invariants.v 4.87 KB
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Require Export algebra.base prelude.countable prelude.co_pset.
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Require Import program_logic.ownership.
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Require Export program_logic.pviewshifts program_logic.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_injective `{Countable A} : Injective2 (=) (=) (=) (@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 <-(associative_L _); f_equal.
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]].
  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.
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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 : positive, (i  nclose N)  ownI i P)%I.
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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).
Proof.
  intros n ? ? EQ. apply exists_ne=>i.
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  apply and_ne; first done.
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  by apply ownI_contractive.
Qed.

Global Instance inv_always_stable N P : AlwaysStable (inv N P) := _.

Lemma always_inv N P : ( inv N P)%I  inv N P.
Proof. by rewrite always_always. Qed.

(* We actually pretty much lose the abolity to deal with mask-changing view
   shifts when using `inv`. This is because we cannot exactly name the invariants
   any more. But that's okay; all this means is that sugar like the atomic
   triples will have to prove its own version of the open_close rule
   by unfolding `inv`. *)
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(* TODO Can we prove something that helps for both open_close lemmas? *)
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Lemma pvs_open_close E N P Q :
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  nclose N  E 
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  (inv N P  (P - pvs (E  nclose N) (E  nclose N) (P  Q)))  pvs E E Q.
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Proof.
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  move=>HN.
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  rewrite /inv and_exist_r. apply exist_elim=>i.
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  rewrite -associative. apply const_elim_l=>HiN.
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  rewrite -(pvs_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_and_sep_l' (always_sep_dup' (ownI _ _)).
  rewrite {1}pvs_openI !pvs_frame_r.
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  apply pvs_mask_frame_mono ; [solve_elem_of..|].
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  rewrite (commutative _ (_)%I) -associative wand_elim_r pvs_frame_l.
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  apply pvs_mask_frame_mono; [solve_elem_of..|].
  rewrite associative -always_and_sep_l' pvs_closeI pvs_frame_r left_id.
  apply pvs_mask_frame'; solve_elem_of.
Qed.

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Lemma wp_open_close E e N P (Q : val Λ  iProp Λ Σ) :
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  atomic e  nclose N  E 
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  (inv N P  (P - wp (E  nclose N) e (λ v, P  Q v)))%I  wp E e Q.
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Proof.
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  move=>He HN.
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  rewrite /inv and_exist_r. apply exist_elim=>i.
  rewrite -associative. apply const_elim_l=>HiN.
  rewrite -(wp_atomic 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_and_sep_l' (always_sep_dup' (ownI _ _)).
  rewrite {1}pvs_openI !pvs_frame_r.
  apply pvs_mask_frame_mono; [solve_elem_of..|].
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  rewrite (commutative _ (_)%I) -associative wand_elim_r wp_frame_l.
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  apply wp_mask_frame_mono; [solve_elem_of..|]=>v.
  rewrite associative -always_and_sep_l' pvs_closeI pvs_frame_r left_id.
  apply pvs_mask_frame'; solve_elem_of.
Qed.

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Lemma pvs_alloc N P :  P  pvs N N (inv N P).
Proof.
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  rewrite /inv (pvs_allocI N); first done.
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  apply coPset_suffixes_infinite.
Qed.
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End inv.