big_op.v 73 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
From iris.algebra Require Export big_op.
2
From iris.bi Require Import derived_laws_sbi plainly.
Robbert Krebbers's avatar
Robbert Krebbers committed
3
From stdpp Require Import countable fin_sets functions.
4
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
5
Import interface.bi derived_laws_bi.bi derived_laws_sbi.bi.
6

Dan Frumin's avatar
Dan Frumin committed
7
(** Notations for unary variants *)
Ralf Jung's avatar
Ralf Jung committed
8
9
10
11
12
13
14
15
16
17
18
19
Notation "'[∗' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_sep (λ k x, P) l) : bi_scope.
Notation "'[∗' 'list]' x ∈ l , P" :=
  (big_opL bi_sep (λ _ x, P) l) : bi_scope.
Notation "'[∗]' Ps" := (big_opL bi_sep (λ _ x, x) Ps) : bi_scope.

Notation "'[∧' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_and (λ k x, P) l) : bi_scope.
Notation "'[∧' 'list]' x ∈ l , P" :=
  (big_opL bi_and (λ _ x, P) l) : bi_scope.
Notation "'[∧]' Ps" := (big_opL bi_and (λ _ x, x) Ps) : bi_scope.

20
21
22
23
24
25
Notation "'[∨' 'list]' k ↦ x ∈ l , P" :=
  (big_opL bi_or (λ k x, P) l) : bi_scope.
Notation "'[∨' 'list]' x ∈ l , P" :=
  (big_opL bi_or (λ _ x, P) l) : bi_scope.
Notation "'[∨]' Ps" := (big_opL bi_or (λ _ x, x) Ps) : bi_scope.

Ralf Jung's avatar
Ralf Jung committed
26
27
28
29
30
31
Notation "'[∗' 'map]' k ↦ x ∈ m , P" := (big_opM bi_sep (λ k x, P) m) : bi_scope.
Notation "'[∗' 'map]' x ∈ m , P" := (big_opM bi_sep (λ _ x, P) m) : bi_scope.

Notation "'[∗' 'set]' x ∈ X , P" := (big_opS bi_sep (λ x, P) X) : bi_scope.

Notation "'[∗' 'mset]' x ∈ X , P" := (big_opMS bi_sep (λ x, P) X) : bi_scope.
32

Dan Frumin's avatar
Dan Frumin committed
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
(** Definitions and notations for binary variants *)
(** A version of the separating big operator that ranges over two lists. This
version also ensures that both lists have the same length. Although this version
can be defined in terms of the unary using a [zip] (see [big_sepL2_alt]), we do
not define it that way to get better computational behavior (for [simpl]). *)
Fixpoint big_sepL2 {PROP : bi} {A B}
    (Φ : nat  A  B  PROP) (l1 : list A) (l2 : list B) : PROP :=
  match l1, l2 with
  | [], [] => emp
  | x1 :: l1, x2 :: l2 => Φ 0 x1 x2  big_sepL2 (λ n, Φ (S n)) l1 l2
  | _, _ => False
  end%I.
Instance: Params (@big_sepL2) 3 := {}.
Arguments big_sepL2 {PROP A B} _ !_ !_ /.
Typeclasses Opaque big_sepL2.
Notation "'[∗' 'list]' k ↦ x1 ; x2 ∈ l1 ; l2 , P" :=
  (big_sepL2 (λ k x1 x2, P) l1 l2) : bi_scope.
Notation "'[∗' 'list]' x1 ; x2 ∈ l1 ; l2 , P" :=
  (big_sepL2 (λ _ x1 x2, P) l1 l2) : bi_scope.

Definition big_sepM2 {PROP : bi} `{Countable K} {A B}
    (Φ : K  A  B  PROP) (m1 : gmap K A) (m2 : gmap K B) : PROP :=
  (  k, is_Some (m1 !! k)  is_Some (m2 !! k)  
   [ map] k  xy  map_zip m1 m2, Φ k xy.1 xy.2)%I.
Instance: Params (@big_sepM2) 6 := {}.
Typeclasses Opaque big_sepM2.
Notation "'[∗' 'map]' k ↦ x1 ; x2 ∈ m1 ; m2 , P" :=
  (big_sepM2 (λ k x1 x2, P) m1 m2) : bi_scope.
Notation "'[∗' 'map]' x1 ; x2 ∈ m1 ; m2 , P" :=
  (big_sepM2 (λ _ x1 x2, P) m1 m2) : bi_scope.

64
(** * Properties *)
Robbert Krebbers's avatar
Robbert Krebbers committed
65
66
Section bi_big_op.
Context {PROP : bi}.
67
Implicit Types P Q : PROP.
Robbert Krebbers's avatar
Robbert Krebbers committed
68
Implicit Types Ps Qs : list PROP.
69
70
Implicit Types A : Type.

71
(** ** Big ops over lists *)
72
Section sep_list.
73
74
  Context {A : Type}.
  Implicit Types l : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
75
  Implicit Types Φ Ψ : nat  A  PROP.
76

Robbert Krebbers's avatar
Robbert Krebbers committed
77
  Lemma big_sepL_nil Φ : ([ list] ky  nil, Φ k y)  emp.
78
  Proof. done. Qed.
79
  Lemma big_sepL_nil' `{BiAffine PROP} P Φ : P  [ list] ky  nil, Φ k y.
Robbert Krebbers's avatar
Robbert Krebbers committed
80
  Proof. apply (affine _). Qed.
81
  Lemma big_sepL_cons Φ x l :
82
    ([ list] ky  x :: l, Φ k y)  Φ 0 x  [ list] ky  l, Φ (S k) y.
83
  Proof. by rewrite big_opL_cons. Qed.
84
  Lemma big_sepL_singleton Φ x : ([ list] ky  [x], Φ k y)  Φ 0 x.
85
86
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_sepL_app Φ l1 l2 :
87
88
    ([ list] ky  l1 ++ l2, Φ k y)
     ([ list] ky  l1, Φ k y)  ([ list] ky  l2, Φ (length l1 + k) y).
89
90
  Proof. by rewrite big_opL_app. Qed.

