weakestpre.v 13.2 KB
Newer Older
1
2
From iris.program_logic Require Export pviewshifts.
From iris.program_logic Require Import wsat.
Robbert Krebbers's avatar
Robbert Krebbers committed
3
Local Hint Extern 10 (_  _) => omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
4
5
Local Hint Extern 100 (_  _) => set_solver.
Local Hint Extern 100 (_ _) => set_solver.
6
Local Hint Extern 100 (@subseteq coPset _ _ _) => set_solver.
Robbert Krebbers's avatar
Robbert Krebbers committed
7
Local Hint Extern 10 ({_} _) =>
8
9
10
  repeat match goal with
  | H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
  end; solve_validN.
Robbert Krebbers's avatar
Robbert Krebbers committed
11

12
Record wp_go {Λ Σ} (E : coPset) (Φ Φfork : expr Λ  nat  iRes Λ Σ  Prop)
13
    (k : nat) (σ1 : state Λ) (rf : iRes Λ Σ) (e1 : expr Λ) := {
14
  wf_safe : reducible e1 σ1;
Robbert Krebbers's avatar
Robbert Krebbers committed
15
16
17
18
  wp_step e2 σ2 ef :
    prim_step e1 σ1 e2 σ2 ef 
     r2 r2',
      wsat k E σ2 (r2  r2'  rf) 
19
20
      Φ e2 k r2 
       e', ef = Some e'  Φfork e' k r2'
Robbert Krebbers's avatar
Robbert Krebbers committed
21
}.
22
CoInductive wp_pre {Λ Σ} (E : coPset)
23
     (Φ : val Λ  iProp Λ Σ) : expr Λ  nat  iRes Λ Σ  Prop :=
24
  | wp_pre_value n r v : (|={E}=> Φ v)%I n r  wp_pre E Φ (of_val v) n r
Robbert Krebbers's avatar
Robbert Krebbers committed
25
26
  | wp_pre_step n r1 e1 :
     to_val e1 = None 
27
     ( k Ef σ1 rf,
Robbert Krebbers's avatar
Robbert Krebbers committed
28
       0 < k < n  E  Ef 
Robbert Krebbers's avatar
Robbert Krebbers committed
29
       wsat (S k) (E  Ef) σ1 (r1  rf) 
30
       wp_go (E  Ef) (wp_pre E Φ)
31
                      (wp_pre  (λ _, True%I)) k σ1 rf e1) 
32
     wp_pre E Φ e1 n r1.
Ralf Jung's avatar
Ralf Jung committed
33
Program Definition wp_def {Λ Σ} (E : coPset) (e : expr Λ)
34
  (Φ : val Λ  iProp Λ Σ) : iProp Λ Σ := {| uPred_holds := wp_pre E Φ e |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
35
Next Obligation.
36
37
38
39
  intros Λ Σ E e Φ n r1 r2; revert Φ E e r1 r2.
  induction n as [n IH] using lt_wf_ind; intros Φ E e r1 r1'.
  destruct 1 as [|n r1 e1 ? Hgo].
  - constructor; eauto using uPred_mono.
40
41
  - intros [rf' Hr]; constructor; [done|intros k Ef σ1 rf ???].
    destruct (Hgo k Ef σ1 (rf'  rf)) as [Hsafe Hstep];
42
      rewrite ?assoc -?(dist_le _ _ _ _ Hr); auto; constructor; [done|].
Robbert Krebbers's avatar
Robbert Krebbers committed
43
    intros e2 σ2 ef ?; destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
44
    exists r2, (r2'  rf'); split_and?; eauto 10 using (IH k), cmra_includedN_l.
45
    by rewrite -!assoc (assoc _ r2).
Robbert Krebbers's avatar
Robbert Krebbers committed
46
Qed.
47
48
Next Obligation. destruct 1; constructor; eauto using uPred_closed. Qed.

