invariants.v 4.42 KB
 Robbert Krebbers committed Feb 13, 2016 1 2 3 4 ``````From algebra Require Export base. From prelude Require Export countable co_pset. From program_logic Require Import ownership. From program_logic Require Export pviewshifts weakestpre. `````` Ralf Jung committed Feb 09, 2016 5 6 7 8 9 10 ``````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. `````` Robbert Krebbers committed Jan 16, 2016 11 `````` `````` Ralf Jung committed Feb 10, 2016 12 `````` `````` Robbert Krebbers committed Jan 16, 2016 13 14 ``````Definition namespace := list positive. Definition nnil : namespace := nil. `````` Ralf Jung committed Feb 08, 2016 15 16 ``````Definition ndot `{Countable A} (N : namespace) (x : A) : namespace := encode x :: N. `````` Ralf Jung committed Feb 08, 2016 17 ``````Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N). `````` Robbert Krebbers committed Jan 16, 2016 18 `````` `````` Robbert Krebbers committed Feb 11, 2016 19 ``````Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _). `````` Ralf Jung committed Feb 08, 2016 20 ``````Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed. `````` Robbert Krebbers committed Jan 16, 2016 21 22 ``````Lemma nclose_nnil : nclose nnil = coPset_all. Proof. by apply (sig_eq_pi _). Qed. `````` Ralf Jung committed Feb 08, 2016 23 ``````Lemma encode_nclose N : encode N ∈ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 24 ``````Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed. `````` Ralf Jung committed Feb 08, 2016 25 ``````Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x) ⊆ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 26 27 ``````Proof. intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->]. `````` Ralf Jung committed Feb 08, 2016 28 `````` destruct (list_encode_suffix N (ndot N x)) as [q' ?]; [by exists [encode x]|]. `````` Robbert Krebbers committed Feb 11, 2016 29 `````` by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal. `````` Robbert Krebbers committed Jan 16, 2016 30 ``````Qed. `````` Ralf Jung committed Feb 08, 2016 31 ``````Lemma ndot_nclose `{Countable A} N x : encode (ndot N x) ∈ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 32 ``````Proof. apply nclose_subseteq with x, encode_nclose. Qed. `````` Ralf Jung committed Feb 08, 2016 33 34 ``````Lemma nclose_disjoint `{Countable A} N (x y : A) : x ≠ y → nclose (ndot N x) ∩ nclose (ndot N y) = ∅. `````` Robbert Krebbers committed Jan 16, 2016 35 36 37 ``````Proof. intros Hxy; apply elem_of_equiv_empty_L=> p; unfold nclose, ndot. rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]]. `````` Robbert Krebbers committed Feb 11, 2016 38 `````` apply Hxy, (inj encode), (inj encode_nat); revert Hq. `````` Robbert Krebbers committed Jan 16, 2016 39 `````` rewrite !(list_encode_cons (encode _)). `````` Robbert Krebbers committed Feb 11, 2016 40 `````` rewrite !(assoc_L _) (inj_iff (++ _)%positive) /=. `````` Robbert Krebbers committed Jan 16, 2016 41 42 `````` generalize (encode_nat (encode y)). induction (encode_nat (encode x)); intros [|?] ?; f_equal'; naive_solver. `````` Ralf Jung committed Feb 08, 2016 43 44 ``````Qed. `````` Ralf Jung committed Feb 09, 2016 45 46 ``````Local Hint Resolve nclose_subseteq ndot_nclose. `````` Ralf Jung committed Feb 08, 2016 47 48 ``````(** Derived forms and lemmas about them. *) Definition inv {Λ Σ} (N : namespace) (P : iProp Λ Σ) : iProp Λ Σ := `````` Robbert Krebbers committed Feb 10, 2016 49 50 51 `````` (∃ i, ■ (i ∈ nclose N) ∧ ownI i P)%I. Instance: Params (@inv) 3. Typeclasses Opaque inv. `````` Ralf Jung committed Feb 09, 2016 52 53 54 55 56 57 58 59 `````` 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). `````` Robbert