upred.v 54.4 KB
Newer Older
1
From iris.algebra Require Export cmra.
2
3
Local Hint Extern 1 (_  _) => etrans; [eassumption|].
Local Hint Extern 1 (_  _) => etrans; [|eassumption].
Robbert Krebbers's avatar
Robbert Krebbers committed
4
5
Local Hint Extern 10 (_  _) => omega.

6
Record uPred (M : ucmraT) : Type := IProp {
Robbert Krebbers's avatar
Robbert Krebbers committed
7
  uPred_holds :> nat  M  Prop;
8
  uPred_mono n x1 x2 : uPred_holds n x1  x1 {n} x2  uPred_holds n x2;
9
  uPred_closed n1 n2 x : uPred_holds n1 x  n2  n1  {n2} x  uPred_holds n2 x
Robbert Krebbers's avatar
Robbert Krebbers committed
10
}.
11
Arguments uPred_holds {_} _ _ _ : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
12
13
Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
14

15
16
17
18
Delimit Scope uPred_scope with I.
Bind Scope uPred_scope with uPred.
Arguments uPred_holds {_} _%I _ _.

19
Section cofe.
20
  Context {M : ucmraT}.
21
22
23
24
25
26
27

  Inductive uPred_equiv' (P Q : uPred M) : Prop :=
    { uPred_in_equiv :  n x, {n} x  P n x  Q n x }.
  Instance uPred_equiv : Equiv (uPred M) := uPred_equiv'.
  Inductive uPred_dist' (n : nat) (P Q : uPred M) : Prop :=
    { uPred_in_dist :  n' x, n'  n  {n'} x  P n' x  Q n' x }.
  Instance uPred_dist : Dist (uPred M) := uPred_dist'.
28
  Program Instance uPred_compl : Compl (uPred M) := λ c,
29
    {| uPred_holds n x := c n n x |}.
30
  Next Obligation. naive_solver eauto using uPred_mono. Qed.
31
  Next Obligation.
32
33
    intros c n1 n2 x ???; simpl in *.
    apply (chain_cauchy c n2 n1); eauto using uPred_closed.
34
35
36
37
  Qed.
  Definition uPred_cofe_mixin : CofeMixin (uPred M).
  Proof.
    split.
38
39
40
    - intros P Q; split.
      + by intros HPQ n; split=> i x ??; apply HPQ.
      + intros HPQ; split=> n x ?; apply HPQ with n; auto.
41
    - intros n; split.
42
43
44
45
46
      + by intros P; split=> x i.
      + by intros P Q HPQ; split=> x i ??; symmetry; apply HPQ.
      + intros P Q Q' HP HQ; split=> i x ??.
        by trans (Q i x);[apply HP|apply HQ].
    - intros n P Q HPQ; split=> i x ??; apply HPQ; auto.
47
    - intros n c; split=>i x ??; symmetry; apply (chain_cauchy c i n); auto.
48
  Qed.
49
  Canonical Structure uPredC : cofeT := CofeT (uPred M) uPred_cofe_mixin.
50
51
52
End cofe.
Arguments uPredC : clear implicits.

Ralf Jung's avatar
Ralf Jung committed
53
Instance uPred_ne {M} (P : uPred M) n : Proper (dist n ==> iff) (P n).
54
55
56
Proof.
  intros x1 x2 Hx; split=> ?; eapply uPred_mono; eauto; by rewrite Hx.
Qed.
57
Instance uPred_proper {M} (P : uPred M) n : Proper (() ==> iff) (P n).
Ralf Jung's avatar
Ralf Jung committed
58
59
60
61
62
63
64
65
Proof. by intros x1 x2 Hx; apply uPred_ne, equiv_dist. Qed.

Lemma uPred_holds_ne {M} (P Q : uPred M) n1 n2 x :
  P {n2} Q  n2  n1  {n2} x  Q n1 x  P n2 x.
Proof.
  intros [Hne] ???. eapply Hne; try done.
  eapply uPred_closed; eauto using cmra_validN_le.
Qed.
66

Robbert Krebbers's avatar
Robbert Krebbers committed
67
(** functor *)
68
Program Definition uPred_map {M1 M2 : ucmraT} (f : M2 -n> M1)
69
70
  `{!CMRAMonotone f} (P : uPred M1) :
  uPred M2 := {| uPred_holds n x := P n (f x) |}.
71
Next Obligation. naive_solver eauto using uPred_mono, includedN_preserving. Qed.
72
73
Next Obligation. naive_solver eauto using uPred_closed, validN_preserving. Qed.

