pviewshifts.v 10.7 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
From iris.proofmode Require Import coq_tactics spec_patterns.
From iris.proofmode Require Export tactics.
From iris.program_logic Require Export pviewshifts.
Import uPred.

Section pvs.
Context {Λ : language} {Σ : iFunctor}.
Implicit Types P Q : iProp Λ Σ.

Global Instance sep_split_pvs E P Q1 Q2 :
  SepSplit P Q1 Q2  SepSplit (|={E}=> P) (|={E}=> Q1) (|={E}=> Q2).
Proof. rewrite /SepSplit=><-. apply pvs_sep. Qed.
Global Instance or_split_pvs E1 E2 P Q1 Q2 :
  OrSplit P Q1 Q2  OrSplit (|={E1,E2}=> P) (|={E1,E2}=> Q1) (|={E1,E2}=> Q2).
Proof. rewrite /OrSplit=><-. apply or_elim; apply pvs_mono; auto with I. Qed.
Global Instance exists_split_pvs {A} E1 E2 P (Φ : A  iProp Λ Σ) :
  ExistSplit P Φ  ExistSplit (|={E1,E2}=> P) (λ a, |={E1,E2}=> Φ a)%I.
Proof.
  rewrite /ExistSplit=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
Global Instance frame_pvs E1 E2 R P Q :
  Frame R P Q  Frame R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof. rewrite /Frame=><-. by rewrite pvs_frame_l. Qed.
Global Instance to_wand_pvs E1 E2 R P Q :
  ToWand R P Q  ToWand R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof. rewrite /ToWand=>->. apply wand_intro_l. by rewrite pvs_wand_r. Qed.

Class FSASplit {A} (P : iProp Λ Σ) (E : coPset)
    (fsa : FSA Λ Σ A) (fsaV : Prop) (Φ : A  iProp Λ Σ) := {
  fsa_split : fsa E Φ  P;
  fsa_split_fsa :> FrameShiftAssertion fsaV fsa;
}.
Global Arguments fsa_split {_} _ _ _ _ _ {_}.
Global Instance fsa_split_pvs E P :
  FSASplit (|={E}=> P)%I E pvs_fsa True (λ _, P).
Proof. split. done. apply _. Qed.

Lemma tac_pvs_intro Δ E Q : Δ  Q  Δ  |={E}=> Q.
Proof. intros ->. apply pvs_intro. Qed.

Lemma tac_pvs_elim Δ Δ' E1 E2 E3 i p P Q :
  envs_lookup i Δ = Some (p, |={E1,E2}=> P)%I 
  envs_replace i p false (Esnoc Enil i P) Δ = Some Δ' 
  E2  E1  E3 
  Δ'  (|={E2,E3}=> Q)  Δ  |={E1,E3}=> Q.
Proof.
  intros ??? HQ. rewrite envs_replace_sound //; simpl. destruct p.
  - by rewrite always_elim right_id pvs_frame_r wand_elim_r HQ pvs_trans.
  - by rewrite right_id pvs_frame_r wand_elim_r HQ pvs_trans.
Qed.

Lemma tac_pvs_elim_fsa {A} (fsa : FSA Λ Σ A) fsaV Δ Δ' E i p P Q Φ :
  envs_lookup i Δ = Some (p, |={E}=> P)%I  FSASplit Q E fsa fsaV Φ 
  envs_replace i p false (Esnoc Enil i P) Δ = Some Δ' 
  Δ'  fsa E Φ  Δ  Q.
Proof.
  intros ???. rewrite -(fsa_split Q) -fsa_pvs_fsa.
  eapply tac_pvs_elim; set_solver.
Qed.

Lemma tac_pvs_timeless Δ Δ' E1 E2 i p P Q :
  envs_lookup i Δ = Some (p,  P)%I  TimelessP P 
  envs_simple_replace i p (Esnoc Enil i P) Δ = Some Δ' 
  Δ'  (|={E1,E2}=> Q)  Δ  (|={E1,E2}=> Q).
Proof.
  intros ??? HQ. rewrite envs_simple_replace_sound //; simpl.
  destruct p.
  - rewrite always_later (pvs_timeless E1 ( P)%I) pvs_frame_r.
    by rewrite right_id wand_elim_r HQ pvs_trans; last set_solver.
  - rewrite (pvs_timeless E1 P) pvs_frame_r right_id wand_elim_r HQ.
    by rewrite pvs_trans; last set_solver.
Qed.

Lemma tac_pvs_timeless_fsa {A} (fsa : FSA Λ Σ A) fsaV Δ Δ' E i p P Q Φ :
  FSASplit Q E fsa fsaV Φ 
  envs_lookup i Δ = Some (p,  P)%I  TimelessP P 
  envs_simple_replace i p (Esnoc Enil i P) Δ = Some Δ' 
  Δ'  fsa E Φ  Δ  Q.
Proof.
  intros ????. rewrite -(fsa_split Q) -fsa_pvs_fsa.
  eauto using tac_pvs_timeless.
Qed.

Lemma tac_pvs_assert {A} (fsa : FSA Λ Σ A) fsaV Δ Δ1 Δ2 Δ2' E lr js j P Q Φ :
  FSASplit Q E fsa fsaV Φ 
  envs_split lr js Δ = Some (Δ1,Δ2) 
  envs_app false (Esnoc Enil j P) Δ2 = Some Δ2' 
  Δ1  (|={E}=> P)  Δ2'  fsa E Φ  Δ  Q.
Proof.
  intros ??? HP HQ. rewrite -(fsa_split Q) -fsa_pvs_fsa -HQ envs_split_sound //.
  rewrite HP envs_app_sound //; simpl.
  by rewrite right_id pvs_frame_r wand_elim_r.
Qed.
End pvs.

