invariants.v 4.41 KB
Newer Older
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's avatar
Ralf Jung committed
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's avatar
Robbert Krebbers committed
11

12

Robbert Krebbers's avatar
Robbert Krebbers committed
13
14
Definition namespace := list positive.
Definition nnil : namespace := nil.
15
16
Definition ndot `{Countable A} (N : namespace) (x : A) : namespace :=
  encode x :: N.
Ralf Jung's avatar
Ralf Jung committed
17
Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N).
Robbert Krebbers's avatar
Robbert Krebbers committed
18

19
Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
20
Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
21
22
Lemma nclose_nnil : nclose nnil = coPset_all.
Proof. by apply (sig_eq_pi _). Qed.
23
Lemma encode_nclose N : encode N  nclose N.
Robbert Krebbers's avatar
Robbert Krebbers committed
24
Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed.
25
Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x)  nclose N.
Robbert Krebbers's avatar
Robbert Krebbers committed
26
27
Proof.
  intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->].
28
  destruct (list_encode_suffix N (ndot N x)) as [q' ?]; [by exists [encode x]|].
29
  by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal.
Robbert Krebbers's avatar
Robbert Krebbers committed
30
Qed.
31
Lemma ndot_nclose `{Countable A} N x : encode (ndot N x)  nclose N.
Robbert Krebbers's avatar
Robbert Krebbers committed
32
Proof. apply nclose_subseteq with x, encode_nclose. Qed.
33
34
Lemma nclose_disjoint `{Countable A} N (x y : A) :
  x  y  nclose (ndot N x)  nclose (ndot N y) = .
Robbert Krebbers's avatar
Robbert Krebbers committed
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]].
38
  apply Hxy, (inj encode), (inj encode_nat); revert Hq.
Robbert Krebbers's avatar
Robbert Krebbers committed
39
  rewrite !(list_encode_cons (encode _)).
40
  rewrite !(assoc_L _) (inj_iff (++ _)%positive) /=.
Robbert Krebbers's avatar
Robbert Krebbers committed
41
42
  generalize (encode_nat (encode y)).
  induction (encode_nat (encode x)); intros [|?] ?; f_equal'; naive_solver.
43
44
Qed.

Ralf Jung's avatar
Ralf Jung committed
45
46
Local Hint Resolve nclose_subseteq ndot_nclose.

47
48
(** Derived forms and lemmas about them. *)
Definition inv {Λ Σ} (N : namespace) (P : iProp Λ Σ) : iProp Λ Σ :=
49
50
51
  ( i,  (i  nclose N)  ownI i P)%I.
Instance: Params (@inv) 3.
Typeclasses Opaque inv.
Ralf Jung's avatar
Ralf Jung committed
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).
60
Proof. intros n ???. apply exists_ne=>i. by apply and_ne, ownI_contractive. Qed.
Ralf Jung's avatar
Ralf Jung committed
61

62
63
Global Instance inv_always_stable N P : AlwaysStable (inv N P).
Proof. rewrite /inv; apply _. Qed.
Ralf Jung's avatar
Ralf Jung committed
64
65
66
67

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

68
(** Invariants can be opened around any frame-shifting assertion. *)
69
70
71
Lemma inv_fsa {A} (fsa : FSA Λ Σ A) `{!FrameShiftAssertion fsaV fsa}
    E N P (Q : A  iProp Λ Σ) R :
  fsaV 
Ralf Jung's avatar
Ralf Jung committed
72
  nclose N  E 
73
  R  inv N P 
74
75
  R  ( P - fsa (E  nclose N) (λ a,  P  Q a)) 
  R  fsa E Q.
Ralf Jung's avatar
Ralf Jung committed
76
Proof.
77
78
  intros ? HN Hinv Hinner.
  rewrite -[R](idemp ()%I) {1}Hinv Hinner =>{Hinv Hinner R}.
79
  rewrite always_and_sep_l /inv sep_exist_r. apply exist_elim=>i.
80
  rewrite always_and_sep_l -assoc. apply const_elim_sep_l=>HiN.
81
  rewrite -(fsa_open_close E (E  {[encode i]})) //; last by solve_elem_of+.
Ralf Jung's avatar
Ralf Jung committed
82
  (* Add this to the local context, so that solve_elem_of finds it. *)
83
  assert ({[encode i]}  nclose N) by eauto.
84
  rewrite (always_sep_dup (ownI _ _)).
Ralf Jung's avatar
Ralf Jung committed
85
  rewrite {1}pvs_openI !pvs_frame_r.
86
  apply pvs_mask_frame_mono; [solve_elem_of..|].
87
  rewrite (comm _ (_)%I) -assoc wand_elim_r fsa_frame_l.
88
  apply fsa_mask_frame_mono; [solve_elem_of..|]. intros a.
89
  rewrite assoc -always_and_sep_l pvs_closeI pvs_frame_r left_id.
Ralf Jung's avatar
Ralf Jung committed
90
91
92
  apply pvs_mask_frame'; solve_elem_of.
Qed.

93
94
(* Derive the concrete forms for pvs and wp, because they are useful. *)

95
Lemma pvs_open_close E N P Q R :
96
  nclose N  E 
97
98
99
  R  inv N P 
  R  (P - pvs (E  nclose N) (E  nclose N) (P  Q)) 
  R  pvs E E Q.
Ralf Jung's avatar
Ralf Jung committed
100
Proof. intros. by apply: (inv_fsa pvs_fsa). Qed.
101

102
Lemma wp_open_close E e N P (Q : val Λ  iProp Λ Σ) R :
Ralf Jung's avatar
Ralf Jung committed
103
  atomic e  nclose N  E 
104
105
106
  R  inv N P 
  R  (P - wp (E  nclose N) e (λ v, P  Q v)) 
  R  wp E e Q.
Ralf Jung's avatar
Ralf Jung committed
107
Proof. intros. by apply: (inv_fsa (wp_fsa e)). Qed.
Ralf Jung's avatar
Ralf Jung committed
108

109
Lemma inv_alloc N P :  P  pvs N N (inv N P).
110
Proof. by rewrite /inv (pvs_allocI N); last apply coPset_suffixes_infinite. Qed.
Ralf Jung's avatar
Ralf Jung committed
111
112

End inv.