91
92
  Lemma big_sepL_mono Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
93
    ([ list] k  y  l, Φ k y)  [ list] k  y  l, Ψ k y.
94
  Proof. apply big_opL_forall; apply _. Qed.
95
96
  Lemma big_sepL_proper Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
97
    ([ list] k  y  l, Φ k y)  ([ list] k  y  l, Ψ k y).
98
  Proof. apply big_opL_proper. Qed.
99
  Lemma big_sepL_submseteq `{BiAffine PROP} (Φ : A  PROP) l1 l2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
100
    l1 + l2  ([ list] y  l2, Φ y)  [ list] y  l1, Φ y.
Robbert Krebbers's avatar
Robbert Krebbers committed
101
102
103
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_sepL_app sep_elim_l.
  Qed.
104

105
106
  Global Instance big_sepL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> (=) ==> ())
Robbert Krebbers's avatar
Robbert Krebbers committed
107
           (big_opL (@bi_sep PROP) (A:=A)).
108
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
109
  Global Instance big_sepL_id_mono' :
110
    Proper (Forall2 () ==> ()) (big_opL (@bi_sep PROP) (λ _ P, P)).
111
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
112

113
  Lemma big_sepL_emp l : ([ list] ky  l, emp) @{PROP} emp.
Robbert Krebbers's avatar
Robbert Krebbers committed
114
115
  Proof. by rewrite big_opL_unit. Qed.

116
117
118
119
  Lemma big_sepL_lookup_acc Φ l i x :
    l !! i = Some x 
    ([ list] ky  l, Φ k y)  Φ i x  (Φ i x - ([ list] ky  l, Φ k y)).
  Proof.
120
121
122
    intros Hli. rewrite -(take_drop_middle l i x) // big_sepL_app /=.
    rewrite Nat.add_0_r take_length_le; eauto using lookup_lt_Some, Nat.lt_le_incl.
    rewrite assoc -!(comm _ (Φ _ _)) -assoc. by apply sep_mono_r, wand_intro_l.
123
124
  Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
125
  Lemma big_sepL_lookup Φ l i x `{!Absorbing (Φ i x)} :
126
    l !! i = Some x  ([ list] ky  l, Φ k y)  Φ i x.
Robbert Krebbers's avatar
Robbert Krebbers committed
127
  Proof. intros. rewrite big_sepL_lookup_acc //. by rewrite sep_elim_l. Qed.
128

Robbert Krebbers's avatar
Robbert Krebbers committed
129
  Lemma big_sepL_elem_of (Φ : A  PROP) l x `{!Absorbing (Φ x)} :
130
    x  l  ([ list] y  l, Φ y)  Φ x.
131
132
133
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_sepL_lookup (λ _, Φ)).
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
134

Robbert Krebbers's avatar
Robbert Krebbers committed
135
  Lemma big_sepL_fmap {B} (f : A  B) (Φ : nat  B  PROP) l :
136
    ([ list] ky  f <$> l, Φ k y)  ([ list] ky  l, Φ k (f y)).
137
  Proof. by rewrite big_opL_fmap. Qed.
138

139
140
141
142
  Lemma big_sepL_bind {B} (f : A  list B) (Φ : B  PROP) l :
    ([ list] y  l = f, Φ y)  ([ list] x  l, [ list] y  f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

143
  Lemma big_sepL_sep Φ Ψ l :
144
145
    ([ list] kx  l, Φ k x  Ψ k x)
     ([ list] kx  l, Φ k x)  ([ list] kx  l, Ψ k x).
146
  Proof. by rewrite big_opL_op. Qed.
147

148
149
150
  Lemma big_sepL_and Φ Ψ l :
    ([ list] kx  l, Φ k x  Ψ k x)
     ([ list] kx  l, Φ k x)  ([ list] kx  l, Ψ k x).
Robbert Krebbers's avatar
Robbert Krebbers committed
151
  Proof. auto using and_intro, big_sepL_mono, and_elim_l, and_elim_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
152

153
  Lemma big_sepL_persistently `{BiAffine PROP} Φ l :
154
    <pers> ([ list] kx  l, Φ k x)  [ list] kx  l, <pers> (Φ k x).
155
  Proof. apply (big_opL_commute _). Qed.
156

157
  Lemma big_sepL_forall `{BiAffine PROP} Φ l :
158
    ( k x, Persistent (Φ k x)) 
Ralf Jung's avatar
Ralf Jung committed
159
    ([ list] kx  l, Φ k x)  ( k x, l !! k = Some x  Φ k x).
160
161
162
  Proof.
    intros HΦ. apply (anti_symm _).
    { apply forall_intro=> k; apply forall_intro=> x.
Robbert Krebbers's avatar
Robbert Krebbers committed
163
164
      apply impl_intro_l, pure_elim_l=> ?; by apply: big_sepL_lookup. }
    revert Φ HΦ. induction l as [|x l IH]=> Φ HΦ; [by auto using big_sepL_nil'|].
165
    rewrite big_sepL_cons. rewrite -persistent_and_sep; apply and_intro.
166
    - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
167
168
169
170
    - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)).
  Qed.

  Lemma big_sepL_impl Φ Ψ l :
Robbert Krebbers's avatar
Robbert Krebbers committed
171
    ([ list] kx  l, Φ k x) -
172
     ( k x, l !! k = Some x  Φ k x - Ψ k x) -
Robbert Krebbers's avatar
Robbert Krebbers committed
173
    [ list] kx  l, Ψ k x.
174
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
175
176
    apply wand_intro_l. revert Φ Ψ. induction l as [|x l IH]=> Φ Ψ /=.
    { by rewrite sep_elim_r. }
177
    rewrite intuitionistically_sep_dup -assoc [( _  _)%I]comm -!assoc assoc.
Robbert Krebbers's avatar
Robbert Krebbers committed
178
179
    apply sep_mono.
    - rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
180
      by rewrite intuitionistically_elim wand_elim_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
181
    - rewrite comm -(IH (Φ  S) (Ψ  S)) /=.
182
      apply sep_mono_l, affinely_mono, persistently_mono.
Robbert Krebbers's avatar
Robbert Krebbers committed
183
      apply forall_intro=> k. by rewrite (forall_elim (S k)).
184
185
  Qed.

186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
  Lemma big_sepL_delete Φ l i x :
    l !! i = Some x 
    ([ list] ky  l, Φ k y)
     Φ i x  [ list] ky  l, if decide (k = i) then emp else Φ k y.
  Proof.
    intros. rewrite -(take_drop_middle l i x) // !big_sepL_app /= Nat.add_0_r.
    rewrite take_length_le; last eauto using lookup_lt_Some, Nat.lt_le_incl.
    rewrite decide_True // left_id.
    rewrite assoc -!(comm _ (Φ _ _)) -assoc. do 2 f_equiv.
    - apply big_sepL_proper=> k y Hk. apply lookup_lt_Some in Hk.
      rewrite take_length in Hk. by rewrite decide_False; last lia.
    - apply big_sepL_proper=> k y _. by rewrite decide_False; last lia.
  Qed.

