interface.v 20.5 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

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).
9
Reserved Notation "'<pers>' P" (at level 20, right associativity).
Robbert Krebbers's avatar
Robbert Krebbers committed
10
11
12
Reserved Notation "▷ P" (at level 20, right associativity).

Section bi_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
13
  Context {PROP : Type} `{Dist PROP, Equiv PROP}.
Robbert Krebbers's avatar
Robbert Krebbers committed
14
15
16
17
18
19
20
21
22
23
24
  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_sep : PROP  PROP  PROP).
  Context (bi_wand : PROP  PROP  PROP).
  Context (bi_persistently : PROP  PROP).
25
  Context (sbi_internal_eq :  A : ofeT, A  A  PROP).
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
26
  Context (sbi_later : PROP  PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
27
28
29
30
31

  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
  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 Infix "∗" := bi_sep.
  Local Infix "-∗" := bi_wand.
42
  Local Notation "'<pers>' P" := (bi_persistently P).
43
  Local Notation "x ≡ y" := (sbi_internal_eq _ x y).
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
44
  Local Notation "▷ P" := (sbi_later P).
Robbert Krebbers's avatar
Robbert Krebbers committed
45

Ralf Jung's avatar
Ralf Jung committed
46
47
48
49
50
51
  (** * Axioms for a general BI (logic of bunched implications) *)

  (** The following axioms are satisifed by both affine and linear BIs, and BIs
  that combine both kinds of resources. In particular, we have an "ordered RA"
  model satisfying all these axioms. For this model, we extend RAs with an
  arbitrary partial order, and up-close resources wrt. that order (instead of
52
  extension order).  We demand composition to be monotone wrt. the order: [x1 ≼
53
54
55
  x2 → x1 ⋅ y ≼ x2 ⋅ y].  We define [emp := λ r, ε ≼ r]; persistently is still
  defined with the core: [persistently P := λ r, P (core r)].  This is uplcosed
  because the core is monotone.  *)
Ralf Jung's avatar
Ralf Jung committed
56

Ralf Jung's avatar
Ralf Jung committed
57
  Record BiMixin := {
Robbert Krebbers's avatar
Robbert Krebbers committed
58
59
60
    bi_mixin_entails_po : PreOrder bi_entails;
    bi_mixin_equiv_spec P Q : equiv P Q  (P  Q)  (Q  P);

61
    (** Non-expansiveness *)
Robbert Krebbers's avatar
Robbert Krebbers committed
62
63
64
65
66
67
68
69
70
71
72
73
    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;
    bi_mixin_persistently_ne : NonExpansive bi_persistently;

74
    (** Higher-order logic *)
Robbert Krebbers's avatar
Robbert Krebbers committed
75
76
    bi_mixin_pure_intro P (φ : Prop) : φ  P   φ ;
    bi_mixin_pure_elim' (φ : Prop) P : (φ  True  P)   φ   P;
77
78
    (* This is actually derivable if we assume excluded middle in Coq,
       via [(∀ a, φ a) ∨ (∃ a, ¬φ a)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
    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;

98
    (** BI connectives *)
Robbert Krebbers's avatar
Robbert Krebbers committed
99
100
101
102
103
104
105
106
    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;

107
    (** Persistently *)
108
    (* In the ordered RA model: Holds without further assumptions. *)
109
    bi_mixin_persistently_mono P Q : (P  Q)  <pers> P  <pers> Q;
110
    (* In the ordered RA model: `core` is idempotent *)
111
    bi_mixin_persistently_idemp_2 P : <pers> P  <pers> <pers> P;
Robbert Krebbers's avatar
Robbert Krebbers committed
112

Ralf Jung's avatar
Ralf Jung committed
113
    (* In the ordered RA model: [ε ≼ core x]. *)
114
    bi_mixin_persistently_emp_2 : emp  <pers> emp;
115

Robbert Krebbers's avatar
Robbert Krebbers committed
116
    bi_mixin_persistently_forall_2 {A} (Ψ : A  PROP) :
117
      ( a, <pers> (Ψ a))  <pers> ( a, Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
118
    bi_mixin_persistently_exist_1 {A} (Ψ : A  PROP) :
119
      <pers> ( a, Ψ a)   a, <pers> (Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
120

121
    (* In the ordered RA model: [core x ≼ core (x ⋅ y)]. *)
122
    bi_mixin_persistently_absorbing P Q : <pers> P  Q  <pers> P;
Ralf Jung's avatar
typo    
Ralf Jung committed
123
    (* In the ordered RA model: [x ⋅ core x = x]. *)
124
    bi_mixin_persistently_and_sep_elim P Q : <pers> P  Q  P  Q;
Robbert Krebbers's avatar
Robbert Krebbers committed
125
126
  }.