74
Instance uPred_map_ne {M1 M2 : ucmraT} (f : M2 -n> M1)
Robbert Krebbers's avatar
Robbert Krebbers committed
75
  `{!CMRAMonotone f} n : Proper (dist n ==> dist n) (uPred_map f).
Robbert Krebbers's avatar
Robbert Krebbers committed
76
Proof.
77
78
  intros x1 x2 Hx; split=> n' y ??.
  split; apply Hx; auto using validN_preserving.
Robbert Krebbers's avatar
Robbert Krebbers committed
79
Qed.
80
Lemma uPred_map_id {M : ucmraT} (P : uPred M): uPred_map cid P  P.
81
Proof. by split=> n x ?. Qed.
82
Lemma uPred_map_compose {M1 M2 M3 : ucmraT} (f : M1 -n> M2) (g : M2 -n> M3)
Robbert Krebbers's avatar
Robbert Krebbers committed
83
    `{!CMRAMonotone f, !CMRAMonotone g} (P : uPred M3):
84
  uPred_map (g  f) P  uPred_map f (uPred_map g P).
85
Proof. by split=> n x Hx. Qed.
86
Lemma uPred_map_ext {M1 M2 : ucmraT} (f g : M1 -n> M2)
87
      `{!CMRAMonotone f} `{!CMRAMonotone g}:
88
  ( x, f x  g x)   x, uPred_map f x  uPred_map g x.
89
Proof. intros Hf P; split=> n x Hx /=; by rewrite /uPred_holds /= Hf. Qed.
90
Definition uPredC_map {M1 M2 : ucmraT} (f : M2 -n> M1) `{!CMRAMonotone f} :
Robbert Krebbers's avatar
Robbert Krebbers committed
91
  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1  uPredC M2).
92
Lemma uPredC_map_ne {M1 M2 : ucmraT} (f g : M2 -n> M1)
Robbert Krebbers's avatar
Robbert Krebbers committed
93
    `{!CMRAMonotone f, !CMRAMonotone g} n :
94
  f {n} g  uPredC_map f {n} uPredC_map g.
Robbert Krebbers's avatar
Robbert Krebbers committed
95
Proof.
96
  by intros Hfg P; split=> n' y ??;
97
    rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
98
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
99

100
101
102
Program Definition uPredCF (F : urFunctor) : cFunctor := {|
  cFunctor_car A B := uPredC (urFunctor_car F B A);
  cFunctor_map A1 A2 B1 B2 fg := uPredC_map (urFunctor_map F (fg.2, fg.1))
103
|}.
104
105
Next Obligation.
  intros F A1 A2 B1 B2 n P Q HPQ.
106
  apply uPredC_map_ne, urFunctor_ne; split; by apply HPQ.
107
Qed.
108
109
Next Obligation.
  intros F A B P; simpl. rewrite -{2}(uPred_map_id P).
110
  apply uPred_map_ext=>y. by rewrite urFunctor_id.
111
112
113
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' P; simpl. rewrite -uPred_map_compose.
114
  apply uPred_map_ext=>y; apply urFunctor_compose.
115
116
Qed.

117
Instance uPredCF_contractive F :
118
  urFunctorContractive F  cFunctorContractive (uPredCF F).
119
120
Proof.
  intros ? A1 A2 B1 B2 n P Q HPQ.
121
  apply uPredC_map_ne, urFunctor_contractive=> i ?; split; by apply HPQ.
122
123
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
124
(** logical entailement *)
125
126
Inductive uPred_entails {M} (P Q : uPred M) : Prop :=
  { uPred_in_entails :  n x, {n} x  P n x  Q n x }.
127
Hint Extern 0 (uPred_entails _ _) => reflexivity.
128
Instance uPred_entails_rewrite_relation M : RewriteRelation (@uPred_entails M).
Robbert Krebbers's avatar
Robbert Krebbers committed
129

