interface.v 23.8 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
5
6
7
8
9
10
From iris.algebra Require Export ofe.

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.
11
  Context {PROP : Type} `{Dist PROP, Equiv PROP} (prop_ofe_mixin : OfeMixin PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
12
13
14
15
16
17
18
19
20
21
22
  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).
23
  Context (bi_plainly : PROP  PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
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
  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).
  Local Notation "'⌜' φ '⌝'" := (bi_pure φ%type%C).
  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;
59
    bi_mixin_plainly_ne : NonExpansive bi_plainly;
Robbert Krebbers's avatar
Robbert Krebbers committed
60
    bi_mixin_persistently_ne : NonExpansive bi_persistently;
61
    bi_mixin_internal_eq_ne (A : ofeT) : NonExpansive2 (bi_internal_eq A);
Robbert Krebbers's avatar
Robbert Krebbers committed
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

    (* 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;
    bi_mixin_fun_ext {A B} (f g : A -c> B) : ( x, f x  g x)  f  g;
    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;

102
103
104
105
106
107
108
109
110
111
112
113
114
    (* 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_plainly_exist_1 {A} (Ψ : A  PROP) :
      bi_plainly ( a, Ψ a)   a, bi_plainly (Ψ a);

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

115
116
117
    (* 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`. *)
118
119
120
121
122
123
124
125
    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
126
    (* Persistently *)
127
128
129
130
    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);
131
132
    bi_mixin_plainly_persistently_1 P :
      bi_plainly (bi_persistently P)  bi_plainly P;
Robbert Krebbers's avatar
Robbert Krebbers committed
133
134

    bi_mixin_persistently_forall_2 {A} (Ψ : A  PROP) :
135
      ( a, bi_persistently (Ψ a))  bi_persistently ( a, Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
136
    bi_mixin_persistently_exist_1 {A} (Ψ : A  PROP) :
137
      bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
138

139
140
141
142
    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
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
  }.

  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);
159
160
    sbi_mixin_later_plainly_1 P :  bi_plainly P  bi_plainly ( P);
    sbi_mixin_later_plainly_2 P : bi_plainly ( P)   bi_plainly P;
161
162
163
164
    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
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184

    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;
185
  bi_plainly : bi_car  bi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
186
187
  bi_persistently : bi_car  bi_car;
  bi_ofe_mixin : OfeMixin bi_car;
188
189
190
  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
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
}.

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.
207
Instance: Params (@bi_plainly) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
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.
Arguments bi_pure {PROP} _%C : simpl never, rename.
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.
225
Arguments bi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
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;
243
  sbi_plainly : sbi_car  sbi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
244
245
246
  sbi_persistently : sbi_car  sbi_car;
  bi_later : sbi_car  sbi_car;
  sbi_ofe_mixin : OfeMixin sbi_car;
247
248
249
  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;
250
  sbi_sbi_mixin : SBIMixin sbi_entails sbi_pure sbi_or sbi_impl
Robbert Krebbers's avatar
Robbert Krebbers committed
251
                           sbi_forall sbi_exist sbi_internal_eq
252
                           sbi_sep sbi_plainly sbi_persistently bi_later;
Robbert Krebbers's avatar
Robbert Krebbers committed
253
254
255
256
257
258
259
260
261
262
263
264
265
266
}.

Arguments sbi_car : simpl never.
Arguments sbi_entails {PROP} _%I _%I : simpl never, rename.
Arguments bi_emp {PROP} : simpl never, rename.
Arguments bi_pure {PROP} _%C : simpl never, rename.
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.
267
Arguments bi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
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.
Arguments sbi_pure {PROP} _%C : simpl never, rename.
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.
290
Arguments sbi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
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
364
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).

Notation "P ⊢ Q" := (bi_entails P%I Q%I) : C_scope.
Notation "(⊢)" := bi_entails (only parsing) : C_scope.

Notation "P ⊣⊢ Q" := (equiv (A:=bi_car _) P%I Q%I)
  (at level 95, no associativity) : C_scope.
Notation "(⊣⊢)" := (equiv (A:=bi_car _)) (only parsing) : C_scope.

Notation "P -∗ Q" := (P  Q) : C_scope.

Notation "'emp'" := (bi_emp) : bi_scope.
Notation "'⌜' φ '⌝'" := (bi_pure φ%type%C) : bi_scope.
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.
365
366
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
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
399
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.
400
Proof. eapply (bi_mixin_forall_elim  _ bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
401
402
403
404
405
406
407

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 *)
408
409
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
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432

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.

Lemma fun_ext {A B} (f g : A -c> B) : ( x, f x  g x)  (f  g : PROP).
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.
433
Proof. eapply (bi_mixin_sep_comm' _ bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
434
435
436
437
438
439
440
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.

441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
(* 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 plainly_exist_1 {A} (Ψ : A  PROP) :
  bi_plainly ( a, Ψ a)   a, bi_plainly (Ψ a).
Proof. eapply bi_mixin_plainly_exist_1, 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
467
(* Persistently *)
468
Lemma persistently_mono P Q : (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
469
Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
470
471
Lemma persistently_idemp_2 P :
  bi_persistently P  bi_persistently (bi_persistently P).
Robbert Krebbers's avatar
Robbert Krebbers committed
472
Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.
473
474
475
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
476

477
478
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
479
Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed.
480
481
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
482
483
Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed.

484
Lemma persistently_absorbing P Q : bi_persistently P  Q  bi_persistently P.
485
Proof. eapply (bi_mixin_persistently_absorbing _ bi_entails), bi_bi_mixin. Qed.
486
Lemma persistently_and_sep_elim P Q : bi_persistently P  Q  (emp  P)  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
487
Proof. eapply bi_mixin_persistently_and_sep_elim, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
513
514
515
516
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.
517
518
519
520
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.
521
Lemma later_persistently_1 P :  bi_persistently P  bi_persistently ( P).
Robbert Krebbers's avatar
Robbert Krebbers committed
522
Proof. eapply (sbi_mixin_later_persistently_1 bi_entails), sbi_sbi_mixin. Qed.
523
Lemma later_persistently_2 P : bi_persistently ( P)   bi_persistently P.
Robbert Krebbers's avatar
Robbert Krebbers committed
524
525
526
527
528
529
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.