list.v 19 KB
Newer Older
1
From iris.algebra Require Export cmra.
2
From iris.prelude Require Export list.
3
From iris.algebra Require Import upred updates local_updates.
Robbert Krebbers's avatar
Robbert Krebbers committed
4 5 6 7 8 9

Section cofe.
Context {A : cofeT}.

Instance list_dist : Dist (list A) := λ n, Forall2 (dist n).

10 11 12
Lemma list_dist_lookup n l1 l2 : l1 {n} l2   i, l1 !! i {n} l2 !! i.
Proof. setoid_rewrite dist_option_Forall2. apply Forall2_lookup. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
13 14 15 16 17 18 19
Global Instance cons_ne n : Proper (dist n ==> dist n ==> dist n) (@cons A) := _.
Global Instance app_ne n : Proper (dist n ==> dist n ==> dist n) (@app A) := _.
Global Instance length_ne n : Proper (dist n ==> (=)) (@length A) := _.
Global Instance tail_ne n : Proper (dist n ==> dist n) (@tail A) := _.
Global Instance take_ne n : Proper (dist n ==> dist n) (@take A n) := _.
Global Instance drop_ne n : Proper (dist n ==> dist n) (@drop A n) := _.
Global Instance list_lookup_ne n i :
20
  Proper (dist n ==> dist n) (lookup (M:=list A) i).
Robbert Krebbers's avatar
Robbert Krebbers committed
21 22 23
Proof. intros ???. by apply dist_option_Forall2, Forall2_lookup. Qed.
Global Instance list_alter_ne n f i :
  Proper (dist n ==> dist n) f 
24
  Proper (dist n ==> dist n) (alter (M:=list A) f i) := _.
Robbert Krebbers's avatar
Robbert Krebbers committed
25
Global Instance list_insert_ne n i :
26
  Proper (dist n ==> dist n ==> dist n) (insert (M:=list A) i) := _.
Robbert Krebbers's avatar
Robbert Krebbers committed
27 28 29
Global Instance list_inserts_ne n i :
  Proper (dist n ==> dist n ==> dist n) (@list_inserts A i) := _.
Global Instance list_delete_ne n i :
30
  Proper (dist n ==> dist n) (delete (M:=list A) i) := _.
Robbert Krebbers's avatar
Robbert Krebbers committed
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Global Instance option_list_ne n : Proper (dist n ==> dist n) (@option_list A).
Proof. intros ???; by apply Forall2_option_list, dist_option_Forall2. Qed.
Global Instance list_filter_ne n P `{ x, Decision (P x)} :
  Proper (dist n ==> iff) P 
  Proper (dist n ==> dist n) (filter (B:=list A) P) := _.
Global Instance replicate_ne n :
  Proper (dist n ==> dist n) (@replicate A n) := _.
Global Instance reverse_ne n : Proper (dist n ==> dist n) (@reverse A) := _.
Global Instance last_ne n : Proper (dist n ==> dist n) (@last A).
Proof. intros ???; by apply dist_option_Forall2, Forall2_last. Qed.
Global Instance resize_ne n :
  Proper (dist n ==> dist n ==> dist n) (@resize A n) := _.

Program Definition list_chain
    (c : chain (list A)) (x : A) (k : nat) : chain A :=
46
  {| chain_car n := from_option id x (c n !! k) |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
Next Obligation. intros c x k n i ?. by rewrite /= (chain_cauchy c n i). Qed.
Instance list_compl : Compl (list A) := λ c,
  match c 0 with
  | [] => []
  | x :: _ => compl  list_chain c x <$> seq 0 (length (c 0))
  end.

Definition list_cofe_mixin : CofeMixin (list A).
Proof.
  split.
  - intros l k. rewrite equiv_Forall2 -Forall2_forall.
    split; induction 1; constructor; intros; try apply equiv_dist; auto.
  - apply _.
  - rewrite /dist /list_dist. eauto using Forall2_impl, dist_S.
  - intros n c; rewrite /compl /list_compl.
    destruct (c 0) as [|x l] eqn:Hc0 at 1.
    { by destruct (chain_cauchy c 0 n); auto with omega. }
    rewrite -(λ H, length_ne _ _ _ (chain_cauchy c 0 n H)); last omega.
65 66
    apply Forall2_lookup=> i. rewrite -dist_option_Forall2 list_lookup_fmap.
    destruct (decide (i < length (c n))); last first.
Robbert Krebbers's avatar
Robbert Krebbers committed
67 68 69 70
    { rewrite lookup_seq_ge ?lookup_ge_None_2; auto with omega. }
    rewrite lookup_seq //= (conv_compl n (list_chain c _ _)) /=.
    by destruct (lookup_lt_is_Some_2 (c n) i) as [? ->].
Qed.
71
Canonical Structure listC := CofeT (list A) list_cofe_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
72 73 74 75 76 77 78 79 80 81 82 83
Global Instance list_discrete : Discrete A  Discrete listC.
Proof. induction 2; constructor; try apply (timeless _); auto. Qed.