  Lemma big_sepL_delete' `{!BiAffine PROP} Φ l i x :
    l !! i = Some x 
    ([ list] ky  l, Φ k y)  Φ i x  [ list] ky  l,  k  i   Φ k y.
  Proof.
    intros. rewrite big_sepL_delete //. (do 2 f_equiv)=> k y.
    rewrite -decide_emp. by repeat case_decide.
  Qed.

208
209
210
211
  Lemma big_sepL_replicate l P :
    [] replicate (length l) P  [ list] y  l, P.
  Proof. induction l as [|x l]=> //=; by f_equiv. Qed.

212
  Global Instance big_sepL_nil_persistent Φ :
213
    Persistent ([ list] kx  [], Φ k x).
214
  Proof. simpl; apply _. Qed.
215
  Global Instance big_sepL_persistent Φ l :
216
    ( k x, Persistent (Φ k x))  Persistent ([ list] kx  l, Φ k x).
217
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
218
  Global Instance big_sepL_persistent_id Ps :
219
    TCForall Persistent Ps  Persistent ([] Ps).
220
  Proof. induction 1; simpl; apply _. Qed.
221

222
223
224
  Global Instance big_sepL_nil_affine Φ :
    Affine ([ list] kx  [], Φ k x).
  Proof. simpl; apply _. Qed.
225
226
227
  Global Instance big_sepL_affine Φ l :
    ( k x, Affine (Φ k x))  Affine ([ list] kx  l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
228
229
  Global Instance big_sepL_affine_id Ps : TCForall Affine Ps  Affine ([] Ps).
  Proof. induction 1; simpl; apply _. Qed.
230
End sep_list.
231

232
Section sep_list_more.
233
234
  Context {A : Type}.
  Implicit Types l : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
235
  Implicit Types Φ Ψ : nat  A  PROP.
236
237
238
  (* Some lemmas depend on the generalized versions of the above ones. *)

  Lemma big_sepL_zip_with {B C} Φ f (l1 : list B) (l2 : list C) :
Robbert Krebbers's avatar
Robbert Krebbers committed
239
    ([ list] kx  zip_with f l1 l2, Φ k x)
Robbert Krebbers's avatar
Robbert Krebbers committed
240
     ([ list] kx  l1, if l2 !! k is Some y then Φ k (f x y) else emp).
241
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
242
243
244
    revert Φ l2; induction l1 as [|x l1 IH]=> Φ [|y l2] //=.
    - by rewrite big_sepL_emp left_id.
    - by rewrite IH.
245
  Qed.
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
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
End sep_list_more.

Lemma big_sepL2_alt {A B} (Φ : nat  A  B  PROP) l1 l2 :
  ([ list] ky1;y2  l1; l2, Φ k y1 y2)
    length l1 = length l2   [ list] k  y  zip l1 l2, Φ k (y.1) (y.2).
Proof.
  apply (anti_symm _).
  - apply and_intro.
    + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=;
        auto using pure_intro, False_elim.
      rewrite IH sep_elim_r. apply pure_mono; auto.
    + revert Φ l2. induction l1 as [|x1 l1 IH]=> Φ -[|x2 l2] /=;
        auto using pure_intro, False_elim.
      by rewrite IH.
  - apply pure_elim_l=> /Forall2_same_length Hl. revert Φ.
    induction Hl as [|x1 l1 x2 l2 _ _ IH]=> Φ //=. by rewrite -IH.
Qed.

(** ** Big ops over two lists *)
Section sep_list2.
  Context {A B : Type}.
  Implicit Types Φ Ψ : nat  A  B  PROP.

  Lemma big_sepL2_nil Φ : ([ list] ky1;y2  []; [], Φ k y1 y2)  emp.
  Proof. done. Qed.
  Lemma big_sepL2_nil' `{BiAffine PROP} P Φ : P  [ list] ky1;y2  [];[], Φ k y1 y2.
  Proof. apply (affine _). Qed.

  Lemma big_sepL2_cons Φ x1 x2 l1 l2 :
    ([ list] ky1;y2  x1 :: l1; x2 :: l2, Φ k y1 y2)
     Φ 0 x1 x2  [ list] ky1;y2  l1;l2, Φ (S k) y1 y2.
  Proof. done. Qed.
  Lemma big_sepL2_cons_inv_l Φ x1 l1 l2 :
    ([ list] ky1;y2  x1 :: l1; l2, Φ k y1 y2) -
     x2 l2',  l2 = x2 :: l2'  
              Φ 0 x1 x2  [ list] ky1;y2  l1;l2', Φ (S k) y1 y2.
  Proof.
    destruct l2 as [|x2 l2]; simpl; auto using False_elim.
    by rewrite -(exist_intro x2) -(exist_intro l2) pure_True // left_id.
  Qed.
  Lemma big_sepL2_cons_inv_r Φ x2 l1 l2 :
    ([ list] ky1;y2  l1; x2 :: l2, Φ k y1 y2) -
     x1 l1',  l1 = x1 :: l1'  
              Φ 0 x1 x2  [ list] ky1;y2  l1';l2, Φ (S k) y1 y2.
  Proof.
    destruct l1 as [|x1 l1]; simpl; auto using False_elim.
    by rewrite -(exist_intro x1) -(exist_intro l1) pure_True // left_id.
  Qed.

  Lemma big_sepL2_singleton Φ x1 x2 :
    ([ list] ky1;y2  [x1];[x2], Φ k y1 y2)  Φ 0 x1 x2.
  Proof. by rewrite /= right_id. Qed.

  Lemma big_sepL2_length Φ l1 l2 :
    ([ list] ky1;y2  l1; l2, Φ k y1 y2) -  length l1 = length l2 .
  Proof. by rewrite big_sepL2_alt and_elim_l. Qed.

