interface.v 23.6 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
From iris.algebra Require Export ofe.
2
Set Primitive Projections.
Robbert Krebbers's avatar
Robbert Krebbers committed
3
4
5
6
7
8
9
10
11

Reserved Notation "P ⊢ Q" (at level 99, Q at level 200, right associativity).
Reserved Notation "'emp'".
Reserved Notation "'⌜' φ '⌝'" (at level 1, φ at level 200, format "⌜ φ ⌝").
Reserved Notation "P ∗ Q" (at level 80, right associativity).
Reserved Notation "P -∗ Q" (at level 99, Q at level 200, right associativity).
Reserved Notation "▷ P" (at level 20, right associativity).

Section bi_mixin.
12
  Context {PROP : Type} `{Dist PROP, Equiv PROP} (prop_ofe_mixin : OfeMixin PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
13
14
15
16
17
18
19
20
21
22
23
  Context (bi_entails : PROP  PROP  Prop).
  Context (bi_emp : PROP).
  Context (bi_pure : Prop  PROP).
  Context (bi_and : PROP  PROP  PROP).
  Context (bi_or : PROP  PROP  PROP).
  Context (bi_impl : PROP  PROP  PROP).
  Context (bi_forall :  A, (A  PROP)  PROP).
  Context (bi_exist :  A, (A  PROP)  PROP).
  Context (bi_internal_eq :  A : ofeT, A  A  PROP).
  Context (bi_sep : PROP  PROP  PROP).
  Context (bi_wand : PROP  PROP  PROP).
24
  Context (bi_plainly : PROP  PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
25
26
27
28
29
30
31
  Context (bi_persistently : PROP  PROP).
  Context (bi_later : PROP  PROP).

  Local Infix "⊢" := bi_entails.
  Local Notation "'emp'" := bi_emp.
  Local Notation "'True'" := (bi_pure True).
  Local Notation "'False'" := (bi_pure False).
32
  Local Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp).
Robbert Krebbers's avatar
Robbert Krebbers committed
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
  Local Infix "∧" := bi_and.
  Local Infix "∨" := bi_or.
  Local Infix "→" := bi_impl.
  Local Notation "∀ x .. y , P" :=
    (bi_forall _ (λ x, .. (bi_forall _ (λ y, P)) ..)).
  Local Notation "∃ x .. y , P" :=
    (bi_exist _ (λ x, .. (bi_exist _ (λ y, P)) ..)).
  Local Notation "x ≡ y" := (bi_internal_eq _ x y).
  Local Infix "∗" := bi_sep.
  Local Infix "-∗" := bi_wand.
  Local Notation "▷ P" := (bi_later P).

  Record BIMixin := {
    bi_mixin_entails_po : PreOrder bi_entails;
    bi_mixin_equiv_spec P Q : equiv P Q  (P  Q)  (Q  P);

    (* Non-expansiveness *)
    bi_mixin_pure_ne n : Proper (iff ==> dist n) bi_pure;
    bi_mixin_and_ne : NonExpansive2 bi_and;
    bi_mixin_or_ne : NonExpansive2 bi_or;
    bi_mixin_impl_ne : NonExpansive2 bi_impl;
    bi_mixin_forall_ne A n :
      Proper (pointwise_relation _ (dist n) ==> dist n) (bi_forall A);
    bi_mixin_exist_ne A n :
      Proper (pointwise_relation _ (dist n) ==> dist n) (bi_exist A);
    bi_mixin_sep_ne : NonExpansive2 bi_sep;
    bi_mixin_wand_ne : NonExpansive2 bi_wand;
60
    bi_mixin_plainly_ne : NonExpansive bi_plainly;
Robbert Krebbers's avatar
Robbert Krebbers committed
61
    bi_mixin_persistently_ne : NonExpansive bi_persistently;
62
    bi_mixin_internal_eq_ne (A : ofeT) : NonExpansive2 (bi_internal_eq A);
Robbert Krebbers's avatar
Robbert Krebbers committed
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