Global Instance nil_timeless : Timeless (@nil A).
Proof. inversion_clear 1; constructor. Qed.
Global Instance cons_timeless x l : Timeless x  Timeless l  Timeless (x :: l).
Proof. intros ??; inversion_clear 1; constructor; by apply timeless. Qed.
End cofe.

Arguments listC : clear implicits.

(** Functor *)
84 85 86
Lemma list_fmap_ext_ne {A} {B : cofeT} (f g : A  B) (l : list A) n :
  ( x, f x {n} g x)  f <$> l {n} g <$> l.
Proof. intros Hf. by apply Forall2_fmap, Forall_Forall2, Forall_true. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
87 88
Instance list_fmap_ne {A B : cofeT} (f : A  B) n:
  Proper (dist n ==> dist n) f  Proper (dist n ==> dist n) (fmap (M:=list) f).
89
Proof. intros Hf l k ?; by eapply Forall2_fmap, Forall2_impl; eauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
90 91 92
Definition listC_map {A B} (f : A -n> B) : listC A -n> listC B :=
  CofeMor (fmap f : listC A  listC B).
Instance listC_map_ne A B n : Proper (dist n ==> dist n) (@listC_map A B).
93
Proof. intros f g ? l. by apply list_fmap_ext_ne. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

Program Definition listCF (F : cFunctor) : cFunctor := {|
  cFunctor_car A B := listC (cFunctor_car F A B);
  cFunctor_map A1 A2 B1 B2 fg := listC_map (cFunctor_map F fg)
|}.
Next Obligation.
  by intros F A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_ne.
Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(list_fmap_id x).
  apply list_fmap_setoid_ext=>y. apply cFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
  apply list_fmap_setoid_ext=>y; apply cFunctor_compose.
Qed.

Instance listCF_contractive F :
  cFunctorContractive F  cFunctorContractive (listCF F).
Proof.
  by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, cFunctor_contractive.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
116 117 118

(* CMRA *)
Section cmra.
119
  Context {A : ucmraT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
120 121 122 123 124 125 126 127 128 129
  Implicit Types l : list A.
  Local Arguments op _ _ !_ !_ / : simpl nomatch.

  Instance list_op : Op (list A) :=
    fix go l1 l2 := let _ : Op _ := @go in
    match l1, l2 with
    | [], _ => l2
    | _, [] => l1
    | x :: l1, y :: l2 => x  y :: l1  l2
    end.
Robbert Krebbers's avatar
Robbert Krebbers committed
130
  Instance list_pcore : PCore (list A) := λ l, Some (core <$> l).
Robbert Krebbers's avatar
Robbert Krebbers committed
131 132 133 134

  Instance list_valid : Valid (list A) := Forall (λ x,  x).
  Instance list_validN : ValidN (list A) := λ n, Forall (λ x, {n} x).

135 136 137 138 139 140 141 142 143
  Lemma cons_valid l x :  (x :: l)   x   l.
  Proof. apply Forall_cons. Qed.
  Lemma cons_validN n l x : {n} (x :: l)  {n} x  {n} l.
  Proof. apply Forall_cons. Qed.
  Lemma app_valid l1 l2 :  (l1 ++ l2)   l1   l2.
  Proof. apply Forall_app. Qed.
  Lemma app_validN n l1 l2 : {n} (l1 ++ l2)  {n} l1  {n} l2.
  Proof. apply Forall_app. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161
  Lemma list_lookup_valid l :  l   i,  (l !! i).
  Proof.
    rewrite {1}/valid /list_valid Forall_lookup; split.
    - intros Hl i. by destruct (l !! i) as [x|] eqn:?; [apply (Hl i)|].
    - intros Hl i x Hi. move: (Hl i); by rewrite Hi.
  Qed.
  Lemma list_lookup_validN n l : {n} l   i, {n} (l !! i).
  Proof.
    rewrite {1}/validN /list_validN Forall_lookup; split.
    - intros Hl i. by destruct (l !! i) as [x|] eqn:?; [apply (Hl i)|].
    - intros Hl i x Hi. move: (Hl i); by rewrite Hi.
  Qed.
  Lemma list_lookup_op l1 l2 i : (l1  l2) !! i = l1 !! i  l2 !! i.
  Proof.
    revert i l2. induction l1 as [|x l1]; intros [|i] [|y l2];
      by rewrite /= ?left_id_L ?right_id_L.
  Qed.
  Lemma list_lookup_core l i : core l !! i = core (l !! i).
Robbert Krebbers's avatar
Robbert Krebbers committed
162 163 164 165
  Proof.
    rewrite /core /= list_lookup_fmap.
    destruct (l !! i); by rewrite /= ?Some_core.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182

