Newer
Older
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
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
From iris.proofmode Require Export classes.
From iris.algebra Require Import upred_big_op gmap upred_tactics.
Import uPred.
Section classes.
Context {M : ucmraT}.
Implicit Types P Q R : uPred M.
(* FromAssumption *)
Global Instance from_assumption_exact p P : FromAssumption p P P.
Proof. destruct p; by rewrite /FromAssumption /= ?always_elim. Qed.
Global Instance from_assumption_always_l p P Q :
FromAssumption p P Q → FromAssumption p (□ P) Q.
Proof. rewrite /FromAssumption=><-. by rewrite always_elim. Qed.
Global Instance from_assumption_always_r P Q :
FromAssumption true P Q → FromAssumption true P (□ Q).
Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
(* IntoPure *)
Global Instance into_pure_pure φ : @IntoPure M (■ φ) φ.
Proof. done. Qed.
Global Instance into_pure_eq {A : cofeT} (a b : A) :
Timeless a → @IntoPure M (a ≡ b) (a ≡ b).
Proof. intros. by rewrite /IntoPure timeless_eq. Qed.
Global Instance into_pure_valid `{CMRADiscrete A} (a : A) : @IntoPure M (✓ a) (✓ a).
Proof. by rewrite /IntoPure discrete_valid. Qed.
(* FromPure *)
Global Instance from_pure_pure φ : @FromPure M (■ φ) φ.
Proof. intros ?. by apply pure_intro. Qed.
Global Instance from_pure_eq {A : cofeT} (a b : A) : @FromPure M (a ≡ b) (a ≡ b).
Proof. intros ->. apply eq_refl. Qed.
Global Instance from_pure_valid {A : cmraT} (a : A) : @FromPure M (✓ a) (✓ a).
Proof. intros ?. by apply valid_intro. Qed.
(* IntoPersistentP *)
Global Instance into_persistentP_always_trans P Q :
IntoPersistentP P Q → IntoPersistentP (□ P) Q | 0.
Proof. rewrite /IntoPersistentP=> ->. by rewrite always_always. Qed.
Global Instance into_persistentP_always P : IntoPersistentP (□ P) P | 1.
Proof. done. Qed.
Global Instance into_persistentP_persistent P :
PersistentP P → IntoPersistentP P P | 100.
Proof. done. Qed.
(* IntoLater *)
Global Instance into_later_default P : IntoLater P P | 1000.
Proof. apply later_intro. Qed.
Global Instance into_later_later P : IntoLater (▷ P) P.
Proof. done. Qed.
Global Instance into_later_and P1 P2 Q1 Q2 :
IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ∧ P2) (Q1 ∧ Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance into_later_or P1 P2 Q1 Q2 :
IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ∨ P2) (Q1 ∨ Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance into_later_sep P1 P2 Q1 Q2 :
IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ★ P2) (Q1 ★ Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.
Global Instance into_later_big_sepM `{Countable K} {A}
(Φ Ψ : K → A → uPred M) (m : gmap K A) :
(∀ x k, IntoLater (Φ k x) (Ψ k x)) →
IntoLater ([★ map] k ↦ x ∈ m, Φ k x) ([★ map] k ↦ x ∈ m, Ψ k x).
Proof.
rewrite /IntoLater=> ?. rewrite big_sepM_later; by apply big_sepM_mono.
Qed.
Global Instance into_later_big_sepS `{Countable A}
(Φ Ψ : A → uPred M) (X : gset A) :
(∀ x, IntoLater (Φ x) (Ψ x)) →
IntoLater ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ x).
Proof.
rewrite /IntoLater=> ?. rewrite big_sepS_later; by apply big_sepS_mono.
Qed.
(* FromLater *)
Global Instance from_later_later P : FromLater (▷ P) P.
Proof. done. Qed.