    (* Higher-order logic *)
    bi_mixin_pure_intro P (φ : Prop) : φ  P   φ ;
    bi_mixin_pure_elim' (φ : Prop) P : (φ  True  P)   φ   P;
    bi_mixin_pure_forall_2 {A} (φ : A  Prop) : ( a,  φ a )    a, φ a ;

    bi_mixin_and_elim_l P Q : P  Q  P;
    bi_mixin_and_elim_r P Q : P  Q  Q;
    bi_mixin_and_intro P Q R : (P  Q)  (P  R)  P  Q  R;

    bi_mixin_or_intro_l P Q : P  P  Q;
    bi_mixin_or_intro_r P Q : Q  P  Q;
    bi_mixin_or_elim P Q R : (P  R)  (Q  R)  P  Q  R;

    bi_mixin_impl_intro_r P Q R : (P  Q  R)  P  Q  R;
    bi_mixin_impl_elim_l' P Q R : (P  Q  R)  P  Q  R;

    bi_mixin_forall_intro {A} P (Ψ : A  PROP) : ( a, P  Ψ a)  P   a, Ψ a;
    bi_mixin_forall_elim {A} {Ψ : A  PROP} a : ( a, Ψ a)  Ψ a;

    bi_mixin_exist_intro {A} {Ψ : A  PROP} a : Ψ a   a, Ψ a;
    bi_mixin_exist_elim {A} (Φ : A  PROP) Q : ( a, Φ a  Q)  ( a, Φ a)  Q;

    (* Equality *)
    bi_mixin_internal_eq_refl {A : ofeT} P (a : A) : P  a  a;
    bi_mixin_internal_eq_rewrite {A : ofeT} a b (Ψ : A  PROP) :
      NonExpansive Ψ  a  b  Ψ a  Ψ b;
90
    bi_mixin_fun_ext {A} {B : A  ofeT} (f g : ofe_fun B) : ( x, f x  g x)  f  g;
Robbert Krebbers's avatar
Robbert Krebbers committed
91
92
93
94
95
96
97
98
99
100
101
102
    bi_mixin_sig_eq {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  x  y;
    bi_mixin_discrete_eq_1 {A : ofeT} (a b : A) : Discrete a  a  b  a  b;

    (* BI connectives *)
    bi_mixin_sep_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q';
    bi_mixin_emp_sep_1 P : P  emp  P;
    bi_mixin_emp_sep_2 P : emp  P  P;
    bi_mixin_sep_comm' P Q : P  Q  Q  P;
    bi_mixin_sep_assoc' P Q R : (P  Q)  R  P  (Q  R);
    bi_mixin_wand_intro_r P Q R : (P  Q  R)  P  Q - R;
    bi_mixin_wand_elim_l' P Q R : (P  Q - R)  P  Q  R;

103
104
105
106
107
108
109
110
111
112
113
    (* Plainly *)
    bi_mixin_plainly_mono P Q : (P  Q)  bi_plainly P  bi_plainly Q;
    bi_mixin_plainly_elim_persistently P : bi_plainly P  bi_persistently P;
    bi_mixin_plainly_idemp_2 P : bi_plainly P  bi_plainly (bi_plainly P);

    bi_mixin_plainly_forall_2 {A} (Ψ : A  PROP) :
      ( a, bi_plainly (Ψ a))  bi_plainly ( a, Ψ a);

    bi_mixin_prop_ext P Q : bi_plainly ((P  Q)  (Q  P)) 
      bi_internal_eq (OfeT PROP prop_ofe_mixin) P Q;

114
115
116
    (* The following two laws are very similar, and indeed they hold
       not just for □ and ■, but for any modality defined as
       `M P n x := ∀ y, R x y → P n y`. *)
117
118
119
120
121
122
123
124
    bi_mixin_persistently_impl_plainly P Q :
      (bi_plainly P  bi_persistently Q)  bi_persistently (bi_plainly P  Q);
    bi_mixin_plainly_impl_plainly P Q :
      (bi_plainly P  bi_plainly Q)  bi_plainly (bi_plainly P  Q);

    bi_mixin_plainly_emp_intro P : P  bi_plainly emp;
    bi_mixin_plainly_absorbing P Q : bi_plainly P  Q  bi_plainly P;