  Lemma list_lookup_included l1 l2 : l1  l2   i, l1 !! i  l2 !! i.
  Proof.
    split.
    { intros [l Hl] i. exists (l !! i). by rewrite Hl list_lookup_op. }
    revert l1. induction l2 as [|y l2 IH]=>-[|x l1] Hl.
    - by exists [].
    - destruct (Hl 0) as [[z|] Hz]; inversion Hz.
    - by exists (y :: l2).
    - destruct (IH l1) as [l3 ?]; first (intros i; apply (Hl (S i))).
      destruct (Hl 0) as [[z|] Hz]; inversion_clear Hz; simplify_eq/=.
      + exists (z :: l3); by constructor.
      + exists (core x :: l3); constructor; by rewrite ?cmra_core_r.
  Qed.

  Definition list_cmra_mixin : CMRAMixin (list A).
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
183 184
    apply cmra_total_mixin.
    - eauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
185 186
    - intros n l l1 l2; rewrite !list_dist_lookup=> Hl i.
      by rewrite !list_lookup_op Hl.
Robbert Krebbers's avatar
Robbert Krebbers committed
187
    - intros n l1 l2 Hl; by rewrite /core /= Hl.
Robbert Krebbers's avatar
Robbert Krebbers committed
188 189 190 191 192 193 194 195 196 197 198 199 200 201
    - intros n l1 l2; rewrite !list_dist_lookup !list_lookup_validN=> Hl ? i.
      by rewrite -Hl.
    - intros l. rewrite list_lookup_valid. setoid_rewrite list_lookup_validN.
      setoid_rewrite cmra_valid_validN. naive_solver.
    - intros n x. rewrite !list_lookup_validN. auto using cmra_validN_S.
    - intros l1 l2 l3; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_op assoc.
    - intros l1 l2; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_op comm.
    - intros l; rewrite list_equiv_lookup=> i.
      by rewrite list_lookup_op list_lookup_core cmra_core_l.
    - intros l; rewrite list_equiv_lookup=> i.
      by rewrite !list_lookup_core cmra_core_idemp.
    - intros l1 l2; rewrite !list_lookup_included=> Hl i.
202
      rewrite !list_lookup_core. by apply cmra_core_mono.
Robbert Krebbers's avatar
Robbert Krebbers committed
203 204
    - intros n l1 l2. rewrite !list_lookup_validN.
      setoid_rewrite list_lookup_op. eauto using cmra_validN_op_l.
205 206 207 208 209 210 211 212
    - intros n l.
      induction l as [|x l IH]=> -[|y1 l1] [|y2 l2] Hl; inversion_clear 1.
      + by exists [], [].
      + exists [], (x :: l); by repeat constructor.
      + exists (x :: l), []; by repeat constructor.
      + inversion_clear Hl. destruct (IH l1 l2) as (l1'&l2'&?&?&?),
          (cmra_extend n x y1 y2) as (y1'&y2'&?&?&?); simplify_eq/=; auto.
        exists (y1' :: l1'), (y2' :: l2'); repeat constructor; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
213
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
214
  Canonical Structure listR := CMRAT (list A) list_cofe_mixin list_cmra_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
215 216

  Global Instance empty_list : Empty (list A) := [].
217
  Definition list_ucmra_mixin : UCMRAMixin (list A).
Robbert Krebbers's avatar
Robbert Krebbers committed
218 219 220 221
  Proof.
    split.
    - constructor.
    - by intros l.
Robbert Krebbers's avatar
Robbert Krebbers committed
222
    - by constructor.
Robbert Krebbers's avatar
Robbert Krebbers committed
223
  Qed.
224 225
  Canonical Structure listUR :=
    UCMRAT (list A) list_cofe_mixin list_cmra_mixin list_ucmra_mixin.
Robbert Krebbers's avatar
Robbert Krebbers committed
226 227 228 229 230 231 232 233 234

  Global Instance list_cmra_discrete : CMRADiscrete A  CMRADiscrete listR.
  Proof.
    split; [apply _|]=> l; rewrite list_lookup_valid list_lookup_validN=> Hl i.
    by apply cmra_discrete_valid.
  Qed.

