)
  ([^o list] k2x2 l2, [^o mset] x1 X1, f x1 k2 x2).
Proof. symmetry. apply big_opL_opMS. Qed.
Lemma big_opMS_opM `{Countable A} `{Countable K} {B} (f : A K B M)
    (X1 : gmultiset A) (m2 : gmap K B) :
  ([^o mset] x1 X1, [^o map] k2x2 m2, f x1 k2 x2)
  ([^o map] k2x2 m2, [^o mset] x1 X1, f x1 k2 x2).
Proof. symmetry. apply big_opM_opMS. Qed.
Lemma big_opMS_opS `{Countable A, Countable B}
    (X : gmultiset A) (Y : gset B) (f : A B M) :
  ([^o mset] x X, [^o set] y Y, f x y) ([^o set] y Y, [^o mset] x X, f x y).
Proof. symmetry. apply big_opS_opMS. Qed.
Lemma big_opMS_opMS `{Countable A, Countable B}
    (X : gmultiset A) (Y : gmultiset B) (f : A B M) :
  ([^o mset] x X, [^o mset] y Y, f x y) ([^o mset] y Y, [^o mset] x X, f x y).
Proof. repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL. Qed.

End big_op.

Section homomorphisms.
  Context `{Monoid M1 o1, Monoid M2 o2}.
  Infix "`o1`" := o1 (at level 50, left associativity).
  Infix "`o2`" := o2 (at level 50, left associativity).
The ssreflect rewrite tactic only works for relations that have a RewriteRelation instance. For the purpose of this section, we want to rewrite with arbitrary relations, so we declare any relation to be a RewriteRelation.
  Local Instance: {A} (R : relation A), RewriteRelation R := {}.

  Lemma big_opL_commute {A} (h : M1 M2) `{!MonoidHomomorphism o1 o2 R h}
      (f : nat A M1) l :
    R (h ([^o1 list] kx l, f k x)) ([^o2 list] kx l, h (f k x)).
  Proof.
    revert f. induction l as [|x l IH]=> f /=.
    - apply monoid_homomorphism_unit.
    - by rewrite monoid_homomorphism IH.
  Qed.
  Lemma big_opL_commute1 {A} (h : M1 M2) `{!WeakMonoidHomomorphism o1 o2 R h}
      (f : nat A M1) l :
    l [] R (h ([^o1 list] kx l, f k x)) ([^o2 list] kx l, h (f k x)).
  Proof.
    intros ?. revert f. induction l as [|x [|x' l'] IH]=> f //.
    - by rewrite !big_opL_singleton.
    - by rewrite !(big_opL_cons _ x) monoid_homomorphism IH.
  Qed.

  Lemma big_opM_commute `{Countable K} {A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 R h} (f : K A M1) m :
    R (h ([^o1 map] kx m, f k x)) ([^o2 map] kx m, h (f k x)).
  Proof.
    intros. induction m as [|i x m ? IH] using map_ind.
    - by rewrite !big_opM_empty monoid_homomorphism_unit.
    - by rewrite !big_opM_insert // monoid_homomorphism -IH.
  Qed.
  Lemma big_opM_commute1 `{Countable K} {A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : K A M1) m :
    m R (h ([^o1 map] kx m, f k x)) ([^o2 map] kx m, h (f k x)).
  Proof.
    intros. induction m as [|i x m ? IH] using map_ind; [done|].
    destruct (decide (m = )) as [->|].
    - by rewrite !big_opM_insert // !big_opM_empty !right_id.
    - by rewrite !big_opM_insert // monoid_homomorphism -IH //.
  Qed.

