classes.v 21 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
From iris.bi Require Export bi.
2
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
3
4
5
Import bi.

Class FromAssumption {PROP : bi} (p : bool) (P Q : PROP) :=
6
  from_assumption : ?p P  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
7
8
9
Arguments FromAssumption {_} _ _%I _%I : simpl never.
Arguments from_assumption {_} _ _%I _%I {_}.
(* No need to restrict Hint Mode, we have a default instance that will always
10
11
be used in case of evars *)
Hint Mode FromAssumption + + - - : typeclass_instances.
12

Robbert Krebbers's avatar
Robbert Krebbers committed
13
14
15
16
Class IntoPure {PROP : bi} (P : PROP) (φ : Prop) :=
  into_pure : P  ⌜φ⌝.
Arguments IntoPure {_} _%I _%type_scope : simpl never.
Arguments into_pure {_} _%I _%type_scope {_}.
17
18
Hint Mode IntoPure + ! - : typeclass_instances.

19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
(* [IntoPureT] is a variant of [IntoPure] with the argument in [Type] to avoid
some shortcoming of unification in Coq's type class search. An example where we
use this workaround is to repair the following instance:

  Global Instance into_exist_and_pure P Q (φ : Prop) :
    IntoPure P φ → IntoExist (P ∧ Q) (λ _ : φ, Q).

Coq is unable to use this instance: [class_apply] -- which is used by type class
search -- fails with the error that it cannot unify [Prop] and [Type]. This is
probably caused because [class_apply] uses an ancient unification algorith. The
[refine] tactic -- which uses a better unification algorithm -- succeeds to
apply the above instance.

Since we do not want to define [Hint Extern] declarations using [refine] for
any instance like [into_exist_and_pure], we factor this out in the class
[IntoPureT]. This way, we only have to declare a [Hint Extern] using [refine]
once, and use [IntoPureT] in any instance like [into_exist_and_pure].

TODO: Report this as a Coq bug, or wait for https://github.com/coq/coq/pull/991
to be finished and merged someday. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
39
Class IntoPureT {PROP : bi} (P : PROP) (φ : Type) :=
40
  into_pureT :  ψ : Prop, φ = ψ  IntoPure P ψ.
Robbert Krebbers's avatar
Robbert Krebbers committed
41
Lemma into_pureT_hint {PROP : bi} (P : PROP) (φ : Prop) : IntoPure P φ  IntoPureT P φ.
42
43
44
45
Proof. by exists φ. Qed.
Hint Extern 0 (IntoPureT _ _) =>
  notypeclasses refine (into_pureT_hint _ _ _) : typeclass_instances.

46
47
48
49
50
51
52
53
54
55
Class FromPure {PROP : bi} (a : bool) (P : PROP) (φ : Prop) :=
  from_pure : bi_affinely_if a ⌜φ⌝  P.
Arguments FromPure {_} _ _%I _%type_scope : simpl never.
Arguments from_pure {_} _ _%I _%type_scope {_}.
Hint Mode FromPure + + ! - : typeclass_instances.

Class FromPureT {PROP : bi} (a : bool) (P : PROP) (φ : Type) :=
  from_pureT :  ψ : Prop, φ = ψ  FromPure a P ψ.
Lemma from_pureT_hint {PROP : bi} (a : bool) (P : PROP) (φ : Prop) :
  FromPure a P φ  FromPureT a P φ.
Robbert Krebbers's avatar
Robbert Krebbers committed
56
Proof. by exists φ. Qed.
57
58
Hint Extern 0 (FromPureT _ _ _) =>
  notypeclasses refine (from_pureT_hint _ _ _ _) : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
59

60
Class IntoInternalEq {PROP : sbi} {A : ofeT} (P : PROP) (x y : A) :=
61
  into_internal_eq : P  x  y.
62
63
Arguments IntoInternalEq {_ _} _%I _%type_scope _%type_scope : simpl never.
Arguments into_internal_eq {_ _} _%I _%type_scope _%type_scope {_}.
64
65
Hint Mode IntoInternalEq + - ! - - : typeclass_instances.