Global Instance from_later_and P1 P2 Q1 Q2 :
FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ∧ P2) (Q1 ∧ Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance from_later_or P1 P2 Q1 Q2 :
FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ∨ P2) (Q1 ∨ Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance from_later_sep P1 P2 Q1 Q2 :
FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ★ P2) (Q1 ★ Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.
(* IntoWand *)
Global Instance into_wand_wand P Q : IntoWand (P -★ Q) P Q.
Proof. done. Qed.
Global Instance into_wand_impl P Q : IntoWand (P → Q) P Q.
Proof. apply impl_wand. Qed.
Global Instance into_wand_iff_l P Q : IntoWand (P ↔ Q) P Q.
Proof. by apply and_elim_l', impl_wand. Qed.
Global Instance into_wand_iff_r P Q : IntoWand (P ↔ Q) Q P.
Proof. apply and_elim_r', impl_wand. Qed.
Global Instance into_wand_always R P Q : IntoWand R P Q → IntoWand (□ R) P Q.
Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
(* FromAnd *)
Global Instance from_and_and P1 P2 : FromAnd (P1 ∧ P2) P1 P2.
Proof. done. Qed.
Global Instance from_and_sep_persistent_l P1 P2 :
PersistentP P1 → FromAnd (P1 ★ P2) P1 P2 | 9.
Proof. intros. by rewrite /FromAnd always_and_sep_l. Qed.
Global Instance from_and_sep_persistent_r P1 P2 :
PersistentP P2 → FromAnd (P1 ★ P2) P1 P2 | 10.
Proof. intros. by rewrite /FromAnd always_and_sep_r. Qed.
Global Instance from_and_always P Q1 Q2 :
FromAnd P Q1 Q2 → FromAnd (□ P) (□ Q1) (□ Q2).
Proof. rewrite /FromAnd=> <-. by rewrite always_and. Qed.
Global Instance from_and_later P Q1 Q2 :
FromAnd P Q1 Q2 → FromAnd (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed.
(* FromSep *)
Global Instance from_sep_sep P1 P2 : FromSep (P1 ★ P2) P1 P2 | 100.
Proof. done. Qed.
Global Instance from_sep_always P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (□ P) (□ Q1) (□ Q2).
Proof. rewrite /FromSep=> <-. by rewrite always_sep. Qed.
Global Instance from_sep_later P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed.
Global Instance from_sep_ownM (a b : M) :
FromSep (uPred_ownM (a ⋅ b)) (uPred_ownM a) (uPred_ownM b) | 99.
Proof. by rewrite /FromSep ownM_op. Qed.
Global Instance from_sep_big_sepM
`{Countable K} {A} (Φ Ψ1 Ψ2 : K → A → uPred M) m :
(∀ k x, FromSep (Φ k x) (Ψ1 k x) (Ψ2 k x)) →
FromSep ([★ map] k ↦ x ∈ m, Φ k x)
([★ map] k ↦ x ∈ m, Ψ1 k x) ([★ map] k ↦ x ∈ m, Ψ2 k x).
Proof.
rewrite /FromSep=> ?. rewrite -big_sepM_sepM. by apply big_sepM_mono.
Qed.
Global Instance from_sep_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A → uPred M) X :
(∀ x, FromSep (Φ x) (Ψ1 x) (Ψ2 x)) →
FromSep ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ1 x) ([★ set] x ∈ X, Ψ2 x).
Proof.
rewrite /FromSep=> ?. rewrite -big_sepS_sepS. by apply big_sepS_mono.
Qed.
(* IntoOp *)
Global Instance into_op_op {A : cmraT} (a b : A) : IntoOp (a ⋅ b) a b.
Proof. by rewrite /IntoOp. Qed.
Global Instance into_op_persistent {A : cmraT} (a : A) :
Persistent a → IntoOp a a a.
Proof. intros; apply (persistent_dup a). Qed.
Global Instance into_op_pair {A B : cmraT} (a b1 b2 : A) (a' b1' b2' : B) :
IntoOp a b1 b2 → IntoOp a' b1' b2' →
IntoOp (a,a') (b1,b1') (b2,b2').