130
Hint Resolve uPred_mono uPred_closed : uPred_def.
131

Robbert Krebbers's avatar
Robbert Krebbers committed
132
(** logical connectives *)
133
Program Definition uPred_pure_def {M} (φ : Prop) : uPred M :=
134
  {| uPred_holds n x := φ |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
135
Solve Obligations with done.
136
137
138
139
Definition uPred_pure_aux : { x | x = @uPred_pure_def }. by eexists. Qed.
Definition uPred_pure {M} := proj1_sig uPred_pure_aux M.
Definition uPred_pure_eq :
  @uPred_pure = @uPred_pure_def := proj2_sig uPred_pure_aux.
140

141
Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_pure True).
Robbert Krebbers's avatar
Robbert Krebbers committed
142

143
Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
144
  {| uPred_holds n x := P n x  Q n x |}.
145
Solve Obligations with naive_solver eauto 2 with uPred_def.
146
147
148
149
150
Definition uPred_and_aux : { x | x = @uPred_and_def }. by eexists. Qed.
Definition uPred_and {M} := proj1_sig uPred_and_aux M.
Definition uPred_and_eq: @uPred_and = @uPred_and_def := proj2_sig uPred_and_aux.

Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
151
  {| uPred_holds n x := P n x  Q n x |}.
152
Solve Obligations with naive_solver eauto 2 with uPred_def.
153
154
155
156
157
Definition uPred_or_aux : { x | x = @uPred_or_def }. by eexists. Qed.
Definition uPred_or {M} := proj1_sig uPred_or_aux M.
Definition uPred_or_eq: @uPred_or = @uPred_or_def := proj2_sig uPred_or_aux.

Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
158
  {| uPred_holds n x :=  n' x',
Robbert Krebbers's avatar
Robbert Krebbers committed
159
       x  x'  n'  n  {n'} x'  P n' x'  Q n' x' |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
160
Next Obligation.
161
162
163
  intros M P Q n1 x1 x1' HPQ [x2 Hx1'] n2 x3 [x4 Hx3] ?; simpl in *.
  rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??.
  eapply HPQ; auto. exists (x2  x4); by rewrite assoc.
Robbert Krebbers's avatar
Robbert Krebbers committed
164
Qed.
165
Next Obligation. intros M P Q [|n1] [|n2] x; auto with lia. Qed.
166
167
168
169
Definition uPred_impl_aux : { x | x = @uPred_impl_def }. by eexists. Qed.
Definition uPred_impl {M} := proj1_sig uPred_impl_aux M.
Definition uPred_impl_eq :
  @uPred_impl = @uPred_impl_def := proj2_sig uPred_impl_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
170

171
Program Definition uPred_forall_def {M A} (Ψ : A  uPred M) : uPred M :=
172
  {| uPred_holds n x :=  a, Ψ a n x |}.
173
Solve Obligations with naive_solver eauto 2 with uPred_def.
174
175
176
177
178
179
Definition uPred_forall_aux : { x | x = @uPred_forall_def }. by eexists. Qed.
Definition uPred_forall {M A} := proj1_sig uPred_forall_aux M A.
Definition uPred_forall_eq :
  @uPred_forall = @uPred_forall_def := proj2_sig uPred_forall_aux.

Program Definition uPred_exist_def {M A} (Ψ : A  uPred M) : uPred M :=
180
  {| uPred_holds n x :=  a, Ψ a n x |}.
181
Solve Obligations with naive_solver eauto 2 with uPred_def.
182
183
184
Definition uPred_exist_aux : { x | x = @uPred_exist_def }. by eexists. Qed.
Definition uPred_exist {M A} := proj1_sig uPred_exist_aux M A.
Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := proj2_sig uPred_exist_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
185

186
Program Definition uPred_eq_def {M} {A : cofeT} (a1 a2 : A) : uPred M :=
187
  {| uPred_holds n x := a1 {n} a2 |}.
188
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
189
190
191
Definition uPred_eq_aux : { x | x = @uPred_eq_def }. by eexists. Qed.
Definition uPred_eq {M A} := proj1_sig uPred_eq_aux M A.
Definition uPred_eq_eq: @uPred_eq = @uPred_eq_def := proj2_sig uPred_eq_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
192