Robbert Krebbers's avatar
Robbert Krebbers committed
125
    (* Persistently *)
126
127
128
129
    bi_mixin_persistently_mono P Q :
      (P  Q)  bi_persistently P  bi_persistently Q;
    bi_mixin_persistently_idemp_2 P :
      bi_persistently P  bi_persistently (bi_persistently P);
130
131
    bi_mixin_plainly_persistently_1 P :
      bi_plainly (bi_persistently P)  bi_plainly P;
Robbert Krebbers's avatar
Robbert Krebbers committed
132
133

    bi_mixin_persistently_forall_2 {A} (Ψ : A  PROP) :
134
      ( a, bi_persistently (Ψ a))  bi_persistently ( a, Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
135
    bi_mixin_persistently_exist_1 {A} (Ψ : A  PROP) :
136
      bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
137

138
139
140
141
    bi_mixin_persistently_absorbing P Q :
      bi_persistently P  Q  bi_persistently P;
    bi_mixin_persistently_and_sep_elim P Q :
      bi_persistently P  Q  (emp  P)  Q;
Robbert Krebbers's avatar
Robbert Krebbers committed
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
  }.

  Record SBIMixin := {
    sbi_mixin_later_contractive : Contractive bi_later;

    sbi_mixin_later_eq_1 {A : ofeT} (x y : A) : Next x  Next y   (x  y);
    sbi_mixin_later_eq_2 {A : ofeT} (x y : A) :  (x  y)  Next x  Next y;

    sbi_mixin_later_mono P Q : (P  Q)   P   Q;
    sbi_mixin_löb P : ( P  P)  P;

    sbi_mixin_later_forall_2 {A} (Φ : A  PROP) : ( a,  Φ a)    a, Φ a;
    sbi_mixin_later_exist_false {A} (Φ : A  PROP) :
      (  a, Φ a)   False  ( a,  Φ a);
    sbi_mixin_later_sep_1 P Q :  (P  Q)   P   Q;
    sbi_mixin_later_sep_2 P Q :  P   Q   (P  Q);
158
159
    sbi_mixin_later_plainly_1 P :  bi_plainly P  bi_plainly ( P);
    sbi_mixin_later_plainly_2 P : bi_plainly ( P)   bi_plainly P;
160
161
162
163
    sbi_mixin_later_persistently_1 P :
       bi_persistently P  bi_persistently ( P);
    sbi_mixin_later_persistently_2 P :
      bi_persistently ( P)   bi_persistently P;
Robbert Krebbers's avatar
Robbert Krebbers committed
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183

    sbi_mixin_later_false_em P :  P   False  ( False  P);
  }.
End bi_mixin.

Structure bi := BI {
  bi_car :> Type;
  bi_dist : Dist bi_car;
  bi_equiv : Equiv bi_car;
  bi_entails : bi_car  bi_car  Prop;
  bi_emp : bi_car;
  bi_pure : Prop  bi_car;
  bi_and : bi_car  bi_car  bi_car;
  bi_or : bi_car  bi_car  bi_car;
  bi_impl : bi_car  bi_car  bi_car;
  bi_forall :  A, (A  bi_car)  bi_car;
  bi_exist :  A, (A  bi_car)  bi_car;
  bi_internal_eq :  A : ofeT, A  A  bi_car;
  bi_sep : bi_car  bi_car  bi_car;
  bi_wand : bi_car  bi_car  bi_car;
184
  bi_plainly : bi_car  bi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
185
186
  bi_persistently : bi_car  bi_car;
  bi_ofe_mixin : OfeMixin bi_car;
187
188
189
  bi_bi_mixin : BIMixin bi_ofe_mixin bi_entails bi_emp bi_pure bi_and bi_or
                        bi_impl bi_forall bi_exist bi_internal_eq
                        bi_sep bi_wand bi_plainly bi_persistently;
Robbert Krebbers's avatar
Robbert Krebbers committed
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
}.

Coercion bi_ofeC (PROP : bi) : ofeT := OfeT PROP (bi_ofe_mixin PROP).
Canonical Structure bi_ofeC.

Instance: Params (@bi_entails) 1.
Instance: Params (@bi_emp) 1.
Instance: Params (@bi_pure) 1.
Instance: Params (@bi_and) 1.
Instance: Params (@bi_or) 1.
Instance: Params (@bi_impl) 1.
Instance: Params (@bi_forall) 2.
Instance: Params (@bi_exist) 2.
Instance: Params (@bi_internal_eq) 2.
Instance: Params (@bi_sep) 1.
Instance: Params (@bi_wand) 1.
206
Instance: Params (@bi_plainly) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
207
208
209
210
211
212
213
214
Instance: Params (@bi_persistently) 1.

