invariants.v 4.3 KB
 Robbert Krebbers committed Feb 04, 2016 1 ``````Require Export algebra.base prelude.countable prelude.co_pset. `````` Ralf Jung committed Feb 08, 2016 2 ``````Require Import program_logic.ownership. `````` Ralf Jung committed Feb 09, 2016 3 ``````Require Export program_logic.pviewshifts program_logic.weakestpre. `````` Ralf Jung committed Feb 09, 2016 4 5 6 7 8 9 ``````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 10 `````` `````` Ralf Jung committed Feb 10, 2016 11 `````` `````` Robbert Krebbers committed Jan 16, 2016 12 13 ``````Definition namespace := list positive. Definition nnil : namespace := nil. `````` Ralf Jung committed Feb 08, 2016 14 15 ``````Definition ndot `{Countable A} (N : namespace) (x : A) : namespace := encode x :: N. `````` Ralf Jung committed Feb 08, 2016 16 ``````Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N). `````` Robbert Krebbers committed Jan 16, 2016 17 18 `````` Instance ndot_injective `{Countable A} : Injective2 (=) (=) (=) (@ndot A _ _). `````` Ralf Jung committed Feb 08, 2016 19 ``````Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed. `````` Robbert Krebbers committed Jan 16, 2016 20 21 ``````Lemma nclose_nnil : nclose nnil = coPset_all. Proof. by apply (sig_eq_pi _). Qed. `````` Ralf Jung committed Feb 08, 2016 22 ``````Lemma encode_nclose N : encode N ∈ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 23 ``````Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed. `````` Ralf Jung committed Feb 08, 2016 24 ``````Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x) ⊆ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 25 26 ``````Proof. intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->]. `````` Ralf Jung committed Feb 08, 2016 27 `````` destruct (list_encode_suffix N (ndot N x)) as [q' ?]; [by exists [encode x]|]. `````` Robbert Krebbers committed Jan 16, 2016 28 29 `````` by exists (q ++ q')%positive; rewrite <-(associative_L _); f_equal. Qed. `````` Ralf Jung committed Feb 08, 2016 30 ``````Lemma ndot_nclose `{Countable A} N x : encode (ndot N x) ∈ nclose N. `````` Robbert Krebbers committed Jan 16, 2016 31 ``````Proof. apply nclose_subseteq with x, encode_nclose. Qed. `````` Ralf Jung committed Feb 08, 2016 32 33 ``````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 34 35 36 37 38 39 40 41 ``````Proof. intros Hxy; apply elem_of_equiv_empty_L=> p; unfold nclose, ndot. rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]]. apply Hxy, (injective encode), (injective encode_nat); revert Hq. rewrite !(list_encode_cons (encode _)). rewrite !(associative_L _) (injective_iff (++ _)%positive) /=. generalize (encode_nat (encode y)). induction (encode_nat (encode x)); intros [|?] ?; f_equal'; naive_solver. `````` Ralf Jung committed Feb 08, 2016 42 43 ``````Qed. `````` Ralf Jung committed Feb 09, 2016 44 45 ``````Local Hint Resolve nclose_subseteq ndot_nclose. `````` Ralf Jung committed Feb 08, 2016 46 47 ``````(** 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 48 49 50 `````` (∃ i, ■ (i ∈ nclose N) ∧ ownI i P)%I. Instance: Params (@inv) 3. Typeclasses Opaque inv. `````` Ralf Jung committed Feb 09, 2016 51 52 53 54 55 56 57 58 `````` 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 