193
Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
194
  {| uPred_holds n x :=  x1 x2, x {n} x1  x2  P n x1  Q n x2 |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
195
Next Obligation.
196
  intros M P Q n x y (x1&x2&Hx&?&?) [z Hy].
197
  exists x1, (x2  z); split_and?; eauto using uPred_mono, cmra_includedN_l.
198
199
200
201
202
203
  by rewrite Hy Hx assoc.
Qed.
Next Obligation.
  intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?; rewrite {1}(dist_le _ _ _ _ Hx) // =>?.
  exists x1, x2; cofe_subst; split_and!;
    eauto using dist_le, uPred_closed, cmra_validN_op_l, cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
204
Qed.
205
206
207
Definition uPred_sep_aux : { x | x = @uPred_sep_def }. by eexists. Qed.
Definition uPred_sep {M} := proj1_sig uPred_sep_aux M.
Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := proj2_sig uPred_sep_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
208

209
Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
210
  {| uPred_holds n x :=  n' x',
Robbert Krebbers's avatar
Robbert Krebbers committed
211
       n'  n  {n'} (x  x')  P n' x'  Q n' (x  x') |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
212
Next Obligation.
213
  intros M P Q n x1 x1' HPQ ? n3 x3 ???; simpl in *.
214
  apply uPred_mono with (x1  x3);
215
    eauto using cmra_validN_includedN, cmra_preservingN_r, cmra_includedN_le.
Robbert Krebbers's avatar
Robbert Krebbers committed
216
Qed.
217
Next Obligation. naive_solver. Qed.
218
219
220
221
Definition uPred_wand_aux : { x | x = @uPred_wand_def }. by eexists. Qed.
Definition uPred_wand {M} := proj1_sig uPred_wand_aux M.
Definition uPred_wand_eq :
  @uPred_wand = @uPred_wand_def := proj2_sig uPred_wand_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
222

223
Program Definition uPred_always_def {M} (P : uPred M) : uPred M :=
Ralf Jung's avatar
Ralf Jung committed
224
  {| uPred_holds n x := P n (core x) |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
225
226
227
228
Next Obligation.
  intros M; naive_solver eauto using uPred_mono, @cmra_core_preservingN.
Qed.
Next Obligation. naive_solver eauto using uPred_closed, @cmra_core_validN. Qed.
229
230
231
232
233
234
Definition uPred_always_aux : { x | x = @uPred_always_def }. by eexists. Qed.
Definition uPred_always {M} := proj1_sig uPred_always_aux M.
Definition uPred_always_eq :
  @uPred_always = @uPred_always_def := proj2_sig uPred_always_aux.

Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
235
  {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
236
237
238
Next Obligation.
  intros M P [|n] x1 x2; eauto using uPred_mono, cmra_includedN_S.
Qed.
239
Next Obligation.
240
  intros M P [|n1] [|n2] x; eauto using uPred_closed, cmra_validN_S with lia.
241
Qed.
242
243
244
245
Definition uPred_later_aux : { x | x = @uPred_later_def }. by eexists. Qed.
Definition uPred_later {M} := proj1_sig uPred_later_aux M.
Definition uPred_later_eq :
  @uPred_later = @uPred_later_def := proj2_sig uPred_later_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
246

247
Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
248
  {| uPred_holds n x := a {n} x |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
249
Next Obligation.
250
251
  intros M a n x1 x [a' Hx1] [x2 ->].
  exists (a'  x2). by rewrite (assoc op) Hx1.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
Qed.
253
Next Obligation. naive_solver eauto using cmra_includedN_le. Qed.
254
255
256
257
258
Definition uPred_ownM_aux : { x | x = @uPred_ownM_def }. by eexists. Qed.
Definition uPred_ownM {M} := proj1_sig uPred_ownM_aux M.
Definition uPred_ownM_eq :
  @uPred_ownM = @uPred_ownM_def := proj2_sig uPred_ownM_aux.