  Lemma big_sepL2_app Φ l1 l2 l1' l2' :
    ([ list] ky1;y2  l1; l1', Φ k y1 y2) -
    ([ list] ky1;y2  l2; l2', Φ (length l1 + k) y1 y2) -
    ([ list] ky1;y2  l1 ++ l2; l1' ++ l2', Φ k y1 y2).
  Proof.
    apply wand_intro_r. revert Φ l1'. induction l1 as [|x1 l1 IH]=> Φ -[|x1' l1'] /=.
    - by rewrite left_id.
    - rewrite left_absorb. apply False_elim.
    - rewrite left_absorb. apply False_elim.
    - by rewrite -assoc IH.
  Qed.
  Lemma big_sepL2_app_inv_l Φ l1' l1'' l2 :
    ([ list] ky1;y2  l1' ++ l1''; l2, Φ k y1 y2) -
     l2' l2'',  l2 = l2' ++ l2''  
                ([ list] ky1;y2  l1';l2', Φ k y1 y2) 
                ([ list] ky1;y2  l1'';l2'', Φ (length l1' + k) y1 y2).
  Proof.
    rewrite -(exist_intro (take (length l1') l2))
      -(exist_intro (drop (length l1') l2)) take_drop pure_True // left_id.
    revert Φ l2. induction l1' as [|x1 l1' IH]=> Φ -[|x2 l2] /=;
       [by rewrite left_id|by rewrite left_id|apply False_elim|].
    by rewrite IH -assoc.
  Qed.
  Lemma big_sepL2_app_inv_r Φ l1 l2' l2'' :
    ([ list] ky1;y2  l1; l2' ++ l2'', Φ k y1 y2) -
     l1' l1'',  l1 = l1' ++ l1''  
                ([ list] ky1;y2  l1';l2', Φ k y1 y2) 
                ([ list] ky1;y2  l1'';l2'', Φ (length l2' + k) y1 y2).
  Proof.
    rewrite -(exist_intro (take (length l2') l1))
      -(exist_intro (drop (length l2') l1)) take_drop pure_True // left_id.
    revert Φ l1. induction l2' as [|x2 l2' IH]=> Φ -[|x1 l1] /=;
       [by rewrite left_id|by rewrite left_id|apply False_elim|].
    by rewrite IH -assoc.
  Qed.
338
339
340
341
342
343
344
345
346
347
  Lemma big_sepL2_app_inv Φ l1 l2 l1' l2' :
    length l1 = length l1' 
    ([ list] ky1;y2  l1 ++ l2; l1' ++ l2', Φ k y1 y2) -
    ([ list] ky1;y2  l1; l1', Φ k y1 y2) 
    ([ list] ky1;y2  l2; l2', Φ (length l1 + k)%nat y1 y2).
  Proof.
    revert Φ l1'. induction l1 as [|x1 l1 IH]=> Φ -[|x1' l1'] //= ?; simplify_eq.
    - by rewrite left_id.
    - by rewrite -assoc IH.
  Qed.
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412

  Lemma big_sepL2_mono Φ Ψ l1 l2 :
    ( k y1 y2, l1 !! k = Some y1  l2 !! k = Some y2  Φ k y1 y2  Ψ k y1 y2) 
    ([ list] k  y1;y2  l1;l2, Φ k y1 y2)  [ list] k  y1;y2  l1;l2, Ψ k y1 y2.
  Proof.
    intros H. rewrite !big_sepL2_alt. f_equiv. apply big_sepL_mono=> k [y1 y2].
    rewrite lookup_zip_with=> ?; simplify_option_eq; auto.
  Qed.
  Lemma big_sepL2_proper Φ Ψ l1 l2 :
    ( k y1 y2, l1 !! k = Some y1  l2 !! k = Some y2  Φ k y1 y2  Ψ k y1 y2) 
    ([ list] k  y1;y2  l1;l2, Φ k y1 y2)  [ list] k  y1;y2  l1;l2, Ψ k y1 y2.
  Proof.
    intros; apply (anti_symm _);
      apply big_sepL2_mono; auto using equiv_entails, equiv_entails_sym.
  Qed.

  Global Instance big_sepL2_ne n :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ (dist n)))
      ==> (=) ==> (=) ==> (dist n))
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof.
    intros Φ1 Φ2 HΦ x1 ? <- x2 ? <-. rewrite !big_sepL2_alt. f_equiv.
    f_equiv=> k [y1 y2]. apply HΦ.
  Qed.
  Global Instance big_sepL2_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ ()))
      ==> (=) ==> (=) ==> ())
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_mono; intros; apply Hf. Qed.
  Global Instance big_sepL2_proper' :
    Proper (pointwise_relation _ (pointwise_relation _ (pointwise_relation _ ()))
      ==> (=) ==> (=) ==> ())
           (big_sepL2 (PROP:=PROP) (A:=A) (B:=B)).
  Proof. intros f g Hf l1 ? <- l2 ? <-. apply big_sepL2_proper; intros; apply Hf. Qed.

  Lemma big_sepL2_lookup_acc Φ l1 l2 i x1 x2 :
    l1 !! i = Some x1  l2 !! i = Some x2 
    ([ list] ky1;y2  l1;l2, Φ k y1 y2) 
    Φ i x1 x2  (Φ i x1 x2 - ([ list] ky1;y2  l1;l2, Φ k y1 y2)).
  Proof.
    intros Hl1 Hl2. rewrite big_sepL2_alt. apply pure_elim_l=> Hl.
    rewrite {1}big_sepL_lookup_acc; last by rewrite lookup_zip_with; simplify_option_eq.
    by rewrite pure_True // left_id.
  Qed.

  Lemma big_sepL2_lookup Φ l1 l2 i x1 x2 `{!Absorbing (Φ i x1 x2)} :
    l1 !! i = Some x1  l2 !! i = Some x2 
    ([ list] ky1;y2  l1;l2, Φ k y1 y2)  Φ i x1 x2.
  Proof. intros. rewrite big_sepL2_lookup_acc //. by rewrite sep_elim_l. Qed.

  Lemma big_sepL2_fmap_l {A'} (f : A  A') (Φ : nat  A'  B  PROP) l1 l2 :
    ([ list] ky1;y2  f <$> l1; l2, Φ k y1 y2)
     ([ list] ky1;y2  l1;l2, Φ k (f y1) y2).
  Proof.
    rewrite !big_sepL2_alt fmap_length zip_with_fmap_l zip_with_zip big_sepL_fmap.
    by f_equiv; f_equiv=> k [??].
  Qed.
  Lemma big_sepL2_fmap_r {B'} (g : B  B') (Φ : nat  A  B'  PROP) l1 l2 :
    ([ list] ky1;y2  l1; g <$> l2, Φ k y1 y2)
     ([ list] ky1;y2  l1;l2, Φ k y1 (g y2)).
  Proof.
    rewrite !big_sepL2_alt fmap_length zip_with_fmap_r zip_with_zip big_sepL_fmap.
    by f_equiv; f_equiv=> k [??].
  Qed.

