invariants.v 4.42 KB
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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.
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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.
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Definition namespace := list positive.
Definition nnil : namespace := nil.
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Definition ndot `{Countable A} (N : namespace) (x : A) : namespace :=
  encode x :: N.
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Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N).
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Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
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Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed.
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Lemma nclose_nnil : nclose nnil = coPset_all.
Proof. by apply (sig_eq_pi _). Qed.
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Lemma encode_nclose N : encode N  nclose N.
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Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed.
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Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x)  nclose N.
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Proof.
  intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->].
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  destruct (list_encode_suffix N (ndot N x)) as [q' ?]; [by exists [encode x]|].
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  by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal.
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Qed.
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Lemma ndot_nclose `{Countable A} N x : encode (ndot N x)  nclose N.
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Proof. apply nclose_subseteq with x, encode_nclose. Qed.
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Lemma nclose_disjoint `{Countable A} N (x y : A) :
  x  y  nclose (ndot N x)  nclose (ndot N y) = .
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Proof.
  intros Hxy; apply elem_of_equiv_empty_L=> p; unfold nclose, ndot.
  rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]].
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  apply Hxy, (inj encode), (inj encode_nat); revert Hq.
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  rewrite !(list_encode_cons (encode _)).
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  rewrite !(assoc_L _) (inj_iff (++ _)%positive) /=.
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  generalize (encode_nat (encode y)).
  induction (encode_nat (encode x)); intros [|?] ?; f_equal'; naive_solver.
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Qed.

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Local Hint Resolve nclose_subseteq ndot_nclose.

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(** Derived forms and lemmas about them. *)
Definition inv {Λ Σ} (N : namespace) (P : iProp Λ Σ) : iProp Λ Σ :=
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  ( i,  (i  nclose N)  ownI i P)%I.
Instance: Params (@inv) 3.
Typeclasses Opaque inv.
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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).
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Proof. intros n ???. apply exists_ne=>i. by apply and_ne, ownI_contractive. Qed.
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Global Instance inv_always_stable N P : AlwaysStable (inv N P).
Proof. rewrite /inv; apply _. Qed.
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Lemma always_inv N P : ( inv N P)%I  inv N P.
Proof. by rewrite always_always. Qed.

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(** Invariants can be opened around any frame-shifting assertion. *)
Lemma inv_fsa {A : Type} {FSA} (FSAs : FrameShiftAssertion (A:=A) FSA)
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      E N P (Q : A  iProp Λ Σ) R :
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  nclose N  E 
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  R  inv N P 
  R  (P - FSA (E  nclose N) (λ a, P  Q a)) 
  R  FSA E Q.
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Proof.
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  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.
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  rewrite always_and_sep_l -assoc. apply const_elim_sep_l=>HiN.
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  rewrite -(fsa_trans3 E (E  {[encode i]})) //; last by solve_elem_of+.
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  (* Add this to the local context, so that solve_elem_of finds it. *)
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  assert ({[encode i]}  nclose N) by eauto.
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  rewrite (always_sep_dup (ownI _ _)).
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  rewrite {1}pvs_openI !pvs_frame_r.
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  apply pvs_mask_frame_mono ; [solve_elem_of..|].
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  rewrite (comm _ (_)%I) -assoc wand_elim_r fsa_frame_l.
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  apply fsa_mask_frame_mono; [solve_elem_of..|]. intros a.
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  rewrite assoc -always_and_sep_l pvs_closeI pvs_frame_r left_id.
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  apply pvs_mask_frame'; solve_elem_of.
Qed.

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(* Derive the concrete forms for pvs and wp, because they are useful. *)

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Lemma pvs_open_close E N P Q R :
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  nclose N  E 
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  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.
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Lemma wp_open_close E e N P (Q : val Λ  iProp Λ Σ) R :
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  atomic e  nclose N  E 
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  R  inv N P 
  R  (P - wp (E  nclose N) e (λ v, P  Q v)) 
  R  wp E e Q.
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Proof.
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  move=>He HN ? ?. apply: (inv_fsa (wp_fsa e _)); eassumption. Qed.
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Lemma inv_alloc N P :  P  pvs N N (inv N P).
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Proof. by rewrite /inv (pvs_allocI N); last apply coPset_suffixes_infinite. Qed.
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End inv.