Ralf Jung's avatar
Ralf Jung committed
49
50
51
52
53
54
(* Perform sealing. *)
Definition wp_aux : { x | x = @wp_def }. by eexists. Qed.
Definition wp := proj1_sig wp_aux.
Definition wp_eq : @wp = @wp_def := proj2_sig wp_aux.

Arguments wp {_ _} _ _ _.
55
Instance: Params (@wp) 4.
Robbert Krebbers's avatar
Robbert Krebbers committed
56

57
Notation "'WP' e @ E {{ Φ } }" := (wp E e Φ)
58
  (at level 20, e, Φ at level 200,
59
60
   format "'WP'  e  @  E  {{  Φ  } }") : uPred_scope.
Notation "'WP' e {{ Φ } }" := (wp  e Φ)
61
  (at level 20, e, Φ at level 200,
62
   format "'WP'  e  {{  Φ  } }") : uPred_scope.
63

64
65
66
67
68
69
70
Notation "'WP' e @ E {{ v , Q } }" := (wp E e (λ v, Q))
  (at level 20, e, Q at level 200,
   format "'WP'  e  @  E  {{  v ,  Q  } }") : uPred_scope.
Notation "'WP' e {{ v , Q } }" := (wp  e (λ v, Q))
  (at level 20, e, Q at level 200,
   format "'WP'  e  {{  v ,  Q  } }") : uPred_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
71
Section wp.
72
Context {Λ : language} {Σ : iFunctor}.
73
Implicit Types P : iProp Λ Σ.
74
Implicit Types Φ : val Λ  iProp Λ Σ.
75
76
Implicit Types v : val Λ.
Implicit Types e : expr Λ.
Robbert Krebbers's avatar
Robbert Krebbers committed
77

78
79
Global Instance wp_ne E e n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@wp Λ Σ E e).
Robbert Krebbers's avatar
Robbert Krebbers committed
80
Proof.
81
  cut ( Φ Ψ, ( v, Φ v {n} Ψ v) 
Robbert Krebbers's avatar
Robbert Krebbers committed
82
     n' r, n'  n  {n'} r  wp E e Φ n' r  wp E e Ψ n' r).
Ralf Jung's avatar
Ralf Jung committed
83
84
  { rewrite wp_eq. intros help Φ Ψ HΦΨ. by do 2 split; apply help. }
  rewrite wp_eq. intros Φ Ψ HΦΨ n' r; revert e r.
85
86
87
  induction n' as [n' IH] using lt_wf_ind=> e r.
  destruct 3 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor. by eapply pvs_ne, HpvsQ; eauto. }
88
89
  constructor; [done|]=> k Ef σ1 rf ???.
  destruct (Hgo k Ef σ1 rf) as [Hsafe Hstep]; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
90
91
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
92
  exists r2, r2'; split_and?; [|eapply IH|]; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
93
94
Qed.
Global Instance wp_proper E e :
95
  Proper (pointwise_relation _ () ==> ()) (@wp Λ Σ E e).
Robbert Krebbers's avatar
Robbert Krebbers committed
96
Proof.
97
  by intros Φ Φ' ?; apply equiv_dist=>n; apply wp_ne=>v; apply equiv_dist.
Robbert Krebbers's avatar
Robbert Krebbers committed
98
Qed.
99

100
Lemma wp_mask_frame_mono E1 E2 e Φ Ψ :
101
  E1  E2  ( v, Φ v  Ψ v)  WP e @ E1 {{ Φ }}  WP e @ E2 {{ Ψ }}.
102
Proof.
Ralf Jung's avatar
Ralf Jung committed
103
  rewrite wp_eq. intros HE HΦ; split=> n r.
104
  revert e r; induction n as [n IH] using lt_wf_ind=> e r.
105
106
  destruct 2 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor; eapply pvs_mask_frame_mono, HpvsQ; eauto. }
107
  constructor; [done|]=> k Ef σ1 rf ???.
108
  assert (E2  Ef = E1  (E2  E1  Ef)) as HE'.
109
  { by rewrite assoc_L -union_difference_L. }
110
  destruct (Hgo k ((E2  E1)  Ef) σ1 rf) as [Hsafe Hstep]; rewrite -?HE'; auto.
111
112
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
113
  exists r2, r2'; split_and?; [rewrite HE'|eapply IH|]; eauto.
114
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
115