Robbert Krebbers's avatar
Robbert Krebbers committed
66
Class IntoPersistent {PROP : bi} (p : bool) (P Q : PROP) :=
67
  into_persistent : bi_persistently_if p P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
68
69
Arguments IntoPersistent {_} _ _%I _%I : simpl never.
Arguments into_persistent {_} _ _%I _%I {_}.
70
Hint Mode IntoPersistent + + ! - : typeclass_instances.
71

72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
(* The `iAlways` tactic is not tied to `persistently` and `affinely`, but can be
instantiated with a variety of comonadic (always-style) modalities.

In order to plug in an always-style modality, one has to decide for both the
persistent and spatial what action should be performed upon introducing the
modality:

- Introduction is only allowed when the context is empty.
- Introduction is only allowed when all hypotheses satisfy some predicate
  `C : PROP → Prop` (where `C` should be a type class).
- Introduction will only keep the hypotheses that satisfy some predicate
  `C : PROP → Prop` (where `C` should be a type class).
- Introduction will clear the context.
- Introduction will keep the context as-if.

Formally, these actions correspond to the following inductive type: *)
Robbert Krebbers's avatar
Robbert Krebbers committed
88
89
90
91
92
93
94
95
96
97
98
99
Inductive always_intro_spec (PROP : bi) :=
  | AIEnvIsEmpty
  | AIEnvForall (C : PROP  Prop)
  | AIEnvFilter (C : PROP  Prop)
  | AIEnvClear
  | AIEnvId.
Arguments AIEnvIsEmpty {_}.
Arguments AIEnvForall {_} _.
Arguments AIEnvFilter {_} _.
Arguments AIEnvClear {_}.
Arguments AIEnvId {_}.

100
101
(* An always-style modality is then a record packing together the modality with
the laws it should satisfy to justify the given actions for both contexts: *)
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Record always_modality_mixin {PROP : bi} (M : PROP  PROP)
    (pspec sspec : always_intro_spec PROP) := {
  always_modality_mixin_persistent :
    match pspec with
    | AIEnvIsEmpty => True
    | AIEnvForall C | AIEnvFilter C =>  P, C P   P  M ( P)
    | AIEnvClear => True
    | AIEnvId =>  P,  P  M ( P)
    end;
  always_modality_mixin_spatial :
    match sspec with
    | AIEnvIsEmpty => True
    | AIEnvForall C =>  P, C P  P  M P
    | AIEnvFilter C => ( P, C P  P  M P)  ( P, Absorbing (M P))
    | AIEnvClear =>  P, Absorbing (M P)
    | AIEnvId => False
    end;
  always_modality_mixin_emp : emp  M emp;
  always_modality_mixin_mono P Q : (P  Q)  M P  M Q;
  always_modality_mixin_and P Q : M P  M Q  M (P  Q);
  always_modality_mixin_sep P Q : M P  M Q  M (P  Q)
}.

Record always_modality (PROP : bi) := AlwaysModality {
  always_modality_car :> PROP  PROP;
  always_modality_persistent_spec : always_intro_spec PROP;
  always_modality_spatial_spec : always_intro_spec PROP;
  always_modality_mixin_of : always_modality_mixin
    always_modality_car
    always_modality_persistent_spec always_modality_spatial_spec
}.
Arguments AlwaysModality {_} _ {_ _} _.
Arguments always_modality_persistent_spec {_} _.
Arguments always_modality_spatial_spec {_} _.

Section always_modality.
  Context {PROP} (M : always_modality PROP).

  Lemma always_modality_persistent_forall C P :
    always_modality_persistent_spec M = AIEnvForall C  C P   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_persistent_filter C P :
    always_modality_persistent_spec M = AIEnvFilter C  C P   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_persistent_id P :
    always_modality_persistent_spec M = AIEnvId   P  M ( P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_forall C P :
    always_modality_spatial_spec M = AIEnvForall C  C P  P  M P.
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_filter C P :
    always_modality_spatial_spec M = AIEnvFilter C  C P  P  M P.
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_filter_absorbing C P :
    always_modality_spatial_spec M = AIEnvFilter C  Absorbing (M P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_clear P :
    always_modality_spatial_spec M = AIEnvClear  Absorbing (M P).
  Proof. destruct M as [??? []]; naive_solver. Qed.
  Lemma always_modality_spatial_id :
    always_modality_spatial_spec M  AIEnvId.
  Proof. destruct M as [??? []]; naive_solver. Qed.