Tactic Notation "iPvsIntro" := apply tac_pvs_intro.

Tactic Notation "iPvsCore" constr(H) :=
  match goal with
  | |- _  |={_,_}=> _ =>
    eapply tac_pvs_elim with _ _ H _ _;
      [env_cbv; reflexivity || fail "iPvs:" H "not found"
      |env_cbv; reflexivity
      |try (rewrite (idemp_L ()); reflexivity)|]
  | |- _ =>
    eapply tac_pvs_elim_fsa with _ _ _ _ H _ _ _;
      [env_cbv; reflexivity || fail "iPvs:" H "not found"
      |let P := match goal with |- FSASplit ?P _ _ _ _ => P end in
       apply _ || fail "iPvs: " P "not a pvs"
      |env_cbv; reflexivity|simpl]
  end.

Tactic Notation "iPvs" open_constr(H) :=
  iPoseProof H as (fun H => iPvsCore H; last iDestructHyp H as "?").
Tactic Notation "iPvs" open_constr(H) "as" constr(pat) :=
  iPoseProof H as (fun H => iPvsCore H; last iDestructHyp H as pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1) "}"
    constr(pat) :=
  iPoseProof H as (fun H => iPvsCore H; last iDestructHyp H as { x1 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) "}" constr(pat) :=
  iPoseProof H as (fun H => iPvsCore H; last iDestructHyp H as { x1 x2 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) "}" constr(pat) :=
  iPoseProof H as (fun H => iPvsCore H; last iDestructHyp H as { x1 x2 x3 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) "}"
    constr(pat) :=
  iPoseProof H as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) "}" constr(pat) :=
  iPoseProof H as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) "}" constr(pat) :=
  iPoseProof H as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7) "}"
    constr(pat) :=
  iPoseProof H as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 x7 } pat).
Tactic Notation "iPvs" open_constr(H) "as" "{" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7)
    simple_intropattern(x8) "}" constr(pat) :=
  iPoseProof H as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 x7 x8 } pat).

Tactic Notation "iPvs" open_constr(lem) constr(Hs) :=
  iPoseProof lem Hs as (fun H => iPvsCore H; last iDestructHyp H as "?").
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" constr(pat) :=
  iPoseProof lem Hs as (fun H => iPvsCore H; last iDestructHyp H as pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H => iPvsCore H; last iDestructHyp H as { x1 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) "}"
    constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3)
    simple_intropattern(x4) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3)
    simple_intropattern(x4) simple_intropattern(x5) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3)
    simple_intropattern(x4) simple_intropattern(x5) simple_intropattern(x6) "}"
    constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3)
    simple_intropattern(x4) simple_intropattern(x5) simple_intropattern(x6)
    simple_intropattern(x7) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 x7 } pat).
Tactic Notation "iPvs" open_constr(lem) constr(Hs) "as" "{"
    simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3)
    simple_intropattern(x4) simple_intropattern(x5) simple_intropattern(x6)
    simple_intropattern(x7) simple_intropattern(x8) "}" constr(pat) :=
  iPoseProof lem Hs as (fun H =>
    iPvsCore H; last iDestructHyp H as { x1 x2 x3 x4 x5 x6 x7 x8 } pat).

Tactic Notation "iTimeless" constr(H) :=
  match goal with
  | |- _  |={_,_}=> _ =>
     eapply tac_pvs_timeless with _ H _ _;
       [env_cbv; reflexivity || fail "iTimeless:" H "not found"
       |let P := match goal with |- TimelessP ?P => P end in
        apply _ || fail "iTimeless: " P "not timeless"
       |env_cbv; reflexivity|simpl]
  | |- _ =>
     eapply tac_pvs_timeless_fsa with _ _ _ _ H _ _ _;
       [let P := match goal with |- FSASplit ?P _ _ _ _ => P end in
        apply _ || fail "iTimeless: " P "not a pvs"
       |env_cbv; reflexivity || fail "iTimeless:" H "not found"
       |let P := match goal with |- TimelessP ?P => P end in
        apply _ || fail "iTimeless: " P "not timeless"
       |env_cbv; reflexivity|simpl]
  end.

Tactic Notation "iTimeless" constr(H) "as" constr(Hs) :=
  iTimeless H; iDestruct H as Hs.

Tactic Notation "iPvsAssert" constr(Q) "as" constr(pat) "with" constr(Hs) :=
  let H := iFresh in
  let Hs := spec_pat.parse_one Hs in
  lazymatch Hs with
  | SSplit ?lr ?Hs =>
     eapply tac_pvs_assert with _ _ _ _ _ _ lr Hs H Q _;
       [let P := match goal with |- FSASplit ?P _ _ _ _ => P end in
        apply _ || fail "iPvsAssert: " P "not a pvs"
       |env_cbv; reflexivity || fail "iPvsAssert:" Hs "not found"
       |env_cbv; reflexivity|
       |simpl; iDestructHyp H as pat]
  | ?pat => fail "iPvsAssert: invalid pattern" pat
  end.
Tactic Notation "iPvsAssert" constr(Q) "as" constr(pat) :=
  iPvsAssert Q as pat with "[]".