Delimit Scope bi_scope with I.
Arguments bi_car : simpl never.
Arguments bi_dist : simpl never.
Arguments bi_equiv : simpl never.
Arguments bi_entails {PROP} _%I _%I : simpl never, rename.
Arguments bi_emp {PROP} : simpl never, rename.
215
Arguments bi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
216
217
218
219
220
221
222
223
Arguments bi_and {PROP} _%I _%I : simpl never, rename.
Arguments bi_or {PROP} _%I _%I : simpl never, rename.
Arguments bi_impl {PROP} _%I _%I : simpl never, rename.
Arguments bi_forall {PROP _} _%I : simpl never, rename.
Arguments bi_exist {PROP _} _%I : simpl never, rename.
Arguments bi_internal_eq {PROP _} _ _ : simpl never, rename.
Arguments bi_sep {PROP} _%I _%I : simpl never, rename.
Arguments bi_wand {PROP} _%I _%I : simpl never, rename.
224
Arguments bi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
Arguments bi_persistently {PROP} _%I : simpl never, rename.

Structure sbi := SBI {
  sbi_car :> Type;
  sbi_dist : Dist sbi_car;
  sbi_equiv : Equiv sbi_car;
  sbi_entails : sbi_car  sbi_car  Prop;
  sbi_emp : sbi_car;
  sbi_pure : Prop  sbi_car;
  sbi_and : sbi_car  sbi_car  sbi_car;
  sbi_or : sbi_car  sbi_car  sbi_car;
  sbi_impl : sbi_car  sbi_car  sbi_car;
  sbi_forall :  A, (A  sbi_car)  sbi_car;
  sbi_exist :  A, (A  sbi_car)  sbi_car;
  sbi_internal_eq :  A : ofeT, A  A  sbi_car;
  sbi_sep : sbi_car  sbi_car  sbi_car;
  sbi_wand : sbi_car  sbi_car  sbi_car;
242
  sbi_plainly : sbi_car  sbi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
243
244
245
  sbi_persistently : sbi_car  sbi_car;
  bi_later : sbi_car  sbi_car;
  sbi_ofe_mixin : OfeMixin sbi_car;
246
247
248
  sbi_bi_mixin : BIMixin sbi_ofe_mixin sbi_entails sbi_emp sbi_pure sbi_and
                         sbi_or sbi_impl sbi_forall sbi_exist sbi_internal_eq
                         sbi_sep sbi_wand sbi_plainly sbi_persistently;
249
  sbi_sbi_mixin : SBIMixin sbi_entails sbi_pure sbi_or sbi_impl
Robbert Krebbers's avatar
Robbert Krebbers committed
250
                           sbi_forall sbi_exist sbi_internal_eq
251
                           sbi_sep sbi_plainly sbi_persistently bi_later;
Robbert Krebbers's avatar
Robbert Krebbers committed
252
253
254
255
256
}.

Arguments sbi_car : simpl never.
Arguments sbi_entails {PROP} _%I _%I : simpl never, rename.
Arguments bi_emp {PROP} : simpl never, rename.
257
Arguments bi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
258
259
260
261
262
263
264
265
Arguments bi_and {PROP} _%I _%I : simpl never, rename.
Arguments bi_or {PROP} _%I _%I : simpl never, rename.
Arguments bi_impl {PROP} _%I _%I : simpl never, rename.
Arguments bi_forall {PROP _} _%I : simpl never, rename.
Arguments bi_exist {PROP _} _%I : simpl never, rename.
Arguments bi_internal_eq {PROP _} _ _ : simpl never, rename.
Arguments bi_sep {PROP} _%I _%I : simpl never, rename.
Arguments bi_wand {PROP} _%I _%I : simpl never, rename.
266
Arguments bi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
267
268
269
270
271
272
273
274
275
276
277
278
279
Arguments bi_persistently {PROP} _%I : simpl never, rename.

