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.