interface.v 23.9 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
  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).
23
  Context (bi_plainly : PROP  PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
24
  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 "x ≡ y" := (sbi_internal_eq _ x y).
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
43
  Local Notation "▷ P" := (sbi_later P).
Robbert Krebbers's avatar
Robbert Krebbers committed
44

Ralf Jung's avatar
Ralf Jung committed
45
46
47
48
49
50
51
52
53
54
  (** * 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
  extension order).  We demand composition to be monotone wrt. the order.  We
  define [emp := λ r, ε ≼ r]; persisently is still defined with the core: [□ P
  := λ r, P (core r)].  *)

Ralf Jung's avatar
Ralf Jung committed
55
  Record BiMixin := {
Robbert Krebbers's avatar
Robbert Krebbers committed
56
57
58
59
60
61
62
63
64
65
66
67
68
69
    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;
70
    bi_mixin_plainly_ne : NonExpansive bi_plainly;
Robbert Krebbers's avatar
Robbert Krebbers committed
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
    bi_mixin_persistently_ne : NonExpansive bi_persistently;

    (* 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;

    (* 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;

104
    (* Plainly *)
105
    (* Note: plainly is about to be removed from this interface *)
106
107
108
109
110
111
112
    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);

113
114
115
    (* 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`. *)
116
117
118
119
120
121
122
123
    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
124
    (* Persistently *)
125
    (* In the ordered RA model: `core` is monotone *)
126
127
    bi_mixin_persistently_mono P Q :
      (P  Q)  bi_persistently P  bi_persistently Q;
128
    (* In the ordered RA model: `core` is idempotent *)
129
130
    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
135
136
    (* In the ordered RA model [P ⊢ □ emp] (which can currently still be derived
    from the plainly axioms, which will be removed): `ε ≼ core x` *)

Robbert Krebbers's avatar
Robbert Krebbers committed
137
    bi_mixin_persistently_forall_2 {A} (Ψ : A  PROP) :
138
      ( a, bi_persistently (Ψ a))  bi_persistently ( a, Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
139
    bi_mixin_persistently_exist_1 {A} (Ψ : A  PROP) :
140
      bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a);
Robbert Krebbers's avatar
Robbert Krebbers committed
141

142
    (* In the ordered RA model: [x ≼ₑₓₜ y → core x ≼ core y] *)
143
144
    bi_mixin_persistently_absorbing P Q :
      bi_persistently P  Q  bi_persistently P;
145
    (* In the ordered RA model: [ε ≼ core x] *)
146
147
    bi_mixin_persistently_and_sep_elim P Q :
      bi_persistently P  Q  (emp  P)  Q;
Robbert Krebbers's avatar
Robbert Krebbers committed
148
149
  }.

Ralf Jung's avatar
Ralf Jung committed
150
  Record SbiMixin := {
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
151
    sbi_mixin_later_contractive : Contractive sbi_later;
152
153
154
155
156
157
158
159
160
161
162
    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;
    sbi_mixin_prop_ext P Q : bi_plainly ((P  Q)  (Q  P)) 
      sbi_internal_eq (OfeT PROP prop_ofe_mixin) P Q;
Robbert Krebbers's avatar
Robbert Krebbers committed
163

164
    (* Later *)
Robbert Krebbers's avatar
Robbert Krebbers committed
165
166
167
168
169
170
171
172
173
174
175
    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);
176
177
    sbi_mixin_later_plainly_1 P :  bi_plainly P  bi_plainly ( P);
    sbi_mixin_later_plainly_2 P : bi_plainly ( P)   bi_plainly P;
178
179
180
181
    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
182
183
184
185
186

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

Ralf Jung's avatar
Ralf Jung committed
187
Structure bi := Bi {
Robbert Krebbers's avatar
Robbert Krebbers committed
188
189
190
191
192
193
194
195
196
197
198
199
200
  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;
201
  bi_plainly : bi_car  bi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
202
203
  bi_persistently : bi_car  bi_car;
  bi_ofe_mixin : OfeMixin bi_car;
204
205
  bi_bi_mixin : BiMixin bi_entails bi_emp bi_pure bi_and bi_or bi_impl bi_forall
                        bi_exist bi_sep bi_wand bi_plainly bi_persistently;
Robbert Krebbers's avatar
Robbert Krebbers committed
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
}.

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.
221
Instance: Params (@bi_plainly) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
222
223
224
225
226
227
228
229
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.
230
Arguments bi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
231
232
233
234
235
236
237
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.
238
Arguments bi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
239
240
Arguments bi_persistently {PROP} _%I : simpl never, rename.

