upred_big_op.v

From algebra Require Export upred.
From prelude Require Import fin_maps fin_collections.

(** * Big ops over lists *)
(* These are the basic building blocks for other big ops *)
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.

(** * Other big ops *)
(** We use a type class to obtain overloaded notations *)
Definition uPred_big_sepM {M} `{FinMapToList K A MA}
    (m : MA) (P : K → A → uPred M) : uPred M :=
  uPred_big_sep (curry P <$> map_to_list m).
Notation "'Π★{map' m } P" := (uPred_big_sepM m P)
  (at level 20, m at level 10, format "Π★{map  m }  P") : uPred_scope.

Definition uPred_big_sepS {M} `{Elements A C}
  (X : C) (P : A → uPred M) : uPred M := uPred_big_sep (P <$> elements X).
Notation "'Π★{set' X } P" := (uPred_big_sepS X P)
  (at level 20, X at level 10, format "Π★{set  X }  P") : uPred_scope.

(** * Always stability for lists *)
Class AlwaysStableL {M} (Ps : list (uPred M)) :=
  always_stableL : Forall AlwaysStable Ps.
Arguments always_stableL {_} _ {_}.

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

(* Big ops *)
Global Instance big_and_proper : Proper ((≡) ==> (≡)) (@uPred_big_and M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
Global Instance big_sep_proper : Proper ((≡) ==> (≡)) (@uPred_big_sep M).
Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
Global Instance big_and_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_and M).
Proof.
  induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
  * by rewrite IH.
  * by rewrite !assoc (comm _ P).
  * etransitivity; eauto.
Qed.
Global Instance big_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_sep M).
Proof.
  induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
  * by rewrite IH.
  * by rewrite !assoc (comm _ P).
  * etransitivity; eauto.
Qed.
Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I.
Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
Lemma big_sep_app Ps Qs : (Π★ (Ps ++ Qs))%I ≡ (Π★ Ps ★ Π★ Qs)%I.
Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
Lemma big_sep_and Ps : (Π★ Ps) ⊑ (Π∧ Ps).
Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed.
Lemma big_and_elem_of Ps P : P ∈ Ps → (Π∧ Ps) ⊑ P.
Proof. induction 1; simpl; auto with I. Qed.
Lemma big_sep_elem_of Ps P : P ∈ Ps → (Π★ Ps) ⊑ P.
Proof. induction 1; simpl; auto with I. Qed.

(* Big ops over finite maps *)
Section fin_map.
  Context `{FinMap K Ma} {A} (P : K → A → uPred M).

  Lemma big_sepM_empty : (Π★{map ∅} P)%I ≡ True%I.
  Proof. by rewrite /uPred_big_sep /uPred_big_sepM map_to_list_empty. Qed.
  Lemma big_sepM_insert (m : Ma 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_sep /uPred_big_sepM map_to_list_insert.
  Qed.
  Lemma big_sepM_singleton i x : (Π★{map {[i := x]}} P)%I ≡ (P i x)%I.
  Proof.
    rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty.
    by rewrite big_sepM_empty right_id.
  Qed.
End fin_map.

(* Always stable *)
Local Notation AS := AlwaysStable.
Local Notation ASL := AlwaysStableL.
Global Instance big_and_always_stable Ps : ASL Ps → AS (Π∧ Ps).
Proof. induction 1; apply _. Qed.
Global Instance big_sep_always_stable Ps : ASL Ps → AS (Π★ Ps).
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.
End big_op.