259
Program Definition uPred_valid_def {M : ucmraT} {A : cmraT} (a : A) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
260
  {| uPred_holds n x := {n} a |}.
261
Solve Obligations with naive_solver eauto 2 using cmra_validN_le.
262
263
264
265
Definition uPred_valid_aux : { x | x = @uPred_valid_def }. by eexists. Qed.
Definition uPred_valid {M A} := proj1_sig uPred_valid_aux M A.
Definition uPred_valid_eq :
  @uPred_valid = @uPred_valid_def := proj2_sig uPred_valid_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
266

267
268
Notation "P ⊢ Q" := (uPred_entails P%I Q%I)
  (at level 99, Q at level 200, right associativity) : C_scope.
269
Notation "(⊢)" := uPred_entails (only parsing) : C_scope.
270
271
Notation "P ⊣⊢ Q" := (equiv (A:=uPred _) P%I Q%I)
  (at level 95, no associativity) : C_scope.
272
Notation "(⊣⊢)" := (equiv (A:=uPred _)) (only parsing) : C_scope.
273
Notation "■ φ" := (uPred_pure φ%C%type)
274
  (at level 20, right associativity) : uPred_scope.
275
276
277
278
Notation "x = y" := (uPred_pure (x%C%type = y%C%type)) : uPred_scope.
Notation "x ⊥ y" := (uPred_pure (x%C%type  y%C%type)) : uPred_scope.
Notation "'False'" := (uPred_pure False) : uPred_scope.
Notation "'True'" := (uPred_pure True) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
279
Infix "∧" := uPred_and : uPred_scope.
280
Notation "(∧)" := uPred_and (only parsing) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
281
Infix "∨" := uPred_or : uPred_scope.
282
Notation "(∨)" := uPred_or (only parsing) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
283
284
Infix "→" := uPred_impl : uPred_scope.
Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope.
285
Notation "(★)" := uPred_sep (only parsing) : uPred_scope.
286
Notation "P -★ Q" := (uPred_wand P Q)
Robbert Krebbers's avatar
Robbert Krebbers committed
287
  (at level 99, Q at level 200, right associativity) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
288
Notation "∀ x .. y , P" :=
289
  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
290
Notation "∃ x .. y , P" :=
291
  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) : uPred_scope.
292
293
Notation "□ P" := (uPred_always P)
  (at level 20, right associativity) : uPred_scope.
294
295
Notation "▷ P" := (uPred_later P)
  (at level 20, right associativity) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
296
Infix "≡" := uPred_eq : uPred_scope.
297
Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
298

299
Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P  Q)  (Q  P))%I.
300
Instance: Params (@uPred_iff) 1.
301
302
Infix "↔" := uPred_iff : uPred_scope.

303
304
305
306
307
308
309
Definition uPred_always_if {M} (p : bool) (P : uPred M) : uPred M :=
  (if p then  P else P)%I.
Instance: Params (@uPred_always_if) 2.
Arguments uPred_always_if _ !_ _/.
Notation "□? p P" := (uPred_always_if p P)
  (at level 20, p at level 0, P at level 20, format "□? p  P").

310
Class TimelessP {M} (P : uPred M) := timelessP :  P  (P   False).
311
Arguments timelessP {_} _ {_}.
312

313
314
Class PersistentP {M} (P : uPred M) := persistentP : P   P.
Arguments persistentP {_} _ {_}.
Robbert Krebbers's avatar
Robbert Krebbers committed
315

316
317
Module uPred.
Definition unseal :=
318
  (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
319
320
  uPred_exist_eq, uPred_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
  uPred_later_eq, uPred_ownM_eq, uPred_valid_eq).
321
Ltac unseal := rewrite !unseal /=.
322
323

Section uPred_logic.
324
Context {M : ucmraT}.
325
Implicit Types φ : Prop.
Robbert Krebbers's avatar
Robbert Krebbers committed
326
Implicit Types P Q : uPred M.
327
Implicit Types A : Type.
328
329
Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *)
Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *)
330
Arguments uPred_holds {_} !_ _ _ /.
331
Hint Immediate uPred_in_entails.
Robbert Krebbers's avatar
Robbert Krebbers committed
332