59 ``````Proof. intros n ???. apply exists_ne=>i. by apply and_ne, ownI_contractive. Qed. `````` Ralf Jung committed Feb 09, 2016 60 `````` `````` Robbert Krebbers committed Feb 10, 2016 61 62 ``````Global Instance inv_always_stable N P : AlwaysStable (inv N P). Proof. rewrite /inv; apply _. Qed. `````` Ralf Jung committed Feb 09, 2016 63 64 65 66 `````` Lemma always_inv N P : (□ inv N P)%I ≡ inv N P. Proof. by rewrite always_always. Qed. `````` Ralf Jung committed Feb 11, 2016 67 68 69 ``````(** Invariants can be opened around any frame-shifting assertion. *) Lemma inv_fsa {A : Type} {FSA} (FSAs : FrameShiftAssertion (A:=A) FSA) E N P (Q : A → iProp Λ Σ) : `````` Ralf Jung committed Feb 09, 2016 70 `````` nclose N ⊆ E → `````` Ralf Jung committed Feb 11, 2016 71 `````` (inv N P ★ (▷P -★ FSA (E ∖ nclose N) (λ a, ▷P ★ Q a))) ⊑ FSA E Q. `````` Ralf Jung committed Feb 09, 2016 72 ``````Proof. `````` Ralf Jung committed Feb 09, 2016 73 `````` move=>HN. `````` Ralf Jung committed Feb 11, 2016 74 75 `````` rewrite /inv sep_exist_r. apply exist_elim=>i. rewrite always_and_sep_l' -associative. apply const_elim_sep_l=>HiN. `````` Ralf Jung committed Feb 11, 2016 76 `````` rewrite -(fsa_trans3 E (E ∖ {[encode i]})) //; last by solve_elem_of+. `````` Ralf Jung committed Feb 09, 2016 77 `````` (* Add this to the local context, so that solve_elem_of finds it. *) `````` Ralf Jung committed Feb 09, 2016 78 `````` assert ({[encode i]} ⊆ nclose N) by eauto. `````` Ralf Jung committed Feb 11, 2016 79 `````` rewrite (always_sep_dup' (ownI _ _)). `````` Ralf Jung committed Feb 09, 2016 80 `````` rewrite {1}pvs_openI !pvs_frame_r. `````` Ralf Jung committed Feb 09, 2016 81 `````` apply pvs_mask_frame_mono ; [solve_elem_of..|]. `````` Ralf Jung committed Feb 11, 2016 82 83 `````` rewrite (commutative _ (▷_)%I) -associative wand_elim_r fsa_frame_l. apply fsa_mask_frame_mono; [solve_elem_of..|]. intros a. `````` Ralf Jung committed Feb 09, 2016 84 85 86 87 `````` rewrite associative -always_and_sep_l' pvs_closeI pvs_frame_r left_id. apply pvs_mask_frame'; solve_elem_of. Qed. `````` Ralf Jung committed Feb 11, 2016 88 89 90 91 92 93 94 ``````(* Derive the concrete forms for pvs and wp, because they are useful. *) Lemma pvs_open_close E N P Q : nclose N ⊆ E → (inv N P ★ (▷P -★ pvs (E ∖ nclose N) (E ∖ nclose N) (▷P ★ Q))) ⊑ pvs E E Q. Proof. move=>HN. by rewrite (inv_fsa pvs_fsa). Qed. `````` Ralf Jung committed Feb 09, 2016 95 ``````Lemma wp_open_close E e N P (Q : val Λ → iProp Λ Σ) : `````` Ralf Jung committed Feb 09, 2016 96 `````` atomic e → nclose N ⊆ E → `````` Ralf Jung committed Feb 11, 2016 97 `````` (inv N P ★ (▷P -★ wp (E ∖ nclose N) e (λ v, ▷P ★ Q v))) ⊑ wp E e Q. `````` Ralf Jung committed Feb 09, 2016 98 ``````Proof. `````` Ralf Jung committed Feb 11, 2016 99 `````` move=>He HN. by rewrite (inv_fsa (wp_fsa e _)). Qed. `````` Ralf Jung committed Feb 09, 2016 100 `````` `````` Ralf Jung committed Feb 10, 2016 101 ``````Lemma inv_alloc N P : ▷ P ⊑ pvs N N (inv N P). `````` Robbert Krebbers committed Feb 10, 2016 102 ``````Proof. by rewrite /inv (pvs_allocI N); last apply coPset_suffixes_infinite. Qed. `````` Ralf Jung committed Feb 09, 2016 103 104 `````` End inv.``````