Coercion sbi_ofeC (PROP : sbi) : ofeT := OfeT PROP (sbi_ofe_mixin PROP).
Canonical Structure sbi_ofeC.
Coercion sbi_bi (PROP : sbi) : bi :=
  {| bi_ofe_mixin := sbi_ofe_mixin PROP; bi_bi_mixin := sbi_bi_mixin PROP |}.
Canonical Structure sbi_bi.

Arguments sbi_car : simpl never.
Arguments sbi_dist : simpl never.
Arguments sbi_equiv : simpl never.
Arguments sbi_entails {PROP} _%I _%I : simpl never, rename.
Arguments sbi_emp {PROP} : simpl never, rename.
280
Arguments sbi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
281
282
283
284
285
286
287
288
Arguments sbi_and {PROP} _%I _%I : simpl never, rename.
Arguments sbi_or {PROP} _%I _%I : simpl never, rename.
Arguments sbi_impl {PROP} _%I _%I : simpl never, rename.
Arguments sbi_forall {PROP _} _%I : simpl never, rename.
Arguments sbi_exist {PROP _} _%I : simpl never, rename.
Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename.
Arguments sbi_sep {PROP} _%I _%I : simpl never, rename.
Arguments sbi_wand {PROP} _%I _%I : simpl never, rename.
289
Arguments sbi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
290
291
292
293
294
295
296
Arguments sbi_persistently {PROP} _%I : simpl never, rename.
Arguments bi_later {PROP} _%I : simpl never, rename.

Hint Extern 0 (bi_entails _ _) => reflexivity.
Instance bi_rewrite_relation (PROP : bi) : RewriteRelation (@bi_entails PROP).
Instance bi_inhabited {PROP : bi} : Inhabited PROP := populate (bi_pure True).

297
298
Notation "P ⊢ Q" := (bi_entails P%I Q%I) : stdpp_scope.
Notation "(⊢)" := bi_entails (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
299
300

Notation "P ⊣⊢ Q" := (equiv (A:=bi_car _) P%I Q%I)
301
302
  (at level 95, no associativity) : stdpp_scope.
Notation "(⊣⊢)" := (equiv (A:=bi_car _)) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
303

304
Notation "P -∗ Q" := (P  Q) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
305
306

Notation "'emp'" := (bi_emp) : bi_scope.
307
Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
Notation "'True'" := (bi_pure True) : bi_scope.
Notation "'False'" := (bi_pure False) : bi_scope.
Infix "∧" := bi_and : bi_scope.
Notation "(∧)" := bi_and (only parsing) : bi_scope.
Infix "∨" := bi_or : bi_scope.
Notation "(∨)" := bi_or (only parsing) : bi_scope.
Infix "→" := bi_impl : bi_scope.
Infix "∗" := bi_sep : bi_scope.
Notation "(∗)" := bi_sep (only parsing) : bi_scope.
Notation "P -∗ Q" := (bi_wand P Q) : bi_scope.
Notation "∀ x .. y , P" :=
  (bi_forall (λ x, .. (bi_forall (λ y, P)) ..)%I) : bi_scope.
Notation "∃ x .. y , P" :=
  (bi_exist (λ x, .. (bi_exist (λ y, P)) ..)%I) : bi_scope.

Infix "≡" := bi_internal_eq : bi_scope.
Notation "▷ P" := (bi_later P) : bi_scope.

Coercion bi_valid {PROP : bi} (P : PROP) : Prop := emp  P.
Coercion sbi_valid {PROP : sbi} : PROP  Prop := bi_valid.

Arguments bi_valid {_} _%I : simpl never.
Typeclasses Opaque bi_valid.

Module bi.
Section bi_laws.
Context {PROP : bi}.
Implicit Types φ : Prop.
Implicit Types P Q R : PROP.
Implicit Types A : Type.

(* About the entailment *)
Global Instance entails_po : PreOrder (@bi_entails PROP).
Proof. eapply bi_mixin_entails_po, bi_bi_mixin. Qed.
Lemma equiv_spec P Q : P  Q  (P  Q)  (Q  P).
Proof. eapply bi_mixin_equiv_spec, bi_bi_mixin. Qed.