116
117
118
119
120
121
122
Lemma wp_zero E e Φ r : wp_def E e Φ 0 r.
Proof.
  case EQ: (to_val e).
  - rewrite -(of_to_val _ _ EQ). constructor. rewrite pvs_eq.
    exact: pvs_zero.
  - constructor; first done. intros ?????. exfalso. omega.
Qed.
123
Lemma wp_value_inv E Φ v n r : wp_def E (of_val v) Φ n r  pvs E E (Φ v) n r.
Robbert Krebbers's avatar
Robbert Krebbers committed
124
Proof.
125
  by inversion 1 as [|??? He]; [|rewrite ?to_of_val in He]; simplify_eq.
Robbert Krebbers's avatar
Robbert Krebbers committed
126
Qed.
127
Lemma wp_step_inv E Ef Φ e k n σ r rf :
Robbert Krebbers's avatar
Robbert Krebbers committed
128
  to_val e = None  0 < k < n  E  Ef 
Ralf Jung's avatar
Ralf Jung committed
129
  wp_def E e Φ n r  wsat (S k) (E  Ef) σ (r  rf) 
130
  wp_go (E  Ef) (λ e, wp_def E e Φ) (λ e, wp_def  e (λ _, True%I)) k σ rf e.
Ralf Jung's avatar
Ralf Jung committed
131
132
133
Proof.
  intros He; destruct 3; [by rewrite ?to_of_val in He|eauto].
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
134