333
Global Instance: PreOrder (@uPred_entails M).
334
335
336
337
338
Proof.
  split.
  * by intros P; split=> x i.
  * by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP.
Qed.
339
Global Instance: AntiSymm () (@uPred_entails M).
340
Proof. intros P Q HPQ HQP; split=> x n; by split; [apply HPQ|apply HQP]. Qed.
341
Lemma equiv_spec P Q : (P  Q)  (P  Q)  (Q  P).
Robbert Krebbers's avatar
Robbert Krebbers committed
342
Proof.
343
  split; [|by intros [??]; apply (anti_symm ())].
344
  intros HPQ; split; split=> x i; apply HPQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
345
Qed.
346
Lemma equiv_entails P Q : (P  Q)  (P  Q).
347
Proof. apply equiv_spec. Qed.
348
Lemma equiv_entails_sym P Q : (Q  P)  (P  Q).
349
Proof. apply equiv_spec. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
350
Global Instance entails_proper :
351
  Proper (() ==> () ==> iff) (() : relation (uPred M)).
Robbert Krebbers's avatar
Robbert Krebbers committed
352
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
353
  move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split; intros.
354
355
  - by trans P1; [|trans Q1].
  - by trans P2; [|trans Q2].
Robbert Krebbers's avatar
Robbert Krebbers committed
356
Qed.
357
Lemma entails_equiv_l (P Q R : uPred M) : (P  Q)  (Q  R)  (P  R).
358
Proof. by intros ->. Qed.
359
Lemma entails_equiv_r (P Q R : uPred M) : (P  Q)  (Q  R)  (P  R).
360
Proof. by intros ? <-. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
361

362
(** Non-expansiveness and setoid morphisms *)
363
Global Instance pure_proper : Proper (iff ==> ()) (@uPred_pure M).
364
Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
365
Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Robbert Krebbers's avatar
Robbert Krebbers committed
366
Proof.
367
  intros P P' HP Q Q' HQ; unseal; split=> x n' ??.
368
  split; (intros [??]; split; [by apply HP|by apply HQ]).
Robbert Krebbers's avatar
Robbert Krebbers committed
369
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
370
Global Instance and_proper :
371
  Proper (() ==> () ==> ()) (@uPred_and M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
372
Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
Robbert Krebbers's avatar
Robbert Krebbers committed
373
Proof.
374
  intros P P' HP Q Q' HQ; split=> x n' ??.
375
  unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]).
Robbert Krebbers's avatar
Robbert Krebbers committed
376
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
377
Global Instance or_proper :
378
  Proper (() ==> () ==> ()) (@uPred_or M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
379
Global Instance impl_ne n :
Robbert Krebbers's avatar
Robbert Krebbers committed
380
  Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
Robbert Krebbers's avatar
Robbert Krebbers committed
381
Proof.
382
  intros P P' HP Q Q' HQ; split=> x n' ??.
383
  unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
384
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
385
Global Instance impl_proper :
386
  Proper (() ==> () ==> ()) (@uPred_impl M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
387
Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
Robbert Krebbers's avatar
Robbert Krebbers committed
388
Proof.
389
  intros P P' HP Q Q' HQ; split=> n' x ??.
390
  unseal; split; intros (x1&x2&?&?&?); cofe_subst x;
391
    exists x1, x2; split_and!; try (apply HP || apply HQ);
392
    eauto using cmra_validN_op_l, cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
393
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
394
Global Instance sep_proper :
395
  Proper (() ==> () ==> ()) (@uPred_sep M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
396
Global Instance wand_ne n :
Robbert Krebbers's avatar
Robbert Krebbers committed
397
  Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
Robbert Krebbers's avatar
Robbert Krebbers committed
398
Proof.
399
  intros P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???;
400
    apply HQ, HPQ, HP; eauto using cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
401
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
402
Global Instance wand_proper :
403
  Proper (() ==> () ==> ()) (@uPred_wand M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
404
Global Instance eq_ne (A : cofeT) n :
Robbert Krebbers's avatar
Robbert Krebbers committed
405
  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
Robbert Krebbers's avatar
Robbert Krebbers committed
406
Proof.
407
  intros x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *.
408
409
  * by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto.
  * by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
410
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
411
Global Instance eq_proper (A : cofeT) :
412
  Proper (() ==> () ==> ()) (@uPred_eq M A) := ne_proper_2 _.
413
Global Instance forall_ne A n :
Robbert Krebbers's avatar
Robbert Krebbers committed
414
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
415
416
417
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
418
Global Instance forall_proper A :
419
  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
420
421
422
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
423
Global Instance exist_ne A n :
Robbert Krebbers's avatar
Robbert Krebbers committed
424
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
425
Proof.
426
427
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ.
428
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
429
Global Instance exist_proper A :
430
  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
431
Proof.
432
433
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ.
434
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
435
Global Instance later_contractive : Contractive (@uPred_later M).
Robbert Krebbers's avatar
Robbert Krebbers committed
436
Proof.
437
  intros n P Q HPQ; unseal; split=> -[|n'] x ??; simpl; [done|].
438
  apply (HPQ n'); eauto using cmra_validN_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
439
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
440
Global Instance later_proper :
441
  Proper (() ==> ()) (@uPred_later M) := ne_proper _.
442
443
Global Instance always_ne n : Proper (dist n ==> dist n) (@uPred_always M).
Proof.
444
  intros P1 P2 HP.
Robbert Krebbers's avatar
Robbert Krebbers committed
445
  unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN.
446
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
447
Global Instance always_proper :
448
  Proper (() ==> ()) (@uPred_always M) := ne_proper _.
449
Global Instance ownM_ne n : Proper (dist n ==> dist n) (@uPred_ownM M).
450
Proof.
451
452
  intros a b Ha.
  unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia.
453
Qed.
454
Global Instance ownM_proper: Proper (() ==> ()) (@uPred_ownM M) := ne_proper _.
455
Global Instance valid_ne {A : cmraT} n :
456
457
Proper (dist n ==> dist n) (@uPred_valid M A).
Proof.
458
459
  intros a b Ha; unseal; split=> n' x ? /=.
  by rewrite (dist_le _ _ _ _ Ha); last lia.
460
Qed.
461
Global Instance valid_proper {A : cmraT} :
462
  Proper (() ==> ()) (@uPred_valid M A) := ne_proper _.
Robbert Krebbers's avatar
Robbert Krebbers committed
463
Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
464
Proof. unfold uPred_iff; solve_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
465
Global Instance iff_proper :
466
  Proper (() ==> () ==> ()) (@uPred_iff M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
467
468