  Global Instance list_persistent l : ( x : A, Persistent x)  Persistent l.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
235 236
    intros ?; constructor; apply list_equiv_lookup=> i.
    by rewrite list_lookup_core (persistent_core (l !! i)).
Robbert Krebbers's avatar
Robbert Krebbers committed
237 238 239
  Qed.

  (** Internalized properties *)
240
  Lemma list_equivI {M} l1 l2 : l1  l2  ( i, l1 !! i  l2 !! i : uPred M).
Robbert Krebbers's avatar
Robbert Krebbers committed
241
  Proof. uPred.unseal; constructor=> n x ?. apply list_dist_lookup. Qed.
242
  Lemma list_validI {M} l :  l  ( i,  (l !! i) : uPred M).
Robbert Krebbers's avatar
Robbert Krebbers committed
243 244 245 246
  Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed.
End cmra.

Arguments listR : clear implicits.
247
Arguments listUR : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
248

249
Instance list_singletonM {A : ucmraT} : SingletonM nat A (list A) := λ n x,
Robbert Krebbers's avatar
Robbert Krebbers committed
250 251 252
  replicate n  ++ [x].

Section properties.
253
  Context {A : ucmraT}.
254
  Implicit Types l : list A.
255
  Implicit Types x y z : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
256 257
  Local Arguments op _ _ !_ !_ / : simpl nomatch.
  Local Arguments cmra_op _ !_ !_ / : simpl nomatch.
258
  Local Arguments ucmra_op _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
259

260
  Lemma list_lookup_opM l mk i : (l ? mk) !! i = l !! i  (mk = (!! i)).
261 262
  Proof. destruct mk; by rewrite /= ?list_lookup_op ?right_id_L. Qed.

263 264 265 266 267
  Global Instance list_op_nil_l : LeftId (=) (@nil A) op.
  Proof. done. Qed.
  Global Instance list_op_nil_r : RightId (=) (@nil A) op.
  Proof. by intros []. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
268
  Lemma list_op_app l1 l2 l3 :
269
    (l1 ++ l3)  l2 = (l1  take (length l1) l2) ++ (l3  drop (length l1) l2).
Robbert Krebbers's avatar
Robbert Krebbers committed
270 271
  Proof.
    revert l2 l3.
272
    induction l1 as [|x1 l1]=> -[|x2 l2] [|x3 l3]; f_equal/=; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
273
  Qed.
274 275 276
  Lemma list_op_app_le l1 l2 l3 :
    length l2  length l1  (l1 ++ l3)  l2 = (l1  l2) ++ l3.
  Proof. intros ?. by rewrite list_op_app take_ge // drop_ge // right_id_L. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
277 278 279 280 281 282 283 284 285 286 287

  Lemma list_lookup_validN_Some n l i x : {n} l  l !! i {n} Some x  {n} x.
  Proof. move=> /list_lookup_validN /(_ i)=> Hl Hi; move: Hl. by rewrite Hi. Qed.
  Lemma list_lookup_valid_Some l i x :  l  l !! i  Some x   x.
  Proof. move=> /list_lookup_valid /(_ i)=> Hl Hi; move: Hl. by rewrite Hi. Qed.

  Lemma list_op_length l1 l2 : length (l1  l2) = max (length l1) (length l2).
  Proof. revert l2. induction l1; intros [|??]; f_equal/=; auto. Qed.