413
414
415
416
417
418
419
420
421
422
423
  Lemma big_sepL2_reverse_2 (Φ : A  B  PROP) l1 l2 :
    ([ list] y1;y2  l1;l2, Φ y1 y2)  ([ list] y1;y2  reverse l1;reverse l2, Φ y1 y2).
  Proof.
    revert l2. induction l1 as [|x1 l1 IH]; intros [|x2 l2]; simpl; auto using False_elim.
    rewrite !reverse_cons (comm bi_sep) IH.
    by rewrite (big_sepL2_app _ _ [x1] _ [x2]) big_sepL2_singleton wand_elim_l.
  Qed.
  Lemma big_sepL2_reverse (Φ : A  B  PROP) l1 l2 :
    ([ list] y1;y2  reverse l1;reverse l2, Φ y1 y2)  ([ list] y1;y2  l1;l2, Φ y1 y2).
  Proof. apply (anti_symm _); by rewrite big_sepL2_reverse_2 ?reverse_involutive. Qed.

424
  Lemma big_sepL2_sep Φ Ψ l1 l2 :
425
426
427
    ([ list] ky1;y2  l1;l2, Φ k y1 y2  Ψ k y1 y2)
     ([ list] ky1;y2  l1;l2, Φ k y1 y2)  ([ list] ky1;y2  l1;l2, Ψ k y1 y2).
  Proof.
428
    rewrite !big_sepL2_alt big_sepL_sep !persistent_and_affinely_sep_l.
429
430
431
432
433
434
435
436
437
438
439
440
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
467
468
469
470
471
472
473
474
475
476
477
    rewrite -assoc (assoc _ _ (<affine> _)%I). rewrite -(comm bi_sep (<affine> _)%I).
    rewrite -assoc (assoc _ _ (<affine> _)%I) -!persistent_and_affinely_sep_l.
    by rewrite affinely_and_r persistent_and_affinely_sep_l idemp.
  Qed.

  Lemma big_sepL2_and Φ Ψ l1 l2 :
    ([ list] ky1;y2  l1;l2, Φ k y1 y2  Ψ k y1 y2)
     ([ list] ky1;y2  l1;l2, Φ k y1 y2)  ([ list] ky1;y2  l1;l2, Ψ k y1 y2).
  Proof. auto using and_intro, big_sepL2_mono, and_elim_l, and_elim_r. Qed.

  Lemma big_sepL2_persistently `{BiAffine PROP} Φ l1 l2 :
    <pers> ([ list] ky1;y2  l1;l2, Φ k y1 y2)
     [ list] ky1;y2  l1;l2, <pers> (Φ k y1 y2).
  Proof.
    by rewrite !big_sepL2_alt persistently_and persistently_pure big_sepL_persistently.
  Qed.

  Lemma big_sepL2_impl Φ Ψ l1 l2 :
    ([ list] ky1;y2  l1;l2, Φ k y1 y2) -
     ( k x1 x2,
      l1 !! k = Some x1  l2 !! k = Some x2  Φ k x1 x2 - Ψ k x1 x2) -
    [ list] ky1;y2  l1;l2, Ψ k y1 y2.
  Proof.
    apply wand_intro_l. revert Φ Ψ l2.
    induction l1 as [|x1 l1 IH]=> Φ Ψ [|x2 l2] /=; [by rewrite sep_elim_r..|].
    rewrite intuitionistically_sep_dup -assoc [( _  _)%I]comm -!assoc assoc.
    apply sep_mono.
    - rewrite (forall_elim 0) (forall_elim x1) (forall_elim x2) !pure_True // !True_impl.
      by rewrite intuitionistically_elim wand_elim_l.
    - rewrite comm -(IH (Φ  S) (Ψ  S)) /=.
      apply sep_mono_l, affinely_mono, persistently_mono.
      apply forall_intro=> k. by rewrite (forall_elim (S k)).
  Qed.

  Global Instance big_sepL2_nil_persistent Φ :
    Persistent ([ list] ky1;y2  []; [], Φ k y1 y2).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL2_persistent Φ l1 l2 :
    ( k x1 x2, Persistent (Φ k x1 x2)) 
    Persistent ([ list] ky1;y2  l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.

  Global Instance big_sepL2_nil_affine Φ :
    Affine ([ list] ky1;y2  []; [], Φ k y1 y2).
  Proof. simpl; apply _. Qed.
  Global Instance big_sepL2_affine Φ l1 l2 :
    ( k x1 x2, Affine (Φ k x1 x2)) 
    Affine ([ list] ky1;y2  l1;l2, Φ k y1 y2).
  Proof. rewrite big_sepL2_alt. apply _. Qed.
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
End sep_list2.

Section and_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat  A  PROP.

  Lemma big_andL_nil Φ : ([ list] ky  nil, Φ k y)  True.
  Proof. done. Qed.
  Lemma big_andL_nil' P Φ : P  [ list] ky  nil, Φ k y.
  Proof. by apply pure_intro. Qed.
  Lemma big_andL_cons Φ x l :
    ([ list] ky  x :: l, Φ k y)  Φ 0 x  [ list] ky  l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_andL_singleton Φ x : ([ list] ky  [x], Φ k y)  Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_andL_app Φ l1 l2 :
    ([ list] ky  l1 ++ l2, Φ k y)
     ([ list] ky  l1, Φ k y)  ([ list] ky  l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.

  Lemma big_andL_mono Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
    ([ list] k  y  l, Φ k y)  [ list] k  y  l, Ψ k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_andL_proper Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
    ([ list] k  y  l, Φ k y)  ([ list] k  y  l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
  Lemma big_andL_submseteq (Φ : A  PROP) l1 l2 :
    l1 + l2  ([ list] y  l2, Φ y)  [ list] y  l1, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_andL_app and_elim_l.
  Qed.

  Global Instance big_andL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> (=) ==> ())
           (big_opL (@bi_and PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
517
  Global Instance big_andL_id_mono' :
518
    Proper (Forall2 () ==> ()) (big_opL (@bi_and PROP) (λ _ P, P)).
519
520
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

521
  Lemma big_andL_lookup Φ l i x :
522
523
524
525
526
527
528
    l !! i = Some x  ([ list] ky  l, Φ k y)  Φ i x.
  Proof.
    intros. rewrite -(take_drop_middle l i x) // big_andL_app /=.
    rewrite Nat.add_0_r take_length_le;
      eauto using lookup_lt_Some, Nat.lt_le_incl, and_elim_l', and_elim_r'.
  Qed.