Proof. by constructor. Qed.
Global Instance into_op_Some {A : cmraT} (a : A) b1 b2 :
IntoOp a b1 b2 → IntoOp (Some a) (Some b1) (Some b2).
Proof. by constructor. Qed.
(* IntoSep *)
Global Instance into_sep_sep p P Q : IntoSep p (P ★ Q) P Q.
Proof. rewrite /IntoSep. by rewrite always_if_sep. Qed.
Global Instance into_sep_ownM p (a b1 b2 : M) :
IntoOp a b1 b2 →
IntoSep p (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof.
rewrite /IntoOp /IntoSep=> ->. by rewrite ownM_op always_if_sep.
Qed.
Global Instance into_sep_and P Q : IntoSep true (P ∧ Q) P Q.
Proof. by rewrite /IntoSep /= always_and_sep. Qed.
Global Instance into_sep_and_persistent_l P Q :
PersistentP P → IntoSep false (P ∧ Q) P Q.
Proof. intros; by rewrite /IntoSep /= always_and_sep_l. Qed.
Global Instance into_sep_and_persistent_r P Q :
PersistentP Q → IntoSep false (P ∧ Q) P Q.
Proof. intros; by rewrite /IntoSep /= always_and_sep_r. Qed.
Global Instance into_sep_later p P Q1 Q2 :
IntoSep p P Q1 Q2 → IntoSep p (▷ P) (▷ Q1) (▷ Q2).
Proof. by rewrite /IntoSep -later_sep !always_if_later=> ->. Qed.
Global Instance into_sep_big_sepM
`{Countable K} {A} (Φ Ψ1 Ψ2 : K → A → uPred M) p m :
(∀ k x, IntoSep p (Φ k x) (Ψ1 k x) (Ψ2 k x)) →
IntoSep p ([★ map] k ↦ x ∈ m, Φ k x)
([★ map] k ↦ x ∈ m, Ψ1 k x) ([★ map] k ↦ x ∈ m, Ψ2 k x).
Proof.
rewrite /IntoSep=> ?. rewrite -big_sepM_sepM !big_sepM_always_if.
by apply big_sepM_mono.
Qed.
Global Instance into_sep_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A → uPred M) p X :
(∀ x, IntoSep p (Φ x) (Ψ1 x) (Ψ2 x)) →
IntoSep p ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ1 x) ([★ set] x ∈ X, Ψ2 x).
Proof.
rewrite /IntoSep=> ?. rewrite -big_sepS_sepS !big_sepS_always_if.
by apply big_sepS_mono.
Qed.
(* Frame *)
Global Instance frame_here R : Frame R R True.
Proof. by rewrite /Frame right_id. Qed.
Class MakeSep (P Q PQ : uPred M) := make_sep : P ★ Q ⊣⊢ PQ.
Global Instance make_sep_true_l P : MakeSep True P P.
Proof. by rewrite /MakeSep left_id. Qed.
Global Instance make_sep_true_r P : MakeSep P True P.
Proof. by rewrite /MakeSep right_id. Qed.
Global Instance make_sep_default P Q : MakeSep P Q (P ★ Q) | 100.
Proof. done. Qed.
Global Instance frame_sep_l R P1 P2 Q Q' :
Frame R P1 Q → MakeSep Q P2 Q' → Frame R (P1 ★ P2) Q' | 9.
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc. Qed.
Global Instance frame_sep_r R P1 P2 Q Q' :
Frame R P2 Q → MakeSep P1 Q Q' → Frame R (P1 ★ P2) Q' | 10.
Proof. rewrite /Frame /MakeSep => <- <-. solve_sep_entails. Qed.
Class MakeAnd (P Q PQ : uPred M) := make_and : P ∧ Q ⊣⊢ PQ.
Global Instance make_and_true_l P : MakeAnd True P P.