Krebbers committed Feb 10, 2016 60 ``````Proof. intros n ???. apply exists_ne=>i. by apply and_ne, ownI_contractive. Qed. `````` Ralf Jung committed Feb 09, 2016 61 `````` `````` Robbert Krebbers committed Feb 10, 2016 62 63 ``````Global Instance inv_always_stable N P : AlwaysStable (inv N P). Proof. rewrite /inv; apply _. Qed. `````` Ralf Jung committed Feb 09, 2016 64 65 66 67 `````` Lemma always_inv N P : (□ inv N P)%I ≡ inv N P. Proof. by rewrite always_always. Qed. `````` Ralf Jung committed Feb 11, 2016 68 69 ``````(** Invariants can be opened around any frame-shifting assertion. *) Lemma inv_fsa {A : Type} {FSA} (FSAs : FrameShiftAssertion (A:=A) FSA) `````` Ralf Jung committed Feb 13, 2016 70 `````` E N P (Q : A → iProp Λ Σ) R : `````` Ralf Jung committed Feb 09, 2016 71 `````` nclose N ⊆ E → `````` Ralf Jung committed Feb 13, 2016 72 73 74 `````` R ⊑ inv N P → R ⊑ (▷P -★ FSA (E ∖ nclose N) (λ a, ▷P ★ Q a)) → R ⊑ FSA E Q. `````` Ralf Jung committed Feb 09, 2016 75 ``````Proof. `````` Ralf Jung committed Feb 13, 2016 76 77 `````` move=>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. `````` Ralf Jung committed Feb 12, 2016 78 `````` rewrite always_and_sep_l -assoc. apply const_elim_sep_l=>HiN. `````` Ralf Jung committed Feb 11, 2016 79 `````` rewrite -(fsa_trans3 E (E ∖ {[encode i]})) //; last by solve_elem_of+. `````` Ralf Jung committed Feb 09, 2016 80 `````` (* Add this to the local context, so that solve_elem_of finds it. *) `````` Ralf Jung committed Feb 09, 2016 81 `````` assert ({[encode i]} ⊆ nclose N) by eauto. `````` Ralf Jung committed Feb 12, 2016 82 `````` rewrite (always_sep_dup (ownI _ _)). `````` Ralf Jung committed Feb 09, 2016 83 `````` rewrite {1}pvs_openI !pvs_frame_r. `````` Ralf Jung committed Feb 09, 2016 84 `````` apply pvs_mask_frame_mono ; [solve_elem_of..|]. `````` Robbert Krebbers committed Feb 11, 2016 85 `````` rewrite (comm _ (▷_)%I) -assoc wand_elim_r fsa_frame_l. `````` Ralf Jung committed Feb 11, 2016 86 `````` apply fsa_mask_frame_mono; [solve_elem_of..|]. intros a. `````` Ralf Jung committed Feb 12, 2016 87 `````` rewrite assoc -always_and_sep_l pvs_closeI pvs_frame_r left_id. `````` Ralf Jung committed Feb 09, 2016 88 89 90 `````` apply pvs_mask_frame'; solve_elem_of. Qed. `````` Ralf Jung committed Feb 11, 2016 91 92 ``````(* Derive the concrete forms for pvs and wp, because they are useful. *) `````` Ralf Jung committed Feb 13, 2016 93 ``````Lemma pvs_open_close E N P Q R : `````` Ralf Jung committed Feb 11, 2016 94 `````` nclose N ⊆ E → `````` Ralf Jung committed Feb 13, 2016 95 96 97 98 `````` R ⊑ inv N P → R ⊑ (▷P -★ pvs (E ∖ nclose N) (E ∖ nclose N) (▷P ★ Q)) → R ⊑ pvs E E Q. Proof. move=>HN ? ?. apply: (inv_fsa pvs_fsa); eassumption. Qed. `````` Ralf Jung committed Feb 11, 2016 99 `````` `````` Ralf Jung committed Feb 13, 2016 100 ``````Lemma wp_open_close E e N P (Q : val Λ → iProp Λ Σ) R : `````` Ralf Jung committed Feb 09, 2016 101 `````` atomic e → nclose N ⊆ E → `````` Ralf Jung committed Feb 13, 2016 102 103 104 `````` R ⊑ inv N P → R ⊑ (▷P -★ wp (E ∖ nclose N) e (λ v, ▷P ★ Q v)) → R ⊑ wp E e Q. `````` Ralf Jung committed Feb 09, 2016 105 ``````Proof. `````` Ralf Jung committed Feb 13, 2016 106 `````` move=>He HN ? ?. apply: (inv_fsa (wp_fsa e _)); eassumption. Qed. `````` Ralf Jung committed Feb 09, 2016 107 `````` `````` Ralf Jung committed Feb 10, 2016 108 ``````Lemma inv_alloc N P : ▷ P ⊑ pvs N N (inv N P). `````` Robbert Krebbers committed Feb 10, 2016 109 ``````Proof. by rewrite /inv (pvs_allocI N); last apply coPset_suffixes_infinite. Qed. `````` Ralf Jung committed Feb 09, 2016 110 111 `````` End inv.``````