Ralf Jung's avatar
Ralf Jung committed
241
Structure sbi := Sbi {
Robbert Krebbers's avatar
Robbert Krebbers committed
242
243
244
245
246
247
248
249
250
251
252
253
254
  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;
255
  sbi_plainly : sbi_car  sbi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
256
  sbi_persistently : sbi_car  sbi_car;
257
  sbi_internal_eq :  A : ofeT, A  A  sbi_car;
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
258
  sbi_later : sbi_car  sbi_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
259
  sbi_ofe_mixin : OfeMixin sbi_car;
260
261
262
263
264
265
  sbi_bi_mixin : BiMixin sbi_entails sbi_emp sbi_pure sbi_and sbi_or sbi_impl
                         sbi_forall sbi_exist sbi_sep sbi_wand sbi_plainly
                         sbi_persistently;
  sbi_sbi_mixin : SbiMixin sbi_ofe_mixin sbi_entails sbi_pure sbi_and sbi_or
                           sbi_impl sbi_forall sbi_exist sbi_sep sbi_plainly
                           sbi_persistently sbi_internal_eq sbi_later;
Robbert Krebbers's avatar
Robbert Krebbers committed
266
267
}.

268
269
270
271
272
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
273
274
275
276
277
278
279
280
281
282
283
284

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.
285
Arguments sbi_pure {PROP} _%stdpp : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
286
287
288
289
290
291
292
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.
293
Arguments sbi_plainly {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
294
Arguments sbi_persistently {PROP} _%I : simpl never, rename.
295
Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
296
Arguments sbi_later {PROP} _%I : simpl never, rename.
Robbert Krebbers's avatar
Robbert Krebbers committed
297
298
299
300
301

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).

302
303
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
304
305

Notation "P ⊣⊢ Q" := (equiv (A:=bi_car _) P%I Q%I)
306
307
  (at level 95, no associativity) : stdpp_scope.
Notation "(⊣⊢)" := (equiv (A:=bi_car _)) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
308

309
Notation "P -∗ Q" := (P  Q) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
310
311

Notation "'emp'" := (bi_emp) : bi_scope.
312
Notation "'⌜' φ '⌝'" := (bi_pure φ%type%stdpp) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
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.

328
Infix "≡" := sbi_internal_eq : bi_scope.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
329
Notation "▷ P" := (sbi_later P) : bi_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
368

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.
369
370
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
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
400
401
402
403
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.
404
Proof. eapply (bi_mixin_forall_elim  bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
405
406
407
408
409
410
411
412
413
414
415
416
417
418

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.
419
Proof. eapply (bi_mixin_sep_comm' bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
420
421
422
423
424
425
426
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.

427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
(* 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 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.
444
Proof. eapply (bi_mixin_plainly_absorbing bi_entails), bi_bi_mixin. Qed.
445
446
447
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
448
(* Persistently *)
449
Lemma persistently_mono P Q : (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
450
Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
451
452
Lemma persistently_idemp_2 P :
  bi_persistently P  bi_persistently (bi_persistently P).
Robbert Krebbers's avatar
Robbert Krebbers committed
453
Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.
454
455
Lemma plainly_persistently_1 P :
  bi_plainly (bi_persistently P)  bi_plainly P.
456
Proof. eapply (bi_mixin_plainly_persistently_1 bi_entails), bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
457

458
459
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
460
Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed.
461
462
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
463
464
Proof. eapply bi_mixin_persistently_exist_1, bi_bi_mixin. Qed.

465
Lemma persistently_absorbing P Q : bi_persistently P  Q  bi_persistently P.
466
Proof. eapply (bi_mixin_persistently_absorbing bi_entails), bi_bi_mixin. Qed.
467
Lemma persistently_and_sep_elim P Q : bi_persistently P  Q  (emp  P)  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
468
Proof. eapply bi_mixin_persistently_and_sep_elim, bi_bi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
469
470
471
472
473
474
475
End bi_laws.

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

476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
(* 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.

Lemma prop_ext P Q : bi_plainly ((P  Q)  (Q  P))  P  Q.
Proof. eapply (sbi_mixin_prop_ext _ bi_entails), sbi_sbi_mixin. Qed.

(* Later *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
498
Global Instance later_contractive : Contractive (@sbi_later PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
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.
520
Lemma later_plainly_1 P :  bi_plainly P  bi_plainly ( P).
521
Proof. eapply (sbi_mixin_later_plainly_1 _ bi_entails), sbi_sbi_mixin. Qed.
522
Lemma later_plainly_2 P : bi_plainly ( P)   bi_plainly P.
523
Proof. eapply (sbi_mixin_later_plainly_2 _ bi_entails), sbi_sbi_mixin. Qed.
524
Lemma later_persistently_1 P :  bi_persistently P  bi_persistently ( P).
525
Proof. eapply (sbi_mixin_later_persistently_1 _ bi_entails), sbi_sbi_mixin. Qed.
526
Lemma later_persistently_2 P : bi_persistently ( P)   bi_persistently P.
527
Proof. eapply (sbi_mixin_later_persistently_2 _ bi_entails), sbi_sbi_mixin. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
528
529
530
531

Lemma later_false_em P :  P   False  ( False  P).
Proof. eapply sbi_mixin_later_false_em, sbi_sbi_mixin. Qed.
End sbi_laws.
532

Robbert Krebbers's avatar
Robbert Krebbers committed
533
End bi.