  Lemma always_modality_emp : emp  M emp.
  Proof. eapply always_modality_mixin_emp, always_modality_mixin_of. Qed.
  Lemma always_modality_mono P Q : (P  Q)  M P  M Q.
  Proof. eapply always_modality_mixin_mono, always_modality_mixin_of. Qed.
  Lemma always_modality_and P Q : M P  M Q  M (P  Q).
  Proof. eapply always_modality_mixin_and, always_modality_mixin_of. Qed.
  Lemma always_modality_sep P Q : M P  M Q  M (P  Q).
  Proof. eapply always_modality_mixin_sep, always_modality_mixin_of. Qed.
  Global Instance always_modality_mono' : Proper (() ==> ()) M.
  Proof. intros P Q. apply always_modality_mono. Qed.
  Global Instance always_modality_flip_mono' : Proper (flip () ==> flip ()) M.
  Proof. intros P Q. apply always_modality_mono. Qed.
  Global Instance always_modality_proper : Proper (() ==> ()) M.
  Proof. intros P Q. rewrite !equiv_spec=> -[??]; eauto using always_modality_mono. Qed.

  Lemma always_modality_persistent_forall_big_and C Ps :
    always_modality_persistent_spec M = AIEnvForall C 
    Forall C Ps   [] Ps  M ( [] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
    - by rewrite persistently_pure affinely_True_emp affinely_emp -always_modality_emp.
    - rewrite affinely_persistently_and -always_modality_and -IH.
      by rewrite {1}(always_modality_persistent_forall _ P).
  Qed.
  Lemma always_modality_spatial_forall_big_sep C Ps :
    always_modality_spatial_spec M = AIEnvForall C 
    Forall C Ps  [] Ps  M ([] Ps).
  Proof.
    induction 2 as [|P Ps ? _ IH]; simpl.
    - by rewrite -always_modality_emp.
    - by rewrite -always_modality_sep -IH {1}(always_modality_spatial_forall _ P).
  Qed.
End always_modality.

Class FromAlways {PROP : bi} (M : always_modality PROP) (P Q : PROP) :=
  from_always : M Q  P.
Arguments FromAlways {_} _ _%I _%I : simpl never.
Arguments from_always {_} _ _%I _%I {_}.
Hint Mode FromAlways + - ! - : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
204

205
206
207
208
209
210
Class FromAffinely {PROP : bi} (P Q : PROP) :=
  from_affinely : bi_affinely Q  P.
Arguments FromAffinely {_} _%I _%type_scope : simpl never.
Arguments from_affinely {_} _%I _%type_scope {_}.
Hint Mode FromAffinely + ! - : typeclass_instances.
Hint Mode FromAffinely + - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
211

212
213
Class IntoAbsorbingly {PROP : bi} (P Q : PROP) :=
  into_absorbingly : P  bi_absorbingly Q.
214
215
216
217
Arguments IntoAbsorbingly {_} _%I _%I.
Arguments into_absorbingly {_} _%I _%I {_}.
Hint Mode IntoAbsorbingly + ! -  : typeclass_instances.
Hint Mode IntoAbsorbingly + - ! : typeclass_instances.
218

Robbert Krebbers's avatar
Robbert Krebbers committed
219
220
221
222
223
224
225
226
227
(*
Converting an assumption [R] into a wand [P -∗ Q] is done in three stages:

- Strip modalities and universal quantifiers of [R] until an arrow or a wand
  has been obtained.
- Balance modalities in the arguments [P] and [Q] to match the goal (which used
  for [iApply]) or the premise (when used with [iSpecialize] and a specific
  hypothesis).
- Instantiate the premise of the wand or implication.
228
*)
Robbert Krebbers's avatar
Robbert Krebbers committed
229
Class IntoWand {PROP : bi} (p q : bool) (R P Q : PROP) :=
230
  into_wand : ?p R  ?q P - Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
Arguments IntoWand {_} _ _ _%I _%I _%I : simpl never.
Arguments into_wand {_} _ _ _%I _%I _%I {_}.
Hint Mode IntoWand + + + ! - - : typeclass_instances.