(** Introduction and elimination rules *)
469
Lemma pure_intro φ P : φ  P   φ.
470
Proof. by intros ?; unseal; split. Qed.
471
Lemma pure_elim φ Q R : (Q   φ)  (φ  Q  R)  Q  R.
472
473
474
Proof.
  unseal; intros HQP HQR; split=> n x ??; apply HQR; first eapply HQP; eauto.
Qed.
475
Lemma and_elim_l P Q : P  Q  P.
476
Proof. by unseal; split=> n x ? [??]. Qed.
477
Lemma and_elim_r P Q : P  Q  Q.
478
Proof. by unseal; split=> n x ? [??]. Qed.
479
Lemma and_intro P Q R : (P  Q)  (P  R)  P  Q  R.
480
Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed.
481
Lemma or_intro_l P Q : P  P  Q.
482
Proof. unseal; split=> n x ??; left; auto. Qed.
483
Lemma or_intro_r P Q : Q  P  Q.
484
Proof. unseal; split=> n x ??; right; auto. Qed.
485
Lemma or_elim P Q R : (P  R)  (Q  R)  P  Q  R.
486
Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed.
487
Lemma impl_intro_r P Q R : (P  Q  R)  P  Q  R.
Robbert Krebbers's avatar
Robbert Krebbers committed
488
Proof.
489
490
  unseal; intros HQ; split=> n x ?? n' x' ????. apply HQ;
    naive_solver eauto using uPred_mono, uPred_closed, cmra_included_includedN.
Robbert Krebbers's avatar
Robbert Krebbers committed
491
Qed.
492
Lemma impl_elim P Q R : (P  Q  R)  (P  Q)  P  R.
493
Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed.
494
Lemma forall_intro {A} P (Ψ : A  uPred M): ( a, P  Ψ a)  P   a, Ψ a.
495
Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed.
496
Lemma forall_elim {A} {Ψ : A  uPred M} a : ( a, Ψ a)  Ψ a.
497
Proof. unseal; split=> n x ? HP; apply HP. Qed.
498
Lemma exist_intro {A} {Ψ : A  uPred M} a : Ψ a   a, Ψ a.
499
Proof. unseal; split=> n x ??; by exists a. Qed.
500
Lemma exist_elim {A} (Φ : A  uPred M) Q : ( a, Φ a  Q)  ( a, Φ a)  Q.
501
Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.
502
Lemma eq_refl {A : cofeT} (a : A) : True  a  a.
503
Proof. unseal; by split=> n x ??; simpl. Qed.
504
Lemma eq_rewrite {A : cofeT} a b (Ψ : A  uPred M) P
505
  {HΨ :  n, Proper (dist n ==> dist n) Ψ} : (P  a  b)  (P  Ψ a)  P  Ψ b.
506
Proof.
507
508
509
  unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
  - by symmetry; apply Hab with x.
  - by apply Ha.
510
Qed.
511
Lemma eq_equiv {A : cofeT} (a b : A) : (True  a  b)  a  b.
512
Proof.
513
  unseal=> Hab; apply equiv_dist; intros n; apply Hab with ; last done.
514
  apply cmra_valid_validN, ucmra_unit_valid.
515
Qed.
516
Lemma eq_rewrite_contractive {A : cofeT} a b (Ψ : A  uPred M) P
517
  {HΨ : Contractive Ψ} : (P   (a  b))  (P  Ψ a)  P  Ψ b.
518
519
520
521
522
523
Proof.
  unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
  - destruct n; intros m ?; first omega. apply (dist_le n); last omega.
    symmetry. by destruct Hab as [Hab]; eapply (Hab (S n)).
  - by apply Ha.
Qed.
524
525

