upred_big_op.v 9.46 KB
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From algebra Require Export upred.
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From prelude Require Import gmap fin_collections.
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Import uPred.
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(** * Big ops over lists *)
(* These are the basic building blocks for other big ops *)
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Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M:=
  match Ps with [] => True | P :: Ps => P  uPred_big_and Ps end%I.
Instance: Params (@uPred_big_and) 1.
Notation "'Π∧' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope.
Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) : uPred M :=
  match Ps with [] => True | P :: Ps => P  uPred_big_sep Ps end%I.
Instance: Params (@uPred_big_sep) 1.
Notation "'Π★' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope.
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(** * Other big ops *)
(** We use a type class to obtain overloaded notations *)
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Definition uPred_big_sepM {M} `{Countable K} {A}
    (m : gmap K A) (P : K  A  uPred M) : uPred M :=
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  uPred_big_sep (curry P <$> map_to_list m).
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Instance: Params (@uPred_big_sepM) 6.
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Notation "'Π★{map' m } P" := (uPred_big_sepM m P)
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  (at level 20, m at level 10, format "Π★{map  m }  P") : uPred_scope.
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Definition uPred_big_sepS {M} `{Countable A}
  (X : gset A) (P : A  uPred M) : uPred M := uPred_big_sep (P <$> elements X).
Instance: Params (@uPred_big_sepS) 5.
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Notation "'Π★{set' X } P" := (uPred_big_sepS X P)
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  (at level 20, X at level 10, format "Π★{set  X }  P") : uPred_scope.
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(** * Always stability for lists *)
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Class AlwaysStableL {M} (Ps : list (uPred M)) :=
  always_stableL : Forall AlwaysStable Ps.
Arguments always_stableL {_} _ {_}.

Section big_op.
Context {M : cmraT}.
Implicit Types Ps Qs : list (uPred M).
Implicit Types A : Type.

(* Big ops *)
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Global Instance big_and_proper : Proper (() ==> ()) (@uPred_big_and M).
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Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
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Global Instance big_sep_proper : Proper (() ==> ()) (@uPred_big_sep M).
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Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
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Global Instance big_and_ne n :
  Proper (Forall2 (dist n) ==> dist n) (@uPred_big_and M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
Global Instance big_sep_ne n :
  Proper (Forall2 (dist n) ==> dist n) (@uPred_big_sep M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

Global Instance big_and_mono' : Proper (Forall2 () ==> ()) (@uPred_big_and M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
Global Instance big_sep_mono' : Proper (Forall2 () ==> ()) (@uPred_big_sep M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

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Global Instance big_and_perm : Proper (() ==> ()) (@uPred_big_and M).
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Proof.
  induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
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  - by rewrite IH.
  - by rewrite !assoc (comm _ P).
  - etransitivity; eauto.
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Qed.
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Global Instance big_sep_perm : Proper (() ==> ()) (@uPred_big_sep M).
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Proof.
  induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
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  - by rewrite IH.
  - by rewrite !assoc (comm _ P).
  - etransitivity; eauto.
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Qed.
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Lemma big_and_app Ps Qs : (Π (Ps ++ Qs))%I  (Π Ps  Π Qs)%I.
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Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
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Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I  (Π★ Ps  Π★ Qs)%I.
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Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
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Lemma big_and_contains Ps Qs : Qs `contains` Ps  (Π Ps)%I  (Π Qs)%I.
Proof.
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  intros [Ps' ->]%contains_Permutation. by rewrite big_and_app and_elim_l.
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Qed.
Lemma big_sep_contains Ps Qs : Qs `contains` Ps  (Π★ Ps)%I  (Π★ Qs)%I.
Proof.
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  intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app sep_elim_l.
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Qed.

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Lemma big_sep_and Ps : (Π★ Ps)  (Π Ps).
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Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed.
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Lemma big_and_elem_of Ps P : P  Ps  (Π Ps)  P.
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Proof. induction 1; simpl; auto with I. Qed.
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Lemma big_sep_elem_of Ps P : P  Ps  (Π★ Ps)  P.
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Proof. induction 1; simpl; auto with I. Qed.

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(* Big ops over finite maps *)
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Section gmap.
  Context `{Countable K} {A : Type}.
  Implicit Types m : gmap K A.
  Implicit Types P : K  A  uPred M.
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  Lemma big_sepM_mono P Q m1 m2 :
    m2  m1  ( x k, m2 !! k = Some x  P k x  Q k x) 
    (Π★{map m1} P)  (Π★{map m2} Q).
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  Proof.
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    intros HX HP. transitivity (Π★{map m2} P)%I.
    - by apply big_sep_contains, fmap_contains, map_to_list_contains.
    - apply big_sep_mono', Forall2_fmap, Forall2_Forall.
      apply Forall_forall=> -[i x] ? /=. by apply HP, elem_of_map_to_list.
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  Qed.
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  Global Instance big_sepM_ne m n :
    Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n))
           (uPred_big_sepM (M:=M) m).
  Proof.
    intros P1 P2 HP. apply big_sep_ne, Forall2_fmap.
    apply Forall2_Forall, Forall_true=> -[i x]; apply HP.
  Qed.
  Global Instance big_sepM_proper m :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> ())
           (uPred_big_sepM (M:=M) m).
  Proof.
    intros P1 P2 HP; apply equiv_dist=> n.
    apply big_sepM_ne=> k x; apply equiv_dist, HP.
  Qed.
  Global Instance big_sepM_mono' m :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> ())
           (uPred_big_sepM (M:=M) m).
  Proof. intros P1 P2 HP. apply big_sepM_mono; intros; [done|apply HP]. Qed.