135
Lemma wp_value' E Φ v : Φ v  WP of_val v @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
136
Proof. rewrite wp_eq. split=> n r; constructor; by apply pvs_intro. Qed.
137
Lemma pvs_wp E e Φ : (|={E}=> WP e @ E {{ Φ }})  WP e @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
138
Proof.
Ralf Jung's avatar
Ralf Jung committed
139
  rewrite wp_eq. split=> n r ? Hvs.
Robbert Krebbers's avatar
Robbert Krebbers committed
140
  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
141
  { constructor; eapply pvs_trans', pvs_mono, Hvs; eauto.
142
    split=> ???; apply wp_value_inv. }
143
144
  constructor; [done|]=> k Ef σ1 rf ???.
  rewrite pvs_eq in Hvs. destruct (Hvs (S k) Ef σ1 rf) as (r'&Hwp&?); auto.
145
146
  eapply wp_step_inv with (S k) r'; eauto.
Qed.
147
Lemma wp_pvs E e Φ : WP e @ E {{ v, |={E}=> Φ v }}  WP e @ E {{ Φ }}.
148
Proof.
Ralf Jung's avatar
Ralf Jung committed
149
150
  rewrite wp_eq. split=> n r; revert e r;
    induction n as [n IH] using lt_wf_ind=> e r Hr HΦ.
151
  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
152
  { constructor; apply pvs_trans', (wp_value_inv _ (pvs E E  Φ)); auto. }
153
  constructor; [done|]=> k Ef σ1 rf ???.
154
  destruct (wp_step_inv E Ef (pvs E E  Φ) e k n σ1 r rf) as [? Hstep]; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
155
156
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); auto.
157
  exists r2, r2'; split_and?; [|apply (IH k)|]; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
158
Qed.
159
Lemma wp_atomic E1 E2 e Φ :
160
  E2  E1  atomic e 
161
  (|={E1,E2}=> WP e @ E2 {{ v, |={E2,E1}=> Φ v }})  WP e @ E1 {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
162
Proof.
Ralf Jung's avatar
Ralf Jung committed
163
  rewrite wp_eq pvs_eq. intros ? He; split=> n r ? Hvs; constructor.
164
165
  eauto using atomic_not_val. intros k Ef σ1 rf ???.
  destruct (Hvs (S k) Ef σ1 rf) as (r'&Hwp&?); auto.
Ralf Jung's avatar
Ralf Jung committed
166
167
  destruct (wp_step_inv E2 Ef (pvs_def E2 E1  Φ) e k (S k) σ1 r' rf)
    as [Hsafe Hstep]; auto using atomic_not_val; [].
168
  split; [done|]=> e2 σ2 ef ?.
Robbert Krebbers's avatar
Robbert Krebbers committed
169
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); clear Hsafe Hstep; auto.
170
171
  destruct Hwp' as [k r2 v Hvs'|k r2 e2 Hgo];
    [|destruct (atomic_step e σ1 e2 σ2 ef); naive_solver].
Ralf Jung's avatar
Ralf Jung committed
172
  rewrite -pvs_eq in Hvs'. apply pvs_trans in Hvs';auto. rewrite pvs_eq in Hvs'.
173
  destruct (Hvs' k Ef σ2 (r2'  rf)) as (r3&[]); rewrite ?assoc; auto.
174
  exists r3, r2'; split_and?; last done.
175
176
  - by rewrite -assoc.
  - constructor; apply pvs_intro; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
177
Qed.
178
Lemma wp_frame_r E e Φ R : WP e @ E {{ Φ }}  R  WP e @ E {{ v, Φ v  R }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
179
Proof.
Ralf Jung's avatar
Ralf Jung committed
180
181
  rewrite wp_eq. uPred.unseal; split; intros n r' Hvalid (r&rR&Hr&Hwp&?).
  revert Hvalid. rewrite Hr; clear Hr; revert e r Hwp.
Robbert Krebbers's avatar
Robbert Krebbers committed
182
  induction n as [n IH] using lt_wf_ind; intros e r1.
183
  destruct 1 as [|n r e ? Hgo]=>?.
184
185
  { constructor. rewrite -uPred_sep_eq; apply pvs_frame_r; auto.
    uPred.unseal; exists r, rR; eauto. }
186
187
  constructor; [done|]=> k Ef σ1 rf ???.
  destruct (Hgo k Ef σ1 (rRrf)) as [Hsafe Hstep]; auto.
188
  { by rewrite assoc. }
Robbert Krebbers's avatar
Robbert Krebbers committed
189
190
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
191