Ralf Jung's avatar
Ralf Jung committed
127
  Record SbiMixin := {
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
128
    sbi_mixin_later_contractive : Contractive sbi_later;
129
130
131
132
133
134
135
136
137
    sbi_mixin_internal_eq_ne (A : ofeT) : NonExpansive2 (sbi_internal_eq A);

    (* Equality *)
    sbi_mixin_internal_eq_refl {A : ofeT} P (a : A) : P  a  a;
    sbi_mixin_internal_eq_rewrite {A : ofeT} a b (Ψ : A  PROP) :
      NonExpansive Ψ  a  b  Ψ a  Ψ b;
    sbi_mixin_fun_ext {A} {B : A  ofeT} (f g : ofe_fun B) : ( x, f x  g x)  f  g;
    sbi_mixin_sig_eq {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  x  y;
    sbi_mixin_discrete_eq_1 {A : ofeT} (a b : A) : Discrete a  a  b  a  b;
Robbert Krebbers's avatar
Robbert Krebbers committed
138

139
    (* Later *)
Robbert Krebbers's avatar
Robbert Krebbers committed
140
141
142
143
144
145
146
147
148
149
150
    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);
151
152
    sbi_mixin_later_persistently_1 P :  <pers> P  <pers>  P;
    sbi_mixin_later_persistently_2 P : <pers>  P   <pers> P;
Robbert Krebbers's avatar
Robbert Krebbers committed
153
154
155
156
157

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

Ralf Jung's avatar
Ralf Jung committed
158
Structure bi := Bi {
Robbert Krebbers's avatar
Robbert Krebbers committed
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
  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_sep : bi_car  bi_car  bi_car;
  bi_wand : bi_car  bi_car  bi_car;
  bi_persistently : bi_car  bi_car;
  bi_ofe_mixin : OfeMixin bi_car;
174
  bi_bi_mixin : BiMixin bi_entails bi_emp bi_pure bi_and bi_or bi_impl bi_forall
Robbert Krebbers's avatar
Robbert Krebbers committed
175
                        bi_exist bi_sep bi_wand bi_persistently;
Robbert Krebbers's avatar
Robbert Krebbers committed
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
}.

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_sep) 1.
Instance: Params (@bi_wand) 1.
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.
199
Arguments bi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
200
201
202
203
204
205
206
207
208
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_sep {PROP} _%I _%I : simpl never, rename.
Arguments bi_wand {PROP} _%I _%I : simpl never, rename.
Arguments bi_persistently {PROP} _%I : simpl never, rename.

Ralf Jung's avatar
Ralf Jung committed
209
Structure sbi := Sbi {
Robbert Krebbers's avatar
Robbert Krebbers committed
210
211
212
213
214
215
216
217
218
219
220
221
222
223
  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_sep : sbi_car  sbi_car  sbi_car;
  sbi_wand : sbi_car  sbi_car  sbi_car;
  sbi_persistently : sbi_car  sbi_car;
224
  sbi_internal_eq :  A : ofeT, A  A  sbi_car;
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
225
  sbi_later : sbi_car  sbi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
226
  sbi_ofe_mixin : OfeMixin sbi_car;
227
  sbi_bi_mixin : BiMixin sbi_entails sbi_emp sbi_pure sbi_and sbi_or sbi_impl
Robbert Krebbers's avatar
Robbert Krebbers committed
228
229
230
231
                         sbi_forall sbi_exist sbi_sep sbi_wand sbi_persistently;
  sbi_sbi_mixin : SbiMixin sbi_entails sbi_pure sbi_or sbi_impl
                           sbi_forall sbi_exist sbi_sep
                           sbi_persistently sbi_internal_eq sbi_later;
Robbert Krebbers's avatar
Robbert Krebbers committed
232
233
}.

234
235
236
237
238
Instance: Params (@sbi_later) 1.
Instance: Params (@sbi_internal_eq) 1.

Arguments sbi_later {PROP} _%I : simpl never, rename.
Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
239
240
241
242
243
244
245
246
247
248
249
250

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.
251
Arguments sbi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
253
254
255
256
257
258
259
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_sep {PROP} _%I _%I : simpl never, rename.
Arguments sbi_wand {PROP} _%I _%I : simpl never, rename.
Arguments sbi_persistently {PROP} _%I : simpl never, rename.
260
Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
261
Arguments sbi_later {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
262
263
264
265
266

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).