(* Non-expansiveness *)
Global Instance pure_ne n : Proper (iff ==> dist n) (@bi_pure PROP).
Proof. eapply bi_mixin_pure_ne, bi_bi_mixin. Qed.
Global Instance and_ne : NonExpansive2 (@bi_and PROP).
Proof. eapply bi_mixin_and_ne, bi_bi_mixin. Qed.
Global Instance or_ne : NonExpansive2 (@bi_or PROP).
Proof. eapply bi_mixin_or_ne, bi_bi_mixin. Qed.
Global Instance impl_ne : NonExpansive2 (@bi_impl PROP).
Proof. eapply bi_mixin_impl_ne, bi_bi_mixin. Qed.
Global Instance forall_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@bi_forall PROP A).
Proof. eapply bi_mixin_forall_ne, bi_bi_mixin. Qed.
Global Instance exist_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@bi_exist PROP A).
Proof. eapply bi_mixin_exist_ne, bi_bi_mixin. Qed.
Global Instance sep_ne : NonExpansive2 (@bi_sep PROP).
Proof. eapply bi_mixin_sep_ne, bi_bi_mixin. Qed.
Global Instance wand_ne : NonExpansive2 (@bi_wand PROP).
Proof. eapply bi_mixin_wand_ne, bi_bi_mixin. Qed.
364
365
Global Instance plainly_ne : NonExpansive (@bi_plainly PROP).
Proof. eapply bi_mixin_plainly_ne, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
Global Instance persistently_ne : NonExpansive (@bi_persistently PROP).
Proof. eapply bi_mixin_persistently_ne, bi_bi_mixin. Qed.

(* Higher-order logic *)
Lemma pure_intro P (φ : Prop) : φ  P   φ .
Proof. eapply bi_mixin_pure_intro, bi_bi_mixin. Qed.
Lemma pure_elim' (φ : Prop) P : (φ  True  P)   φ   P.
Proof. eapply bi_mixin_pure_elim', bi_bi_mixin. Qed.
Lemma pure_forall_2 {A} (φ : A  Prop) : ( a,  φ a  : PROP)    a, φ a .
Proof. eapply bi_mixin_pure_forall_2, bi_bi_mixin. Qed.

Lemma and_elim_l P Q : P  Q  P.
Proof. eapply bi_mixin_and_elim_l, bi_bi_mixin. Qed.
Lemma and_elim_r P Q : P  Q  Q.
Proof. eapply bi_mixin_and_elim_r, bi_bi_mixin. Qed.
Lemma and_intro P Q R : (P  Q)  (P  R)  P  Q  R.
Proof. eapply bi_mixin_and_intro, bi_bi_mixin. Qed.

Lemma or_intro_l P Q : P  P  Q.
Proof. eapply bi_mixin_or_intro_l, bi_bi_mixin. Qed.
Lemma or_intro_r P Q : Q  P  Q.
Proof. eapply bi_mixin_or_intro_r, bi_bi_mixin. Qed.
Lemma or_elim P Q R : (P  R)  (Q  R)  P  Q  R.
Proof. eapply bi_mixin_or_elim, bi_bi_mixin. Qed.

Lemma impl_intro_r P Q R : (P  Q  R)  P  Q  R.
Proof. eapply bi_mixin_impl_intro_r, bi_bi_mixin. Qed.
Lemma impl_elim_l' P Q R : (P  Q  R)  P  Q  R.
Proof. eapply bi_mixin_impl_elim_l', bi_bi_mixin. Qed.

Lemma forall_intro {A} P (Ψ : A  PROP) : ( a, P  Ψ a)  P   a, Ψ a.
Proof. eapply bi_mixin_forall_intro, bi_bi_mixin. Qed.
Lemma forall_elim {A} {Ψ : A  PROP} a : ( a, Ψ a)  Ψ a.
399
Proof. eapply (bi_mixin_forall_elim  _ bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
400
401
402
403
404
405
406

Lemma exist_intro {A} {Ψ : A  PROP} a : Ψ a   a, Ψ a.
Proof. eapply bi_mixin_exist_intro, bi_bi_mixin. Qed.
Lemma exist_elim {A} (Φ : A  PROP) Q : ( a, Φ a  Q)  ( a, Φ a)  Q.
Proof. eapply bi_mixin_exist_elim, bi_bi_mixin. Qed.