  exists (r2  rR), r2'; split_and?; auto.
192
  - by rewrite -(assoc _ r2) (comm _ rR) !assoc -(assoc _ _ rR).
193
  - apply IH; eauto using uPred_closed.
Robbert Krebbers's avatar
Robbert Krebbers committed
194
Qed.
Ralf Jung's avatar
Ralf Jung committed
195
196
Lemma wp_frame_step_r E E1 E2 e Φ R :
  to_val e = None  E  E1  E2  E1 
197
  WP e @ E {{ Φ }}  (|={E1,E2}=>  |={E2,E1}=> R)
Robbert Krebbers's avatar
Robbert Krebbers committed
198
199
   WP e @ E  E1 {{ v, Φ v  R }}.
Proof.
Ralf Jung's avatar
Ralf Jung committed
200
201
  rewrite wp_eq pvs_eq=> He ??.
  uPred.unseal; split; intros n r' Hvalid (r&rR&Hr&Hwp&HR); cofe_subst.
202
  constructor; [done|]=> k Ef σ1 rf ?? Hws1.
203
  destruct Hwp as [|n r e ? Hgo]; [by rewrite to_of_val in He|].
Ralf Jung's avatar
Ralf Jung committed
204
  (* "execute" HR *)
205
  destruct (HR (S k) (E  Ef) σ1 (r  rf)) as (s&Hvs&Hws2); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
206
207
208
  { eapply wsat_proper, Hws1; first by set_solver+.
    by rewrite assoc [rR  _]comm. }
  clear Hws1 HR.
Ralf Jung's avatar
Ralf Jung committed
209
  (* Take a step *)
210
  destruct (Hgo k (E2  Ef) σ1 (s  rf)) as [Hsafe Hstep]; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
211
212
213
  { eapply wsat_proper, Hws2; first by set_solver+.
    by rewrite !assoc [s  _]comm. }
  clear Hgo.
Robbert Krebbers's avatar
Robbert Krebbers committed
214
  split; [done|intros e2 σ2 ef ?].
Ralf Jung's avatar
Ralf Jung committed
215
216
  destruct (Hstep e2 σ2 ef) as (r2&r2'&Hws3&?&?); auto. clear Hws2.
  (* Execute 2nd part of the view shift *)
217
  destruct (Hvs k (E  Ef) σ2 (r2  r2'  rf)) as (t&HR&Hws4); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
218
219
220
  { eapply wsat_proper, Hws3; first by set_solver+.
    by rewrite !assoc [_  s]comm !assoc. }
  clear Hvs Hws3.
Ralf Jung's avatar
Ralf Jung committed
221
  (* Execute the rest of e *)
222
  exists (r2  t), r2'; split_and?; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
223
224
225
  - eapply wsat_proper, Hws4; first by set_solver+.
    by rewrite !assoc [_  t]comm.
  - rewrite -uPred_sep_eq. move: wp_frame_r. rewrite wp_eq=>Hframe.
Ralf Jung's avatar
Ralf Jung committed
226
    apply Hframe; first by auto. uPred.unseal; exists r2, t; split_and?; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
227
    move: wp_mask_frame_mono. rewrite wp_eq=>Hmask.
Ralf Jung's avatar
Ralf Jung committed
228
    eapply (Hmask E); by auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
229
Qed.
230
Lemma wp_bind `{LanguageCtx Λ K} E e Φ :
231
  WP e @ E {{ v, WP K (of_val v) @ E {{ Φ }} }}  WP K e @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
232
Proof.
Ralf Jung's avatar
Ralf Jung committed
233
234
235
236
  rewrite wp_eq. split=> n r; revert e r;
    induction n as [n IH] using lt_wf_ind=> e r ?.
  destruct 1 as [|n r e ? Hgo].
  { rewrite -wp_eq. apply pvs_wp; rewrite ?wp_eq; done. }
237
238
  constructor; auto using fill_not_val=> k Ef σ1 rf ???.
  destruct (Hgo k Ef σ1 rf) as [Hsafe Hstep]; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
239
240
  split.
  { destruct Hsafe as (e2&σ2&ef&?).
241
    by exists (K e2), σ2, ef; apply fill_step. }
Robbert Krebbers's avatar
Robbert Krebbers committed
242
  intros e2 σ2 ef ?.
243
  destruct (fill_step_inv e σ1 e2 σ2 ef) as (e2'&->&?); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
244
  destruct (Hstep e2' σ2 ef) as (r2&r2'&?&?&?); auto.
245
  exists r2, r2'; split_and?; try eapply IH; eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
246
247
Qed.