  Lemma big_sepM_empty P : (Π★{map } P)%I  True%I.
  Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed.
  Lemma big_sepM_insert P (m : gmap K A) i x :
    m !! i = None  (Π★{map <[i:=x]> m} P)%I  (P i x  Π★{map m} P)%I.
  Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed.
  Lemma big_sepM_singleton P i x : (Π★{map {[i := x]}} P)%I  (P i x)%I.
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  Proof.
    rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty.
    by rewrite big_sepM_empty right_id.
  Qed.
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  Lemma big_sepM_sepM P Q m :
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    (Π★{map m} (λ i x, P i x  Q i x))%I  (Π★{map m} P  Π★{map m} Q)%I.
  Proof.
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    rewrite /uPred_big_sepM.
    induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //.
    by rewrite IH -!assoc (assoc _ (Q _ _)) [(Q _ _  _)%I]comm -!assoc.
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  Qed.
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  Lemma big_sepM_later P m : ( Π★{map m} P)%I  (Π★{map m} (λ i x,  P i x))%I.
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  Proof.
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    rewrite /uPred_big_sepM.
    induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //.
    by rewrite later_sep IH.
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  Qed.
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End gmap.

(* Big ops over finite sets *)
Section gset.
  Context `{Countable A}.
  Implicit Types X : gset A.
  Implicit Types P : A  uPred M.

  Lemma big_sepS_mono P Q X Y :
    Y  X  ( x, x  Y  P x  Q x)  (Π★{set X} P)  (Π★{set Y} Q).
  Proof.
    intros HX HP. transitivity (Π★{set Y} P)%I.
    - by apply big_sep_contains, fmap_contains, elements_contains.
    - apply big_sep_mono', Forall2_fmap, Forall2_Forall.
      apply Forall_forall=> x ? /=. by apply HP, elem_of_elements.
  Qed.

  Lemma big_sepS_ne X n :
    Proper (pointwise_relation _ (dist n) ==> dist n) (uPred_big_sepS (M:=M) X).
  Proof.
    intros P1 P2 HP. apply big_sep_ne, Forall2_fmap.
    apply Forall2_Forall, Forall_true=> x; apply HP.
  Qed.
  Lemma big_sepS_proper X :
    Proper (pointwise_relation _ () ==> ()) (uPred_big_sepS (M:=M) X).
  Proof.
    intros P1 P2 HP; apply equiv_dist=> n.
    apply big_sepS_ne=> x; apply equiv_dist, HP.
  Qed.
  Lemma big_sepS_mono' X :
    Proper (pointwise_relation _ () ==> ()) (uPred_big_sepS (M:=M) X).
  Proof. intros P1 P2 HP. apply big_sepS_mono; naive_solver. Qed.

  Lemma big_sepS_empty P : (Π★{set } P)%I  True%I.
  Proof. by rewrite /uPred_big_sepS elements_empty. Qed.
  Lemma big_sepS_insert P X x :
    x  X  (Π★{set {[ x ]}  X} P)%I  (P x  Π★{set X} P)%I.
  Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed.
  Lemma big_sepS_delete P X x :
    x  X  (Π★{set X} P)%I  (P x  Π★{set X  {[ x ]}} P)%I.
  Proof.
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    intros. rewrite -big_sepS_insert; last set_solver.
    by rewrite -union_difference_L; last set_solver.
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  Qed.
  Lemma big_sepS_singleton P x : (Π★{set {[ x ]}} P)%I  (P x)%I.
  Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed.
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  Lemma big_sepS_sepS P Q X :
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    (Π★{set X} (λ x, P x  Q x))%I  (Π★{set X} P  Π★{set X} Q)%I.
  Proof.
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    rewrite /uPred_big_sepS.
    induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id.
    by rewrite IH -!assoc (assoc _ (Q _)) [(Q _  _)%I]comm -!assoc.
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  Qed.

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  Lemma big_sepS_later P X : ( Π★{set X} P)%I  (Π★{set X} (λ x,  P x))%I.
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  Proof.
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    rewrite /uPred_big_sepS.
    induction (elements X) as [|x l IH]; csimpl; first by rewrite ?later_True.
    by rewrite later_sep IH.
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  Qed.
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End gset.
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(* Always stable *)
Local Notation AS := AlwaysStable.
Local Notation ASL := AlwaysStableL.
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Global Instance big_and_always_stable Ps : ASL Ps  AS (Π Ps).
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Proof. induction 1; apply _. Qed.
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Global Instance big_sep_always_stable Ps : ASL Ps  AS (Π★ Ps).
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Proof. induction 1; apply _. Qed.

Global Instance nil_always_stable : ASL (@nil (uPred M)).
Proof. constructor. Qed.
Global Instance cons_always_stable P Ps : AS P  ASL Ps  ASL (P :: Ps).
Proof. by constructor. Qed.
Global Instance app_always_stable Ps Ps' : ASL Ps  ASL Ps'  ASL (Ps ++ Ps').
Proof. apply Forall_app_2. Qed.
Global Instance zip_with_always_stable {A B} (f : A  B  uPred M) xs ys :
  ( x y, AS (f x y))  ASL (zip_with f xs ys).
Proof. unfold ASL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed.
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End big_op.