From algebra Require Export base. From program_logic Require Import ownership. From program_logic Require Export namespaces pviewshifts weakestpre. Import uPred. Local Hint Extern 100 (@eq coPset _ _) => solve_elem_of. Local Hint Extern 100 (@subseteq coPset _ _) => solve_elem_of. Local Hint Extern 100 (_ ∉ _) => solve_elem_of. Local Hint Extern 99 ({[ _ ]} ⊆ _) => apply elem_of_subseteq_singleton. (** Derived forms and lemmas about them. *) Definition inv {Λ Σ} (N : namespace) (P : iProp Λ Σ) : iProp Λ Σ := (∃ i, ■ (i ∈ nclose N) ∧ ownI i P)%I. Instance: Params (@inv) 3. Typeclasses Opaque inv. Section inv. Context {Λ : language} {Σ : iFunctor}. Implicit Types i : positive. Implicit Types N : namespace. Implicit Types P Q R : iProp Λ Σ. Global Instance inv_contractive N : Contractive (@inv Λ Σ N). Proof. intros n ???. apply exists_ne=>i. by apply and_ne, ownI_contractive. Qed. Global Instance inv_always_stable N P : AlwaysStable (inv N P). Proof. rewrite /inv; apply _. Qed. Lemma always_inv N P : (□ inv N P)%I ≡ inv N P. Proof. by rewrite always_always. Qed. (** Invariants can be opened around any frame-shifting assertion. *) Lemma inv_fsa {A} (fsa : FSA Λ Σ A) `{!FrameShiftAssertion fsaV fsa} E N P (Q : A → iProp Λ Σ) R : fsaV → nclose N ⊆ E → R ⊑ inv N P → R ⊑ (▷ P -★ fsa (E ∖ nclose N) (λ a, ▷ P ★ Q a)) → R ⊑ fsa E Q. Proof. intros ? HN Hinv Hinner. rewrite -[R](idemp (∧)%I) {1}Hinv Hinner =>{Hinv Hinner R}. rewrite always_and_sep_l /inv sep_exist_r. apply exist_elim=>i. rewrite always_and_sep_l -assoc. apply const_elim_sep_l=>HiN. rewrite -(fsa_open_close E (E ∖ {[encode i]})) //; last by solve_elem_of+. (* Add this to the local context, so that solve_elem_of finds it. *) assert ({[encode i]} ⊆ nclose N) by eauto. rewrite (always_sep_dup (ownI _ _)). rewrite {1}pvs_openI !pvs_frame_r. apply pvs_mask_frame_mono; [solve_elem_of..|]. rewrite (comm _ (▷_)%I) -assoc wand_elim_r fsa_frame_l. apply fsa_mask_frame_mono; [solve_elem_of..|]. intros a. rewrite assoc -always_and_sep_l pvs_closeI pvs_frame_r left_id. apply pvs_mask_frame'; solve_elem_of. Qed. (* Derive the concrete forms for pvs and wp, because they are useful. *) Lemma pvs_open_close E N P Q R : nclose N ⊆ E → R ⊑ inv N P → R ⊑ (▷P -★ pvs (E ∖ nclose N) (E ∖ nclose N) (▷P ★ Q)) → R ⊑ pvs E E Q. Proof. intros. by apply: (inv_fsa pvs_fsa). Qed. Lemma wp_open_close E e N P (Q : val Λ → iProp Λ Σ) R : atomic e → nclose N ⊆ E → R ⊑ inv N P → R ⊑ (▷ P -★ wp (E ∖ nclose N) e (λ v, ▷P ★ Q v)) → R ⊑ wp E e Q. Proof. intros. by apply: (inv_fsa (wp_fsa e)). Qed. Lemma inv_alloc N P : ▷ P ⊑ pvs N N (inv N P). Proof. by rewrite /inv (pvs_allocI N); last apply coPset_suffixes_infinite. Qed. End inv.