248
(** * Derived rules *)
Robbert Krebbers's avatar
Robbert Krebbers committed
249
Import uPred.
250
Lemma wp_mono E e Φ Ψ : ( v, Φ v  Ψ v)  WP e @ E {{ Φ }}  WP e @ E {{ Ψ }}.
251
Proof. by apply wp_mask_frame_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
Global Instance wp_mono' E e :
253
  Proper (pointwise_relation _ () ==> ()) (@wp Λ Σ E e).
254
Proof. by intros Φ Φ' ?; apply wp_mono. Qed.
255
Lemma wp_strip_pvs E e P Φ :
256
  (P  WP e @ E {{ Φ }})  (|={E}=> P)  WP e @ E {{ Φ }}.
257
Proof. move=>->. by rewrite pvs_wp. Qed.
258
Lemma wp_value E Φ e v : to_val e = Some v  Φ v  WP e @ E {{ Φ }}.
259
Proof. intros; rewrite -(of_to_val e v) //; by apply wp_value'. Qed.
260
Lemma wp_value_pvs E Φ e v :
261
  to_val e = Some v  (|={E}=> Φ v)  WP e @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
262
Proof. intros. rewrite -wp_pvs. rewrite -wp_value //. Qed.
263
Lemma wp_frame_l E e Φ R : R  WP e @ E {{ Φ }}  WP e @ E {{ v, R  Φ v }}.
264
Proof. setoid_rewrite (comm _ R); apply wp_frame_r. Qed.
Ralf Jung's avatar
Ralf Jung committed
265
Lemma wp_frame_step_r' E e Φ R :
266
  to_val e = None  WP e @ E {{ Φ }}   R  WP e @ E {{ v, Φ v  R }}.
Ralf Jung's avatar
Ralf Jung committed
267
268
269
270
271
272
273
Proof.
  intros. rewrite {2}(_ : E = E  ); last by set_solver.
  rewrite -(wp_frame_step_r E  ); [|auto|set_solver..].
  apply sep_mono_r. rewrite -!pvs_intro. done.
Qed.
Lemma wp_frame_step_l E E1 E2 e Φ R :
  to_val e = None  E  E1  E2  E1 
274
  (|={E1,E2}=>  |={E2,E1}=> R)  WP e @ E {{ Φ }}
275
   WP e @ (E  E1) {{ v, R  Φ v }}.
Ralf Jung's avatar
Ralf Jung committed
276
277
278
279
280
Proof.
  rewrite [((|={E1,E2}=> _)  _)%I]comm; setoid_rewrite (comm _ R).
  apply wp_frame_step_r.
Qed.
Lemma wp_frame_step_l' E e Φ R :
281
  to_val e = None   R  WP e @ E {{ Φ }}  WP e @ E {{ v, R  Φ v }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
282
Proof.
283
  rewrite (comm _ ( R)%I); setoid_rewrite (comm _ R).
Ralf Jung's avatar
Ralf Jung committed
284
  apply wp_frame_step_r'.
Robbert Krebbers's avatar
Robbert Krebbers committed
285
Qed.
286
Lemma wp_always_l E e Φ R `{!PersistentP R} :
287
  R  WP e @ E {{ Φ }}  WP e @ E {{ v, R  Φ v }}.
288
Proof. by setoid_rewrite (always_and_sep_l _ _); rewrite wp_frame_l. Qed.
289
Lemma wp_always_r E e Φ R `{!PersistentP R} :
290
  WP e @ E {{ Φ }}  R  WP e @ E {{ v, Φ v  R }}.
291
Proof. by setoid_rewrite (always_and_sep_r _ _); rewrite wp_frame_r. Qed.
292
Lemma wp_wand_l E e Φ Ψ :
293
  ( v, Φ v - Ψ v)  WP e @ E {{ Φ }}  WP e @ E {{ Ψ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
294
Proof.
295
  rewrite wp_frame_l. apply wp_mono=> v. by rewrite (forall_elim v) wand_elim_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
296
Qed.
297
Lemma wp_wand_r E e Φ Ψ :
298
  WP e @ E {{ Φ }}  ( v, Φ v - Ψ v)  WP e @ E {{ Ψ }}.
299
300
Proof. by rewrite comm wp_wand_l. Qed.

301
Lemma wp_mask_weaken E1 E2 e Φ :
302
  E1  E2  WP e @ E1 {{ Φ }}  WP e @ E2 {{ Φ }}.
303
304
305
Proof. auto using wp_mask_frame_mono. Qed.

(** * Weakest-pre is a FSA. *)
306
Definition wp_fsa (e : expr Λ) : FSA Λ Σ (val Λ) := λ E, wp E e.
307
Global Arguments wp_fsa _ _ / _.
308
Global Instance wp_fsa_prf : FrameShiftAssertion (atomic e) (wp_fsa e).
309
Proof.
310
  rewrite /wp_fsa; split; auto using wp_mask_frame_mono, wp_atomic, wp_frame_r.
311
  intros E Φ. by rewrite -(pvs_wp E e Φ) -(wp_pvs E e Φ).
312
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
313
End wp.