Proof. by rewrite /MakeAnd left_id. Qed.
Global Instance make_and_true_r P : MakeAnd P True P.
Proof. by rewrite /MakeAnd right_id. Qed.
Global Instance make_and_default P Q : MakeSep P Q (P ★ Q) | 100.
Proof. done. Qed.
Global Instance frame_and_l R P1 P2 Q Q' :
Frame R P1 Q → MakeAnd Q P2 Q' → Frame R (P1 ∧ P2) Q' | 9.
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.
Global Instance frame_and_r R P1 P2 Q Q' :
Frame R P2 Q → MakeAnd P1 Q Q' → Frame R (P1 ∧ P2) Q' | 10.
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.
Class MakeOr (P Q PQ : uPred M) := make_or : P ∨ Q ⊣⊢ PQ.
Global Instance make_or_true_l P : MakeOr True P True.
Proof. by rewrite /MakeOr left_absorb. Qed.
Global Instance make_or_true_r P : MakeOr P True True.
Proof. by rewrite /MakeOr right_absorb. Qed.
Global Instance make_or_default P Q : MakeOr P Q (P ∨ Q) | 100.
Proof. done. Qed.
Global Instance frame_or R P1 P2 Q1 Q2 Q :
Frame R P1 Q1 → Frame R P2 Q2 → MakeOr Q1 Q2 Q → Frame R (P1 ∨ P2) Q.
Proof. rewrite /Frame /MakeOr => <- <- <-. by rewrite -sep_or_l. Qed.
Global Instance frame_wand R P1 P2 Q2 :
Frame R P2 Q2 → Frame R (P1 -★ P2) (P1 -★ Q2).
Proof.
rewrite /Frame=> ?. apply wand_intro_l.
by rewrite assoc (comm _ P1) -assoc wand_elim_r.
Qed.
Class MakeLater (P lP : uPred M) := make_later : ▷ P ⊣⊢ lP.
Global Instance make_later_true : MakeLater True True.
Proof. by rewrite /MakeLater later_True. Qed.
Global Instance make_later_default P : MakeLater P (▷ P) | 100.
Proof. done. Qed.
Global Instance frame_later R P Q Q' :
Frame R P Q → MakeLater Q Q' → Frame R (▷ P) Q'.
Proof.
rewrite /Frame /MakeLater=><- <-. by rewrite later_sep -(later_intro R).
Qed.
Global Instance frame_exist {A} R (Φ Ψ : A → uPred M) :
(∀ a, Frame R (Φ a) (Ψ a)) → Frame R (∃ x, Φ x) (∃ x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
Global Instance frame_forall {A} R (Φ Ψ : A → uPred M) :
(∀ a, Frame R (Φ a) (Ψ a)) → Frame R (∀ x, Φ x) (∀ x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.
(* FromOr *)
Global Instance from_or_or P1 P2 : FromOr (P1 ∨ P2) P1 P2.
Proof. done. Qed.
(* IntoOr *)
Global Instance into_or_or P Q : IntoOr (P ∨ Q) P Q.
Proof. done. Qed.
Global Instance into_or_later P Q1 Q2 :
IntoOr P Q1 Q2 → IntoOr (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed.
(* FromExist *)
Global Instance from_exist_exist {A} (Φ: A → uPred M): FromExist (∃ a, Φ a) Φ.
Proof. done. Qed.
(* IntoExist *)
Global Instance into_exist_exist {A} (Φ : A → uPred M) : IntoExist (∃ a, Φ a) Φ.
Proof. done. Qed.
Global Instance into_exist_later {A} P (Φ : A → uPred M) :
IntoExist P Φ → Inhabited A → IntoExist (▷ P) (λ a, ▷ (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP later_exist. Qed.
Global Instance into_exist_always {A} P (Φ : A → uPred M) :
IntoExist P Φ → IntoExist (□ P) (λ a, □ (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP always_exist. Qed.
End classes.