Class IntoWand' {PROP : bi} (p q : bool) (R P Q : PROP) :=
  into_wand' : IntoWand p q R P Q.
Arguments IntoWand' {_} _ _ _%I _%I _%I : simpl never.
Hint Mode IntoWand' + + + ! ! - : typeclass_instances.
Hint Mode IntoWand' + + + ! - ! : typeclass_instances.

Instance into_wand_wand' {PROP : bi} p q (P Q P' Q' : PROP) :
  IntoWand' p q (P - Q) P' Q'  IntoWand p q (P - Q) P' Q' | 100.
Proof. done. Qed.
Instance into_wand_impl' {PROP : bi} p q (P Q P' Q' : PROP) :
  IntoWand' p q (P  Q) P' Q'  IntoWand p q (P  Q) P' Q' | 100.
Proof. done. Qed.
247

248
249
250
251
252
253
254
255
256
257
Class FromWand {PROP : bi} (P Q1 Q2 : PROP) := from_wand : (Q1 - Q2)  P.
Arguments FromWand {_} _%I _%I _%I : simpl never.
Arguments from_wand {_} _%I _%I _%I {_}.
Hint Mode FromWand + ! - - : typeclass_instances.

Class FromImpl {PROP : bi} (P Q1 Q2 : PROP) := from_impl : (Q1  Q2)  P.
Arguments FromImpl {_} _%I _%I _%I : simpl never.
Arguments from_impl {_} _%I _%I _%I {_}.
Hint Mode FromImpl + ! - - : typeclass_instances.

Robbert Krebbers's avatar
Robbert Krebbers committed
258
259
260
261
262
263
264
265
266
267
268
269
270
Class FromSep {PROP : bi} (P Q1 Q2 : PROP) := from_sep : Q1  Q2  P.
Arguments FromSep {_} _%I _%I _%I : simpl never.
Arguments from_sep {_} _%I _%I _%I {_}.
Hint Mode FromSep + ! - - : typeclass_instances.
Hint Mode FromSep + - ! ! : typeclass_instances. (* For iCombine *)

Class FromAnd {PROP : bi} (P Q1 Q2 : PROP) := from_and : Q1  Q2  P.
Arguments FromAnd {_} _%I _%I _%I : simpl never.
Arguments from_and {_} _%I _%I _%I {_}.
Hint Mode FromAnd + ! - - : typeclass_instances.
Hint Mode FromAnd + - ! ! : typeclass_instances. (* For iCombine *)

Class IntoAnd {PROP : bi} (p : bool) (P Q1 Q2 : PROP) :=
271
  into_and : ?p P  ?p (Q1  Q2).
Robbert Krebbers's avatar
Robbert Krebbers committed
272
273
Arguments IntoAnd {_} _ _%I _%I _%I : simpl never.
Arguments into_and {_} _ _%I _%I _%I {_}.
274
Hint Mode IntoAnd + + ! - - : typeclass_instances.
275

276
277
278
279
280
Class IntoSep {PROP : bi} (P Q1 Q2 : PROP) :=
  into_sep : P  Q1  Q2.
Arguments IntoSep {_} _%I _%I _%I : simpl never.
Arguments into_sep {_} _%I _%I _%I {_}.
Hint Mode IntoSep + ! - - : typeclass_instances.
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
281

Robbert Krebbers's avatar
Robbert Krebbers committed
282
283
284
Class FromOr {PROP : bi} (P Q1 Q2 : PROP) := from_or : Q1  Q2  P.
Arguments FromOr {_} _%I _%I _%I : simpl never.
Arguments from_or {_} _%I _%I _%I {_}.
285
Hint Mode FromOr + ! - - : typeclass_instances.
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
286

Robbert Krebbers's avatar
Robbert Krebbers committed
287
288
289
Class IntoOr {PROP : bi} (P Q1 Q2 : PROP) := into_or : P  Q1  Q2.
Arguments IntoOr {_} _%I _%I _%I : simpl never.
Arguments into_or {_} _%I _%I _%I {_}.
290
Hint Mode IntoOr + ! - - : typeclass_instances.
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
291