529
  Lemma big_andL_elem_of (Φ : A  PROP) l x :
530
531
532
533
534
535
536
537
538
    x  l  ([ list] y  l, Φ y)  Φ x.
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_andL_lookup (λ _, Φ)).
  Qed.

  Lemma big_andL_fmap {B} (f : A  B) (Φ : nat  B  PROP) l :
    ([ list] ky  f <$> l, Φ k y)  ([ list] ky  l, Φ k (f y)).
  Proof. by rewrite big_opL_fmap. Qed.

539
540
541
542
  Lemma big_andL_bind {B} (f : A  list B) (Φ : B  PROP) l :
    ([ list] y  l = f, Φ y)  ([ list] x  l, [ list] y  f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

543
544
  Lemma big_andL_and Φ Ψ l :
    ([ list] kx  l, Φ k x  Ψ k x)
545
546
     ([ list] kx  l, Φ k x)  ([ list] kx  l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.
547
548

  Lemma big_andL_persistently Φ l :
549
    <pers> ([ list] kx  l, Φ k x)  [ list] kx  l, <pers> (Φ k x).
550
551
  Proof. apply (big_opL_commute _). Qed.

552
  Lemma big_andL_forall Φ l :
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
    ([ list] kx  l, Φ k x)  ( k x, l !! k = Some x  Φ k x).
  Proof.
    apply (anti_symm _).
    { apply forall_intro=> k; apply forall_intro=> x.
      apply impl_intro_l, pure_elim_l=> ?; by apply: big_andL_lookup. }
    revert Φ. induction l as [|x l IH]=> Φ; [by auto using big_andL_nil'|].
    rewrite big_andL_cons. apply and_intro.
    - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
    - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)).
  Qed.

  Global Instance big_andL_nil_persistent Φ :
    Persistent ([ list] kx  [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_andL_persistent Φ l :
    ( k x, Persistent (Φ k x))  Persistent ([ list] kx  l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
End and_list.
571

572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
Section or_list.
  Context {A : Type}.
  Implicit Types l : list A.
  Implicit Types Φ Ψ : nat  A  PROP.

  Lemma big_orL_nil Φ : ([ list] ky  nil, Φ k y)  False.
  Proof. done. Qed.
  Lemma big_orL_cons Φ x l :
    ([ list] ky  x :: l, Φ k y)  Φ 0 x  [ list] ky  l, Φ (S k) y.
  Proof. by rewrite big_opL_cons. Qed.
  Lemma big_orL_singleton Φ x : ([ list] ky  [x], Φ k y)  Φ 0 x.
  Proof. by rewrite big_opL_singleton. Qed.
  Lemma big_orL_app Φ l1 l2 :
    ([ list] ky  l1 ++ l2, Φ k y)
     ([ list] ky  l1, Φ k y)  ([ list] ky  l2, Φ (length l1 + k) y).
  Proof. by rewrite big_opL_app. Qed.

  Lemma big_orL_mono Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
    ([ list] k  y  l, Φ k y)  [ list] k  y  l, Ψ k y.
  Proof. apply big_opL_forall; apply _. Qed.
  Lemma big_orL_proper Φ Ψ l :
    ( k y, l !! k = Some y  Φ k y  Ψ k y) 
    ([ list] k  y  l, Φ k y)  ([ list] k  y  l, Ψ k y).
  Proof. apply big_opL_proper. Qed.
  Lemma big_orL_submseteq (Φ : A  PROP) l1 l2 :
    l1 + l2  ([ list] y  l1, Φ y)  [ list] y  l2, Φ y.
  Proof.
    intros [l ->]%submseteq_Permutation. by rewrite big_orL_app -or_intro_l.
  Qed.

  Global Instance big_orL_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> (=) ==> ())
           (big_opL (@bi_or PROP) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_opL_forall; apply _ || intros; apply Hf. Qed.
  Global Instance big_orL_id_mono' :
    Proper (Forall2 () ==> ()) (big_opL (@bi_or PROP) (λ _ P, P)).
  Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.

  Lemma big_orL_lookup Φ l i x :
    l !! i = Some x  Φ i x  ([ list] ky  l, Φ k y).
  Proof.
    intros. rewrite -(take_drop_middle l i x) // big_orL_app /=.
    rewrite Nat.add_0_r take_length_le;
      eauto using lookup_lt_Some, Nat.lt_le_incl, or_intro_l', or_intro_r'.
  Qed.

  Lemma big_orL_elem_of (Φ : A  PROP) l x :
    x  l  Φ x  ([ list] y  l, Φ y).
  Proof.
    intros [i ?]%elem_of_list_lookup; eauto using (big_orL_lookup (λ _, Φ)).
  Qed.

  Lemma big_orL_fmap {B} (f : A  B) (Φ : nat  B  PROP) l :
    ([ list] ky  f <$> l, Φ k y)  ([ list] ky  l, Φ k (f y)).
  Proof. by rewrite big_opL_fmap. Qed.

629
630
631
632
  Lemma big_orL_bind {B} (f : A  list B) (Φ : B  PROP) l :
    ([ list] y  l = f, Φ y)  ([ list] x  l, [ list] y  f x, Φ y).
  Proof. by rewrite big_opL_bind. Qed.

633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
  Lemma big_orL_or Φ Ψ l :
    ([ list] kx  l, Φ k x  Ψ k x)
     ([ list] kx  l, Φ k x)  ([ list] kx  l, Ψ k x).
  Proof. by rewrite big_opL_op. Qed.

  Lemma big_orL_persistently Φ l :
    <pers> ([ list] kx  l, Φ k x)  [ list] kx  l, <pers> (Φ k x).
  Proof. apply (big_opL_commute _). Qed.

  Lemma big_orL_exist Φ l :
    ([ list] kx  l, Φ k x)  ( k x, l !! k = Some x  Φ k x).
  Proof.
    apply (anti_symm _).
    { revert Φ. induction l as [|x l IH]=> Φ.
      { rewrite big_orL_nil. apply False_elim. }
      rewrite big_orL_cons. apply or_elim.
      - by rewrite -(exist_intro 0) -(exist_intro x) pure_True // left_id.
      - rewrite IH. apply exist_elim=> k. by rewrite -(exist_intro (S k)). }
    apply exist_elim=> k; apply exist_elim=> x. apply pure_elim_l=> ?.
    by apply: big_orL_lookup.
  Qed.