267
268
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
269
270

Notation "P ⊣⊢ Q" := (equiv (A:=bi_car _) P%I Q%I)
271
272
  (at level 95, no associativity) : stdpp_scope.
Notation "(⊣⊢)" := (equiv (A:=bi_car _)) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
273

274
Notation "P -∗ Q" := (P  Q) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
275
276

Notation "'emp'" := (bi_emp) : bi_scope.
277
Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
278
279
280
281
282
283
284
285
286
287
288
289
290
291
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.
292
Notation "'<pers>' P" := (bi_persistently P) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
293

294
Infix "≡" := sbi_internal_eq : bi_scope.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
295
Notation "▷ P" := (sbi_later P) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
296

Ralf Jung's avatar
Ralf Jung committed
297
298
Coercion bi_emp_valid {PROP : bi} (P : PROP) : Prop := emp  P.
Coercion sbi_emp_valid {PROP : sbi} : PROP  Prop := bi_emp_valid.
Robbert Krebbers's avatar
Robbert Krebbers committed
299

Ralf Jung's avatar
Ralf Jung committed
300
301
Arguments bi_emp_valid {_} _%I : simpl never.
Typeclasses Opaque bi_emp_valid.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
365
366
367

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.
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.
368
Proof. eapply (bi_mixin_forall_elim  bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
369
370
371
372
373
374
375
376
377
378
379
380
381
382

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.

(* 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.
383
Proof. eapply (bi_mixin_sep_comm' bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
384
385
386
387
388
389
390
391
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.

(* Persistently *)
392
Lemma persistently_mono P Q : (P  Q)  <pers> P  <pers> Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
393
Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
394
Lemma persistently_idemp_2 P : <pers> P  <pers> <pers> P.
Robbert Krebbers's avatar
Robbert Krebbers committed
395
396
Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.

397
398
Lemma persistently_emp_2 : (emp : PROP)  <pers> emp.
Proof. eapply bi_mixin_persistently_emp_2, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
399

400
Lemma persistently_forall_2 {A} (Ψ : A  PROP) :
401
  ( a, <pers> (Ψ a))  <pers> ( a, Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
402
Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed.
403
Lemma persistently_exist_1 {A} (Ψ : A  PROP) :
404
  <pers> ( a, Ψ a)   a, <pers> (Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
405
406
Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed.

407
Lemma persistently_absorbing P Q : <pers> P  Q  <pers> P.
408
Proof. eapply (bi_mixin_persistently_absorbing bi_entails), bi_bi_mixin. Qed.
409
Lemma persistently_and_sep_elim P Q : <pers> P  Q  P  Q.
410
Proof. eapply (bi_mixin_persistently_and_sep_elim bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
411
412
413
414
415
416
417
End bi_laws.

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

418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
(* Equality *)
Global Instance internal_eq_ne (A : ofeT) : NonExpansive2 (@sbi_internal_eq PROP A).
Proof. eapply sbi_mixin_internal_eq_ne, sbi_sbi_mixin. Qed.

Lemma internal_eq_refl {A : ofeT} P (a : A) : P  a  a.
Proof. eapply sbi_mixin_internal_eq_refl, sbi_sbi_mixin. Qed.
Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A  PROP) :
  NonExpansive Ψ  a  b  Ψ a  Ψ b.
Proof. eapply sbi_mixin_internal_eq_rewrite, sbi_sbi_mixin. Qed.

Lemma fun_ext {A} {B : A  ofeT} (f g : ofe_fun B) : ( x, f x  g x)  (f  g : PROP).
Proof. eapply sbi_mixin_fun_ext, sbi_sbi_mixin. Qed.
Lemma sig_eq {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  (x  y : PROP).
Proof. eapply sbi_mixin_sig_eq, sbi_sbi_mixin. Qed.
Lemma discrete_eq_1 {A : ofeT} (a b : A) :
  Discrete a  a  b  (a  b : PROP).
Proof. eapply sbi_mixin_discrete_eq_1, sbi_sbi_mixin. Qed.

(* Later *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
437
Global Instance later_contractive : Contractive (@sbi_later PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
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.
459
Lemma later_persistently_1 P :  <pers> P  <pers>  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
460
Proof. eapply (sbi_mixin_later_persistently_1 bi_entails), sbi_sbi_mixin. Qed.
461
Lemma later_persistently_2 P : <pers>  P   <pers> P.
Robbert Krebbers's avatar
Robbert Krebbers committed
462
Proof. eapply (sbi_mixin_later_persistently_2 bi_entails), sbi_sbi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
463
464
465
466

Lemma later_false_em P :  P   False  ( False  P).
Proof. eapply sbi_mixin_later_false_em, sbi_sbi_mixin. Qed.
End sbi_laws.
467

Robbert Krebbers's avatar
Robbert Krebbers committed
468
End bi.