Robbert Krebbers's avatar
Robbert Krebbers committed
292
Class FromExist {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
293
  from_exist : ( x, Φ x)  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
294
295
Arguments FromExist {_ _} _%I _%I : simpl never.
Arguments from_exist {_ _} _%I _%I {_}.
296
Hint Mode FromExist + - ! - : typeclass_instances.
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
297

Robbert Krebbers's avatar
Robbert Krebbers committed
298
Class IntoExist {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
Robbert Krebbers's avatar
Oops!    
Robbert Krebbers committed
299
  into_exist : P   x, Φ x.
Robbert Krebbers's avatar
Robbert Krebbers committed
300
301
Arguments IntoExist {_ _} _%I _%I : simpl never.
Arguments into_exist {_ _} _%I _%I {_}.
302
Hint Mode IntoExist + - ! - : typeclass_instances.
303

Robbert Krebbers's avatar
Robbert Krebbers committed
304
Class IntoForall {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
305
  into_forall : P   x, Φ x.
Robbert Krebbers's avatar
Robbert Krebbers committed
306
307
Arguments IntoForall {_ _} _%I _%I : simpl never.
Arguments into_forall {_ _} _%I _%I {_}.
308
309
Hint Mode IntoForall + - ! - : typeclass_instances.

Robbert Krebbers's avatar
Robbert Krebbers committed
310
Class FromForall {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :=
311
312
313
314
  from_forall : ( x, Φ x)  P.
Arguments from_forall {_ _} _ _ {_}.
Hint Mode FromForall + - ! - : typeclass_instances.

Robbert Krebbers's avatar
Robbert Krebbers committed
315
316
317
318
319
320
321
322
Class IsExcept0 {PROP : sbi} (Q : PROP) := is_except_0 :  Q  Q.
Arguments IsExcept0 {_} _%I : simpl never.
Arguments is_except_0 {_} _%I {_}.
Hint Mode IsExcept0 + ! : typeclass_instances.

Class FromModal {PROP : bi} (P Q : PROP) := from_modal : Q  P.
Arguments FromModal {_} _%I _%I : simpl never.
Arguments from_modal {_} _%I _%I {_}.
323
Hint Mode FromModal + ! - : typeclass_instances.
324

Robbert Krebbers's avatar
Robbert Krebbers committed
325
Class ElimModal {PROP : bi} (P P' : PROP) (Q Q' : PROP) :=
326
  elim_modal : P  (P' - Q')  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
327
328
Arguments ElimModal {_} _%I _%I _%I _%I : simpl never.
Arguments elim_modal {_} _%I _%I _%I _%I {_}.
329
Hint Mode ElimModal + ! - ! - : typeclass_instances.
330

331
332
(* Used by the specialization pattern [ > ] in [iSpecialize] and [iAssert] to
add a modality to the goal corresponding to a premise/asserted proposition. *)
333
Class AddModal {PROP : bi} (P P' : PROP) (Q : PROP) :=
334
  add_modal : P  (P' - Q)  Q.
335
336
Arguments AddModal {_} _%I _%I _%I : simpl never.
Arguments add_modal {_} _%I _%I _%I {_}.
337
338
Hint Mode AddModal + - ! ! : typeclass_instances.

339
Lemma add_modal_id {PROP : bi} (P Q : PROP) : AddModal P P Q.
340
Proof. by rewrite /AddModal wand_elim_r. Qed.
341

342
343
344
345
346
347
348
349
350
Class IsCons {A} (l : list A) (x : A) (k : list A) := is_cons : l = x :: k.
Class IsApp {A} (l k1 k2 : list A) := is_app : l = k1 ++ k2.
Global Hint Mode IsCons + ! - - : typeclass_instances.
Global Hint Mode IsApp + ! - - : typeclass_instances.

Instance is_cons_cons {A} (x : A) (l : list A) : IsCons (x :: l) x l.
Proof. done. Qed.
Instance is_app_app {A} (l1 l2 : list A) : IsApp (l1 ++ l2) l1 l2.
Proof. done. Qed.
351

352
Class Frame {PROP : bi} (p : bool) (R P Q : PROP) := frame : ?p R  Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
353
354
355
356
357
Arguments Frame {_} _ _%I _%I _%I.
Arguments frame {_ _} _%I _%I _%I {_}.
Hint Mode Frame + + ! ! - : typeclass_instances.