  Lemma big_orL_sep_l P Φ l :
    P  ([ list] kx  l, Φ k x)  ([ list] kx  l, P  Φ k x).
  Proof.
    rewrite !big_orL_exist sep_exist_l.
    f_equiv=> k. rewrite sep_exist_l. f_equiv=> x.
    by rewrite !persistent_and_affinely_sep_l !assoc (comm _ P).
 Qed.
  Lemma big_orL_sep_r Q Φ l :
    ([ list] kx  l, Φ k x)  Q  ([ list] kx  l, Φ k x  Q).
  Proof. setoid_rewrite (comm bi_sep). apply big_orL_sep_l. Qed.

  Global Instance big_orL_nil_persistent Φ :
    Persistent ([ list] kx  [], Φ k x).
  Proof. simpl; apply _. Qed.
  Global Instance big_orL_persistent Φ l :
    ( k x, Persistent (Φ k x))  Persistent ([ list] kx  l, Φ k x).
  Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
End or_list.

674
(** ** Big ops over finite maps *)
Dan Frumin's avatar
Dan Frumin committed
675
Section map.
676
677
  Context `{Countable K} {A : Type}.
  Implicit Types m : gmap K A.
Robbert Krebbers's avatar
Robbert Krebbers committed
678
  Implicit Types Φ Ψ : K  A  PROP.
679

Robbert Krebbers's avatar
Robbert Krebbers committed
680
681
682
683
  Lemma big_sepM_mono Φ Ψ m :
    ( k x, m !! k = Some x  Φ k x  Ψ k x) 
    ([ map] k  x  m, Φ k x)  [ map] k  x  m, Ψ k x.
  Proof. apply big_opM_forall; apply _ || auto. Qed.
684
685
  Lemma big_sepM_proper Φ Ψ m :
    ( k x, m !! k = Some x  Φ k x  Ψ k x) 
686
    ([ map] k  x  m, Φ k x)  ([ map] k  x  m, Ψ k x).
687
  Proof. apply big_opM_proper. Qed.
688
  Lemma big_sepM_subseteq `{BiAffine PROP} Φ m1 m2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
689
690
    m2  m1  ([ map] k  x  m1, Φ k x)  [ map] k  x  m2, Φ k x.
  Proof. intros. by apply big_sepL_submseteq, map_to_list_submseteq. Qed.
691

692
693
  Global Instance big_sepM_mono' :
    Proper (pointwise_relation _ (pointwise_relation _ ()) ==> (=) ==> ())
Robbert Krebbers's avatar
Robbert Krebbers committed
694
695
           (big_opM (@bi_sep PROP) (K:=K) (A:=A)).
  Proof. intros f g Hf m ? <-. apply big_sepM_mono=> ???; apply Hf. Qed.
696

Robbert Krebbers's avatar
Robbert Krebbers committed
697
  Lemma big_sepM_empty Φ : ([ map] kx  , Φ k x)  emp.
698
  Proof. by rewrite big_opM_empty. Qed.
699
  Lemma big_sepM_empty' `{BiAffine PROP} P Φ : P  [ map] kx  , Φ k x.
Robbert Krebbers's avatar
Robbert Krebbers committed
700
  Proof. rewrite big_sepM_empty. apply: affine. Qed.
701

702
  Lemma big_sepM_insert Φ m i x :
703
    m !! i = None 
704
    ([ map] ky  <[i:=x]> m, Φ k y)  Φ i x  [ map] ky  m, Φ k y.
705
  Proof. apply big_opM_insert. Qed.
706

707
  Lemma big_sepM_delete Φ m i x :
708
    m !! i = Some x 
709
    ([ map] ky  m, Φ k y)  Φ i x  [ map] ky  delete i m, Φ k y.
710
  Proof. apply big_opM_delete. Qed.
711

712
713
714
715
716
717
718
719
720
721
722
723
724
  Lemma big_sepM_insert_2 Φ m i x :
    TCOr ( x, Affine (Φ i x)) (Absorbing (Φ i x)) 
    Φ i x - ([ map] ky  m, Φ k y) - [ map] ky  <[i:=x]> m, Φ k y.
  Proof.
    intros Ha. apply wand_intro_r. destruct (m !! i) as [y|] eqn:Hi; last first.
    { by rewrite -big_sepM_insert. }
    assert (TCOr (Affine (Φ i y)) (Absorbing (Φ i x))).
    { destruct Ha; try apply _. }
    rewrite big_sepM_delete // assoc.
    rewrite (sep_elim_l (Φ i x)) -big_sepM_insert ?lookup_delete //.
    by rewrite insert_delete.
  Qed.

725
726
727
728
729
730
731
  Lemma big_sepM_lookup_acc Φ m i x :
    m !! i = Some x 
    ([ map] ky  m, Φ k y)  Φ i x  (Φ i x - ([ map] ky  m, Φ k y)).
  Proof.
    intros. rewrite big_sepM_delete //. by apply sep_mono_r, wand_intro_l.
  Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
732
  Lemma big_sepM_lookup Φ m i x `{!Absorbing (Φ i x)} :
733
    m !! i = Some x  ([ map] ky  m, Φ k y)  Φ i x.
Robbert Krebbers's avatar
Robbert Krebbers committed
734
  Proof. intros. rewrite big_sepM_lookup_acc //. by rewrite sep_elim_l. Qed.
735

Robbert Krebbers's avatar
Robbert Krebbers committed
736
  Lemma big_sepM_lookup_dom (Φ : K  PROP) m i `{!Absorbing (Φ i)} :
Robbert Krebbers's avatar
Robbert Krebbers committed
737
738
    is_Some (m !! i)  ([ map] k_  m, Φ k)  Φ i.
  Proof. intros [x ?]. by eapply (big_sepM_lookup (λ i x, Φ i)). Qed.
739

740
  Lemma big_sepM_singleton Φ i x : ([ map] ky  {[i:=x]}, Φ k y)  Φ i x.
741
  Proof. by rewrite big_opM_singleton. Qed.
742

Robbert Krebbers's avatar
Robbert Krebbers committed
743
  Lemma big_sepM_fmap {B} (f : A  B) (Φ : K  B  PROP) m :
744
    ([ map] ky  f <$> m, Φ k y)  ([ map] ky  m, Φ k (f y)).
745
  Proof. by rewrite big_opM_fmap. Qed.
746

Robbert Krebbers's avatar
Robbert Krebbers committed
747
748
749
  Lemma big_sepM_insert_override Φ m i x x' :
    m !! i = Some x  (Φ i x  Φ i x') 
    ([ map] ky  <[i:=x']> m, Φ k y)  ([ map] ky  m, Φ k y).
750
  Proof. apply big_opM_insert_override. Qed.
751

Robbert Krebbers's avatar
Robbert Krebbers committed
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
  Lemma big_sepM_insert_override_1 Φ m i x x' :
    m !! i = Some x 
    ([ map] ky  <[i:=x']> m, Φ k y) 
      (Φ i x' - Φ i x) - ([ map] ky  m, Φ k y).
  Proof.
    intros ?. apply wand_intro_l.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
    by rewrite assoc wand_elim_l -big_sepM_delete.
  Qed.