(* Derived logical stuff *)
526
Lemma False_elim P : False  P.
527
Proof. by apply (pure_elim False). Qed.
528
Lemma True_intro P : P  True.
529
Proof. by apply pure_intro. Qed.
530
Lemma and_elim_l' P Q R : (P  R)  P  Q  R.
531
Proof. by rewrite and_elim_l. Qed.
532
Lemma and_elim_r' P Q R : (Q  R)  P  Q  R.
533
Proof. by rewrite and_elim_r. Qed.
534
Lemma or_intro_l' P Q R : (P  Q)  P  Q  R.
535
Proof. intros ->; apply or_intro_l. Qed.
536
Lemma or_intro_r' P Q R : (P  R)  P  Q  R.
537
Proof. intros ->; apply or_intro_r. Qed.
538
Lemma exist_intro' {A} P (Ψ : A  uPred M) a : (P  Ψ a)  P   a, Ψ a.
539
Proof. intros ->; apply exist_intro. Qed.
540
Lemma forall_elim' {A} P (Ψ : A  uPred M) : (P   a, Ψ a)   a, P  Ψ a.
541
Proof. move=> HP a. by rewrite HP forall_elim. Qed.
542

543
Hint Resolve or_elim or_intro_l' or_intro_r'.
Robbert Krebbers's avatar
Robbert Krebbers committed
544
545
Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.
546

547
Lemma impl_intro_l P Q R : (Q  P  R)  P  Q  R.
548
Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed.
549
Lemma impl_elim_l P Q : (P  Q)  P  Q.
550
Proof. apply impl_elim with P; auto. Qed.
551
Lemma impl_elim_r P Q : P  (P  Q)  Q.
552
Proof. apply impl_elim with P; auto. Qed.
553
Lemma impl_elim_l' P Q R : (P  Q  R)  P  Q  R.
554
Proof. intros; apply impl_elim with Q; auto. Qed.
555
Lemma impl_elim_r' P Q R : (Q  P  R)  P  Q  R.
556
Proof. intros; apply impl_elim with P; auto. Qed.
557
Lemma impl_entails P Q : (True  P  Q)  P  Q.
558
Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
559
Lemma entails_impl P Q : (P