Class MaybeFrame {PROP : bi} (p : bool) (R P Q : PROP) :=
358
  maybe_frame : ?p R  Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
359
360
361
362
363
364
365
366
367
368
369
370
371
372
Arguments MaybeFrame {_} _ _%I _%I _%I.
Arguments maybe_frame {_} _%I _%I _%I {_}.
Hint Mode MaybeFrame + + ! ! - : typeclass_instances.

Instance maybe_frame_frame {PROP : bi} p (R P Q : PROP) :
  Frame p R P Q  MaybeFrame p R P Q.
Proof. done. Qed.
Instance maybe_frame_default_persistent {PROP : bi} (R P : PROP) :
  MaybeFrame true R P P | 100.
Proof. intros. rewrite /MaybeFrame /=. by rewrite sep_elim_r. Qed.
Instance maybe_frame_default {PROP : bi} (R P : PROP) :
  TCOr (Affine R) (Absorbing P)  MaybeFrame false R P P | 100.
Proof. intros. rewrite /MaybeFrame /=. apply: sep_elim_r. Qed.

373
374
375
376
377
378
Class IntoExcept0 {PROP : sbi} (P Q : PROP) := into_except_0 : P   Q.
Arguments IntoExcept0 {_} _%I _%I : simpl never.
Arguments into_except_0 {_} _%I _%I {_}.
Hint Mode IntoExcept0 + ! - : typeclass_instances.
Hint Mode IntoExcept0 + - ! : typeclass_instances.

Robbert Krebbers's avatar
Robbert Krebbers committed
379
380
381
382
383
384
385
386
(* The class [IntoLaterN] has only two instances:

- The default instance [IntoLaterN n P P], i.e. [▷^n P -∗ P]
- The instance [IntoLaterN' n P Q → IntoLaterN n P Q], where [IntoLaterN']
  is identical to [IntoLaterN], but computationally is supposed to make
  progress, i.e. its instances should actually strip a later.

The point of using the auxilary class [IntoLaterN'] is to ensure that the
387
default instance is not applied deeply in the term, which may result in too many
Robbert Krebbers's avatar
Robbert Krebbers committed
388
389
390
391
392
393
394
395
396
397
398
399
definitions being unfolded (see issue #55).

For binary connectives we have the following instances:

<<
IntoLaterN' n P P'       IntoLaterN n Q Q'
------------------------------------------
     IntoLaterN' n (P /\ Q) (P' /\ Q')


      IntoLaterN' n Q Q'
-------------------------------
400
IntoLaterN' n (P /\ Q) (P /\ Q')
Robbert Krebbers's avatar
Robbert Krebbers committed
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
>>
*)
Class IntoLaterN {PROP : sbi} (n : nat) (P Q : PROP) := into_laterN : P  ^n Q.
Arguments IntoLaterN {_} _%nat_scope _%I _%I.
Arguments into_laterN {_} _%nat_scope _%I _%I {_}.
Hint Mode IntoLaterN + - - - : typeclass_instances.

Class IntoLaterN' {PROP : sbi} (n : nat) (P Q : PROP) :=
  into_laterN' :> IntoLaterN n P Q.
Arguments IntoLaterN' {_} _%nat_scope _%I _%I.
Hint Mode IntoLaterN' + - ! - : typeclass_instances.

Instance into_laterN_default {PROP : sbi} n (P : PROP) : IntoLaterN n P P | 1000.
Proof. apply laterN_intro. Qed.

Class FromLaterN {PROP : sbi} (n : nat) (P Q : PROP) := from_laterN : ^n Q  P.
Arguments FromLaterN {_} _%nat_scope _%I _%I.
Arguments from_laterN {_} _%nat_scope _%I _%I {_}.
Hint Mode FromLaterN + - ! - : typeclass_instances.

421
422
423
424
425
426
427
428
429
430
431
432
433
Class AsValid {PROP : bi} (φ : Prop) (P : PROP) := as_valid : φ  P.
Arguments AsValid {_} _%type _%I.

Class AsValid0 {PROP : bi} (φ : Prop) (P : PROP) :=
  as_valid_here : AsValid φ P.
Arguments AsValid0 {_} _%type _%I.
Existing Instance as_valid_here | 0.