  Lemma big_sepM_insert_override_2 Φ m i x x' :
    m !! i = Some x 
    ([ map] ky  m, Φ k y) 
      (Φ i x - Φ i x') - ([ map] ky  <[i:=x']> m, Φ k y).
  Proof.
    intros ?. apply wand_intro_l.
    rewrite {1}big_sepM_delete //; rewrite assoc wand_elim_l.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
  Qed.

Dan Frumin's avatar
Dan Frumin committed
772
773
774
775
776
777
778
779
780
781
782
  Lemma big_sepM_insert_acc Φ m i x :
    m !! i = Some x 
    ([ map] ky  m, Φ k y) 
      Φ i x  ( x', Φ i x' - ([ map] ky  <[i:=x']> m, Φ k y)).
  Proof.
    intros ?. rewrite {1}big_sepM_delete //. apply sep_mono; [done|].
    apply forall_intro=> x'.
    rewrite -insert_delete big_sepM_insert ?lookup_delete //.
    by apply wand_intro_l.
  Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
783
  Lemma big_sepM_fn_insert {B} (Ψ : K  A  B  PROP) (f : K  B) m i x b :
784
    m !! i = None 
785
786
       ([ map] ky  <[i:=x]> m, Ψ k y (<[i:=b]> f k))
     (Ψ i x b  [ map] ky  m, Ψ k y (f k)).
787
  Proof. apply big_opM_fn_insert. Qed.
788

Robbert Krebbers's avatar
Robbert Krebbers committed
789
  Lemma big_sepM_fn_insert' (Φ : K  PROP) m i x P :
790
    m !! i = None 
791
    ([ map] ky  <[i:=x]> m, <[i:=P]> Φ k)  (P  [ map] ky  m, Φ k).
792
  Proof. apply big_opM_fn_insert'. Qed.
793

794
795
796
797
798
799
  Lemma big_sepM_union Φ m1 m2 :
    m1 ## m2 
    ([ map] ky  m1  m2, Φ k y)
     ([ map] ky  m1, Φ k y)  ([ map] ky  m2, Φ k y).
  Proof. apply big_opM_union. Qed.

800
  Lemma big_sepM_sep Φ Ψ m :
801
    ([ map] kx  m, Φ k x  Ψ k x)
802
     ([ map] kx  m, Φ k x)  ([ map] kx  m, Ψ k x).
803
  Proof. apply big_opM_op. Qed.
804

805
806
807
  Lemma big_sepM_and Φ Ψ m :
    ([ map] kx  m, Φ k x  Ψ k x)
     ([ map] kx  m, Φ k x)  ([ map] kx  m, Ψ k x).
Robbert Krebbers's avatar
Robbert Krebbers committed
808
  Proof. auto using and_intro, big_sepM_mono, and_elim_l, and_elim_r. Qed.
809

810
  Lemma big_sepM_persistently `{BiAffine PROP} Φ m :
811
    (<pers> ([ map] kx  m, Φ k x))  ([ map] kx  m, <pers> (Φ k x)).
812
  Proof. apply (big_opM_commute _). Qed.
813

814
  Lemma big_sepM_forall `{BiAffine PROP} Φ m :
815
    ( k x, Persistent (Φ k x)) 
Ralf Jung's avatar
Ralf Jung committed
816
    ([ map] kx  m, Φ k x)  ( k x, m !! k = Some x  Φ k x).
817
818
819
  Proof.
    intros. apply (anti_symm _).
    { apply forall_intro=> k; apply forall_intro=> x.
Robbert Krebbers's avatar
Robbert Krebbers committed
820
821
      apply impl_intro_l, pure_elim_l=> ?; by apply: big_sepM_lookup. }
    induction m as [|i x m ? IH] using map_ind; auto using big_sepM_empty'.
822
    rewrite big_sepM_insert // -persistent_and_sep. apply and_intro.
823
    - rewrite (forall_elim i) (forall_elim x) lookup_insert.
824
      by rewrite pure_True // True_impl.
825
    - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y.
826
827
      apply impl_intro_l, pure_elim_l=> ?.
      rewrite lookup_insert_ne; last by intros ?; simplify_map_eq.
828
      by rewrite pure_True // True_impl.
829
830
831
  Qed.

  Lemma big_sepM_impl Φ Ψ m :
Robbert Krebbers's avatar
Robbert Krebbers committed
832
    ([ map] kx  m, Φ k x) -
833
     ( k x, m !! k = Some x  Φ k x - Ψ k x) -
Robbert Krebbers's avatar
Robbert Krebbers committed
834
    [ map] kx  m, Ψ k x.
835
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
836
837
    apply wand_intro_l. induction m as [|i x m ? IH] using map_ind.
    { by rewrite sep_elim_r. }
838
    rewrite !big_sepM_insert // intuitionistically_sep_dup.
839
    rewrite -assoc [( _  _)%I]comm -!assoc assoc. apply sep_mono.
Robbert Krebbers's avatar
Robbert Krebbers committed
840
    - rewrite (forall_elim i) (forall_elim x) pure_True ?lookup_insert //.
841
      by rewrite True_impl intuitionistically_elim wand_elim_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
842
    - rewrite comm -IH /=.
843
      apply sep_mono_l, affinely_mono, persistently_mono, forall_mono=> k.
Robbert Krebbers's avatar
Robbert Krebbers committed
844
845
846
      apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?.
      rewrite lookup_insert_ne; last by intros ?; simplify_map_eq.
      by rewrite pure_True // True_impl.
847
  Qed.
848

849
  Global Instance big_sepM_empty_persistent Φ :
850
    Persistent ([ map] kx  , Φ k x).
851
  Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.