(* Equality *)
407
408
Global Instance internal_eq_ne (A : ofeT) : NonExpansive2 (@bi_internal_eq PROP A).
Proof. eapply bi_mixin_internal_eq_ne, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
409
410
411
412
413
414
415

Lemma internal_eq_refl {A : ofeT} P (a : A) : P  a  a.
Proof. eapply bi_mixin_internal_eq_refl, bi_bi_mixin. Qed.
Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A  PROP) :
  NonExpansive Ψ  a  b  Ψ a  Ψ b.
Proof. eapply bi_mixin_internal_eq_rewrite, bi_bi_mixin. Qed.

416
Lemma fun_ext {A} {B : A  ofeT} (f g : ofe_fun B) : ( x, f x  g x)  (f  g : PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
Proof. eapply bi_mixin_fun_ext, bi_bi_mixin. Qed.
Lemma sig_eq {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  (x  y : PROP).
Proof. eapply bi_mixin_sig_eq, bi_bi_mixin. Qed.
Lemma discrete_eq_1 {A : ofeT} (a b : A) :
  Discrete a  a  b  (a  b : PROP).
Proof. eapply bi_mixin_discrete_eq_1, bi_bi_mixin. Qed.

(* BI connectives *)
Lemma sep_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
Proof. eapply bi_mixin_sep_mono, bi_bi_mixin. Qed.
Lemma emp_sep_1 P : P  emp  P.
Proof. eapply bi_mixin_emp_sep_1, bi_bi_mixin. Qed.
Lemma emp_sep_2 P : emp  P  P.
Proof. eapply bi_mixin_emp_sep_2, bi_bi_mixin. Qed.
Lemma sep_comm' P Q : P  Q  Q  P.
432
Proof. eapply (bi_mixin_sep_comm' _ bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
433
434
435
436
437
438
439
Lemma sep_assoc' P Q R : (P  Q)  R  P  (Q  R).
Proof. eapply bi_mixin_sep_assoc', bi_bi_mixin. Qed.
Lemma wand_intro_r P Q R : (P  Q  R)  P  Q - R.
Proof. eapply bi_mixin_wand_intro_r, bi_bi_mixin. Qed.
Lemma wand_elim_l' P Q R : (P  Q - R)  P  Q  R.
Proof. eapply bi_mixin_wand_elim_l', bi_bi_mixin. Qed.

440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
(* Plainly *)
Lemma plainly_mono P Q : (P  Q)  bi_plainly P  bi_plainly Q.
Proof. eapply bi_mixin_plainly_mono, bi_bi_mixin. Qed.
Lemma plainly_elim_persistently P : bi_plainly P  bi_persistently P.
Proof. eapply bi_mixin_plainly_elim_persistently, bi_bi_mixin. Qed.
Lemma plainly_idemp_2 P : bi_plainly P  bi_plainly (bi_plainly P).
Proof. eapply bi_mixin_plainly_idemp_2, bi_bi_mixin. Qed.
Lemma plainly_forall_2 {A} (Ψ : A  PROP) :
  ( a, bi_plainly (Ψ a))  bi_plainly ( a, Ψ a).
Proof. eapply bi_mixin_plainly_forall_2, bi_bi_mixin. Qed.
Lemma prop_ext P Q : bi_plainly ((P  Q)  (Q  P))  P  Q.
Proof. eapply (bi_mixin_prop_ext _ bi_entails), bi_bi_mixin. Qed.
Lemma persistently_impl_plainly P Q :
  (bi_plainly P  bi_persistently Q)  bi_persistently (bi_plainly P  Q).
Proof. eapply bi_mixin_persistently_impl_plainly, bi_bi_mixin. Qed.
Lemma plainly_impl_plainly P Q :
  (bi_plainly P  bi_plainly Q)  bi_plainly (bi_plainly P  Q).
Proof. eapply bi_mixin_plainly_impl_plainly, bi_bi_mixin. Qed.
Lemma plainly_absorbing P Q : bi_plainly P  Q  bi_plainly P.
Proof. eapply (bi_mixin_plainly_absorbing _ bi_entails), bi_bi_mixin. Qed.
Lemma plainly_emp_intro P : P  bi_plainly emp.
Proof. eapply bi_mixin_plainly_emp_intro, bi_bi_mixin. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
463
(* Persistently *)
464
Lemma persistently_mono P Q : (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
465
Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
466
467
Lemma persistently_idemp_2 P :
  bi_persistently P  bi_persistently (bi_persistently P).
Robbert Krebbers's avatar
Robbert Krebbers committed
468
Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.
469
470
471
Lemma plainly_persistently_1 P :
  bi_plainly (bi_persistently P)  bi_plainly P.
Proof. eapply (bi_mixin_plainly_persistently_1 _ bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
472