  Lemma big_opS_commute `{Countable A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 R h} (f : A M1) X :
    R (h ([^o1 set] x X, f x)) ([^o2 set] x X, h (f x)).
  Proof.
    intros. induction X as [|x X ? IH] using set_ind_L.
    - by rewrite !big_opS_empty monoid_homomorphism_unit.
    - by rewrite !big_opS_insert // monoid_homomorphism -IH.
  Qed.
  Lemma big_opS_commute1 `{Countable A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : A M1) X :
    X R (h ([^o1 set] x X, f x)) ([^o2 set] x X, h (f x)).
  Proof.
    intros. induction X as [|x X ? IH] using set_ind_L; [done|].
    destruct (decide (X = )) as [->|].
    - by rewrite !big_opS_insert // !big_opS_empty !right_id.
    - by rewrite !big_opS_insert // monoid_homomorphism -IH //.
  Qed.

  Lemma big_opMS_commute `{Countable A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 R h} (f : A M1) X :
    R (h ([^o1 mset] x X, f x)) ([^o2 mset] x X, h (f x)).
  Proof.
    intros. induction X as [|x X IH] using gmultiset_ind.
    - by rewrite !big_opMS_empty monoid_homomorphism_unit.
    - by rewrite !big_opMS_disj_union !big_opMS_singleton monoid_homomorphism -IH.
  Qed.
  Lemma big_opMS_commute1 `{Countable A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 R h} (f : A M1) X :
    X R (h ([^o1 mset] x X, f x)) ([^o2 mset] x X, h (f x)).
  Proof.
    intros. induction X as [|x X IH] using gmultiset_ind; [done|].
    destruct (decide (X = )) as [->|].
    - by rewrite !big_opMS_disj_union !big_opMS_singleton !big_opMS_empty !right_id.
    - by rewrite !big_opMS_disj_union !big_opMS_singleton monoid_homomorphism -IH //.
  Qed.

  Context `{!LeibnizEquiv M2}.

  Lemma big_opL_commute_L {A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 (≡) h} (f : nat A M1) l :
    h ([^o1 list] kx l, f k x) = ([^o2 list] kx l, h (f k x)).
  Proof using Type×. unfold_leibniz. by apply big_opL_commute. Qed.
  Lemma big_opL_commute1_L {A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : nat A M1) l :
    l [] h ([^o1 list] kx l, f k x) = ([^o2 list] kx l, h (f k x)).
  Proof using Type×. unfold_leibniz. by apply big_opL_commute1. Qed.

  Lemma big_opM_commute_L `{Countable K} {A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 (≡) h} (f : K A M1) m :
    h ([^o1 map] kx m, f k x) = ([^o2 map] kx m, h (f k x)).
  Proof using Type×. unfold_leibniz. by apply big_opM_commute. Qed.
  Lemma big_opM_commute1_L `{Countable K} {A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : K A M1) m :
    m h ([^o1 map] kx m, f k x) = ([^o2 map] kx m, h (f k x)).
  Proof using Type×. unfold_leibniz. by apply big_opM_commute1. Qed.

  Lemma big_opS_commute_L `{Countable A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 (≡) h} (f : A M1) X :
    h ([^o1 set] x X, f x) = ([^o2 set] x X, h (f x)).
  Proof using Type×. unfold_leibniz. by apply big_opS_commute. Qed.
  Lemma big_opS_commute1_L `{ Countable A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : A M1) X :
    X h ([^o1 set] x X, f x) = ([^o2 set] x X, h (f x)).
  Proof using Type×. intros. rewrite <-leibniz_equiv_iff. by apply big_opS_commute1. Qed.

  Lemma big_opMS_commute_L `{Countable A} (h : M1 M2)
      `{!MonoidHomomorphism o1 o2 (≡) h} (f : A M1) X :
    h ([^o1 mset] x X, f x) = ([^o2 mset] x X, h (f x)).
  Proof using Type×. unfold_leibniz. by apply big_opMS_commute. Qed.
  Lemma big_opMS_commute1_L `{Countable A} (h : M1 M2)
      `{!WeakMonoidHomomorphism o1 o2 (≡) h} (f : A M1) X :
    X h ([^o1 mset] x X, f x) = ([^o2 mset] x X, h (f x)).
  Proof using Type×. intros. rewrite <-leibniz_equiv_iff. by apply big_opMS_commute1. Qed.
End homomorphisms.