Lemma as_valid_1 (φ : Prop) {PROP : bi} (P : PROP) `{!AsValid φ P} : φ  P.
Proof. by apply as_valid. Qed.
Lemma as_valid_2 (φ : Prop) {PROP : bi} (P : PROP) `{!AsValid φ P} : P  φ.
Proof. by apply as_valid. Qed.

434
435
436
437
438
439
440
441
442
443
444
445
(* We make sure that tactics that perform actions on *specific* hypotheses or
parts of the goal look through the [tc_opaque] connective, which is used to make
definitions opaque for type class search. For example, when using `iDestruct`,
an explicit hypothesis is affected, and as such, we should look through opaque
definitions. However, when using `iFrame` or `iNext`, arbitrary hypotheses or
parts of the goal are affected, and as such, type class opacity should be
respected.

This means that there are [tc_opaque] instances for all proofmode type classes
with the exception of:

- [FromAssumption] used by [iAssumption]
Robbert Krebbers's avatar
Robbert Krebbers committed
446
- [Frame] and [MaybeFrame] used by [iFrame]
447
- [IntoLaterN] and [FromLaterN] used by [iNext]
Robbert Krebbers's avatar
Robbert Krebbers committed
448
- [IntoPersistent] used by [iPersistent]
449
*)
Robbert Krebbers's avatar
Robbert Krebbers committed
450
Instance into_pure_tc_opaque {PROP : bi} (P : PROP) φ :
451
  IntoPure P φ  IntoPure (tc_opaque P) φ := id.
452
453
Instance from_pure_tc_opaque {PROP : bi} (a : bool) (P : PROP) φ :
  FromPure a P φ  FromPure a (tc_opaque P) φ := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
454
Instance from_laterN_tc_opaque {PROP : sbi} n (P Q : PROP) :
455
  FromLaterN n P Q  FromLaterN n (tc_opaque P) Q := id.
456
457
Instance from_wand_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
  FromWand P Q1 Q2  FromWand (tc_opaque P) Q1 Q2 := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
458
459
Instance into_wand_tc_opaque {PROP : bi} p q (R P Q : PROP) :
  IntoWand p q R P Q  IntoWand p q (tc_opaque R) P Q := id.
460
(* Higher precedence than [from_and_sep] so that [iCombine] does not loop. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
461
462
463
Instance from_and_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
  FromAnd P Q1 Q2  FromAnd (tc_opaque P) Q1 Q2 | 102 := id.
Instance into_and_tc_opaque {PROP : bi} p (P Q1 Q2 : PROP) :
464
  IntoAnd p P Q1 Q2  IntoAnd p (tc_opaque P) Q1 Q2 := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
465
Instance from_or_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
466
  FromOr P Q1 Q2  FromOr (tc_opaque P) Q1 Q2 := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
467
Instance into_or_tc_opaque {PROP : bi} (P Q1 Q2 : PROP) :
468
  IntoOr P Q1 Q2  IntoOr (tc_opaque P) Q1 Q2 := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
469
Instance from_exist_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
470
  FromExist P Φ  FromExist (tc_opaque P) Φ := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
471
Instance into_exist_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
472
  IntoExist P Φ  IntoExist (tc_opaque P) Φ := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
473
Instance into_forall_tc_opaque {PROP : bi} {A} (P : PROP) (Φ : A  PROP) :
474
  IntoForall P Φ  IntoForall (tc_opaque P) Φ := id.
Robbert Krebbers's avatar
Robbert Krebbers committed
475
Instance from_modal_tc_opaque {PROP : bi} (P Q : PROP) :
476
  FromModal P Q  FromModal (tc_opaque P) Q := id.
477
478
(* Higher precedence than [elim_modal_timeless], so that [iAssert] does not
   loop (see test [test_iAssert_modality] in proofmode.v). *)
Robbert Krebbers's avatar
Robbert Krebbers committed
479
Instance elim_modal_tc_opaque {PROP : bi} (P P' Q Q' : PROP) :
480
  ElimModal P P' Q Q'  ElimModal (tc_opaque P) P' Q Q' | 100 := id.