473
474
Lemma persistently_forall_2 {A} (Ψ : A  PROP) :
  ( a, bi_persistently (Ψ a))  bi_persistently ( a, Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
475
Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed.
476
477
Lemma persistently_exist_1 {A} (Ψ : A  PROP) :
  bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
478
479
Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed.

480
Lemma persistently_absorbing P Q : bi_persistently P  Q  bi_persistently P.
481
Proof. eapply (bi_mixin_persistently_absorbing _ bi_entails), bi_bi_mixin. Qed.
482
Lemma persistently_and_sep_elim P Q : bi_persistently P  Q  (emp  P)  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
483
Proof. eapply bi_mixin_persistently_and_sep_elim, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
End bi_laws.

Section sbi_laws.
Context {PROP : sbi}.
Implicit Types φ : Prop.
Implicit Types P Q R : PROP.

Global Instance later_contractive : Contractive (@bi_later PROP).
Proof. eapply sbi_mixin_later_contractive, sbi_sbi_mixin. Qed.

Lemma later_eq_1 {A : ofeT} (x y : A) : Next x  Next y   (x  y : PROP).
Proof. eapply sbi_mixin_later_eq_1, sbi_sbi_mixin. Qed.
Lemma later_eq_2 {A : ofeT} (x y : A) :  (x  y)  (Next x  Next y : PROP).
Proof. eapply sbi_mixin_later_eq_2, sbi_sbi_mixin. Qed.

Lemma later_mono P Q : (P  Q)   P   Q.
Proof. eapply sbi_mixin_later_mono, sbi_sbi_mixin. Qed.
Lemma löb P : ( P  P)  P.
Proof. eapply sbi_mixin_löb, sbi_sbi_mixin. Qed.

Lemma later_forall_2 {A} (Φ : A  PROP) : ( a,  Φ a)    a, Φ a.
Proof. eapply sbi_mixin_later_forall_2, sbi_sbi_mixin. Qed.
Lemma later_exist_false {A} (Φ : A  PROP) :
  (  a, Φ a)   False  ( a,  Φ a).
Proof. eapply sbi_mixin_later_exist_false, sbi_sbi_mixin. Qed.
Lemma later_sep_1 P Q :  (P  Q)   P   Q.
Proof. eapply sbi_mixin_later_sep_1, sbi_sbi_mixin. Qed.
Lemma later_sep_2 P Q :  P   Q   (P  Q).
Proof. eapply sbi_mixin_later_sep_2, sbi_sbi_mixin. Qed.
513
514
515
516
Lemma later_plainly_1 P :  bi_plainly P  bi_plainly ( P).
Proof. eapply (sbi_mixin_later_plainly_1 bi_entails), sbi_sbi_mixin. Qed.
Lemma later_plainly_2 P : bi_plainly ( P)   bi_plainly P.
Proof. eapply (sbi_mixin_later_plainly_2 bi_entails), sbi_sbi_mixin. Qed.
517
Lemma later_persistently_1 P :  bi_persistently P  bi_persistently ( P).
Robbert Krebbers's avatar
Robbert Krebbers committed
518
Proof. eapply (sbi_mixin_later_persistently_1 bi_entails), sbi_sbi_mixin. Qed.
519
Lemma later_persistently_2 P : bi_persistently ( P)   bi_persistently P.
Robbert Krebbers's avatar
Robbert Krebbers committed
520
521
522
523
524
525
Proof. eapply (sbi_mixin_later_persistently_2 bi_entails), sbi_sbi_mixin. Qed.

Lemma later_false_em P :  P   False  ( False  P).
Proof. eapply sbi_mixin_later_false_em, sbi_sbi_mixin. Qed.
End sbi_laws.
End bi.