  Lemma replicate_valid n (x : A) :  x   replicate n x.
  Proof. apply Forall_replicate. Qed.
288 289
  Global Instance list_singletonM_ne n i :
    Proper (dist n ==> dist n) (@list_singletonM A i).
290
  Proof. intros l1 l2 ?. apply Forall2_app; by repeat constructor. Qed.
291 292
  Global Instance list_singletonM_proper i :
    Proper (() ==> ()) (list_singletonM i) := ne_proper _.
Robbert Krebbers's avatar
Robbert Krebbers committed
293

294
  Lemma elem_of_list_singletonM i z x : z  {[i := x]}  z =   z = x.
295 296 297
  Proof.
    rewrite elem_of_app elem_of_list_singleton elem_of_replicate. naive_solver.
  Qed.
298
  Lemma list_lookup_singletonM i x : {[ i := x ]} !! i = Some x.
299
  Proof. induction i; by f_equal/=. Qed.
300 301
  Lemma list_lookup_singletonM_ne i j x :
    i  j  {[ i := x ]} !! j = None  {[ i := x ]} !! j = Some .
302
  Proof. revert j; induction i; intros [|j]; naive_solver auto with omega. Qed.
303
  Lemma list_singletonM_validN n i x : {n} {[ i := x ]}  {n} x.
304 305
  Proof.
    rewrite list_lookup_validN. split.
306
    { move=> /(_ i). by rewrite list_lookup_singletonM. }
307
    intros Hx j; destruct (decide (i = j)); subst.
308 309
    - by rewrite list_lookup_singletonM.
    - destruct (list_lookup_singletonM_ne i j x) as [Hi|Hi]; first done;
310 311
        rewrite Hi; by try apply (ucmra_unit_validN (A:=A)).
  Qed.
312 313 314 315 316
  Lemma list_singleton_valid  i x :  {[ i := x ]}   x.
  Proof.
    rewrite !cmra_valid_validN. by setoid_rewrite list_singletonM_validN.
  Qed.
  Lemma list_singletonM_length i x : length {[ i := x ]} = S i.
317
  Proof.
318
    rewrite /singletonM /list_singletonM app_length replicate_length /=; lia.
319 320
  Qed.

321
  Lemma list_core_singletonM i (x : A) : core {[ i := x ]}  {[ i := core x ]}.
322
  Proof.
323
    rewrite /singletonM /list_singletonM.
Robbert Krebbers's avatar
Robbert Krebbers committed
324
    by rewrite {1}/core /= fmap_app fmap_replicate (persistent_core ).
325
  Qed.
326 327 328 329 330 331 332 333 334 335 336 337 338
  Lemma list_op_singletonM i (x y : A) :
    {[ i := x ]}  {[ i := y ]}  {[ i := x  y ]}.
  Proof.
    rewrite /singletonM /list_singletonM /=.
    induction i; constructor; rewrite ?left_id; auto.
  Qed.
  Lemma list_alter_singletonM f i x : alter f i {[i := x]} = {[i := f x]}.
  Proof.
    rewrite /singletonM /list_singletonM /=. induction i; f_equal/=; auto.
  Qed.
  Global Instance list_singleton_persistent i (x : A) :
    Persistent x  Persistent {[ i := x ]}.
  Proof. by rewrite !persistent_total list_core_singletonM=> ->. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
339 340

  (* Update *)
341 342
  Lemma list_singleton_updateP (P : A  Prop) (Q : list A  Prop) x :
    x ~~>: P  ( y, P y  Q [y])  [x] ~~>: Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
343
  Proof.
344 345 346 347
    rewrite !cmra_total_updateP=> Hup HQ n lf /list_lookup_validN Hv.
    destruct (Hup n (from_option id  (lf !! 0))) as (y&?&Hv').
    { move: (Hv 0). by destruct lf; rewrite /= ?right_id. }
    exists [y]; split; first by auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
348
    apply list_lookup_validN=> i.
349 350 351 352 353 354 355 356
    move: (Hv i) Hv'. by destruct i, lf; rewrite /= ?right_id.
  Qed.
  Lemma list_singleton_updateP' (P : A  Prop) x :
    x ~~>: P  [x] ~~>: λ k,  y, k = [y]  P y.
  Proof. eauto using list_singleton_updateP. Qed.
  Lemma list_singleton_update x y : x ~~> y  [x] ~~> [y].
  Proof.
    rewrite !cmra_update_updateP; eauto using list_singleton_updateP with subst.
Robbert Krebbers's avatar
Robbert Krebbers committed
357 358
  Qed.

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
  Lemma app_updateP (P1 P2 Q : list A  Prop) l1 l2 :
    l1 ~~>: P1  l2 ~~>: P2 
    ( k1 k2, P1 k1  P2 k2  length l1 = length k1  Q (k1 ++ k2)) 
    l1 ++ l2 ~~>: Q.
  Proof.
    rewrite !cmra_total_updateP=> Hup1 Hup2 HQ n lf.
    rewrite list_op_app app_validN=> -[??].
    destruct (Hup1 n (take (length l1) lf)) as (k1&?&?); auto.
    destruct (Hup2 n (drop (length l1) lf)) as (k2&?&?); auto.
    exists (k1 ++ k2). rewrite list_op_app app_validN.
    by destruct (HQ k1 k2) as [<- ?].
  Qed.
  Lemma app_update l1 l2 k1 k2 :
    length l1 = length k1 
    l1 ~~> k1  l2 ~~> k2  l1 ++ l2 ~~> k1 ++ k2.
  Proof. rewrite !cmra_update_updateP; eauto using app_updateP with subst. Qed.

  Lemma cons_updateP (P1 : A  Prop) (P2 Q : list A  Prop) x l :
    x ~~>: P1  l ~~>: P2  ( y k, P1 y  P2 k  Q (y :: k))  x :: l ~~>: Q.
  Proof.
    intros. eapply (app_updateP _ _ _ [x]);
      naive_solver eauto using list_singleton_updateP'.
  Qed.
  Lemma cons_updateP' (P1 : A  Prop) (P2 : list A  Prop) x l :
    x ~~>: P1  l ~~>: P2  x :: l ~~>: λ k,  y k', k = y :: k'  P1 y  P2 k'.
  Proof. eauto 10 using cons_updateP. Qed.
  Lemma cons_update x y l k : x ~~> y  l ~~> k  x :: l ~~> y :: k.
  Proof. rewrite !cmra_update_updateP; eauto using cons_updateP with subst. Qed.

  Lemma list_middle_updateP (P : A  Prop) (Q : list A  Prop) l1 x l2 :
    x ~~>: P  ( y, P y  Q (l1 ++ y :: l2))  l1 ++ x :: l2 ~~>: Q.
  Proof.
    intros. eapply app_updateP.
    - by apply cmra_update_updateP.
    - by eapply cons_updateP', cmra_update_updateP.
    - naive_solver.
  Qed.
396
  Lemma list_middle_update l1 l2 x y : x ~~> y  l1 ++ x :: l2 ~~> l1 ++ y :: l2.
Robbert Krebbers's avatar
Robbert Krebbers committed
397
  Proof.
398
    rewrite !cmra_update_updateP=> ?; eauto using list_middle_updateP with subst.
Robbert Krebbers's avatar
Robbert Krebbers committed
399 400
  Qed.

401 402 403
  Lemma list_middle_local_update l1 l2 x y ml :
    x ~l~> y @ ml = (!! length l1) 
    l1 ++ x :: l2 ~l~> l1 ++ y :: l2 @ ml.
Robbert Krebbers's avatar
Robbert Krebbers committed
404
  Proof.
405 406 407 408 409 410 411 412 413 414 415 416 417 418 419
    intros [Hxy Hxy']; split.
    - intros n; rewrite !list_lookup_validN=> Hl i; move: (Hl i).
      destruct (lt_eq_lt_dec i (length l1)) as [[?|?]|?]; subst.
      + by rewrite !list_lookup_opM !lookup_app_l.
      + rewrite !list_lookup_opM !list_lookup_middle // !Some_op_opM; apply (Hxy n).
      + rewrite !(cons_middle _ l1 l2) !assoc.
        rewrite !list_lookup_opM !lookup_app_r !app_length //=; lia.
    - intros n mk; rewrite !list_lookup_validN !list_dist_lookup => Hl Hl' i.
      move: (Hl i) (Hl' i).
      destruct (lt_eq_lt_dec i (length l1)) as [[?|?]|?]; subst.
      + by rewrite !list_lookup_opM !lookup_app_l.
      + rewrite !list_lookup_opM !list_lookup_middle // !Some_op_opM !inj_iff.
        apply (Hxy' n).
      + rewrite !(cons_middle _ l1 l2) !assoc.
        rewrite !list_lookup_opM !lookup_app_r !app_length //=; lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
420
  Qed.
421
  Lemma list_singleton_local_update i x y ml :
422
    x ~l~> y @ ml = (!! i)  {[ i := x ]} ~l~> {[ i := y ]} @ ml.
423
  Proof. intros; apply list_middle_local_update. by rewrite replicate_length. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
424 425 426
End properties.

(** Functor *)
427
Instance list_fmap_cmra_monotone {A B : ucmraT} (f : A  B)
Robbert Krebbers's avatar
Robbert Krebbers committed
428 429 430 431 432 433
  `{!CMRAMonotone f} : CMRAMonotone (fmap f : list A  list B).
Proof.
  split; try apply _.
  - intros n l. rewrite !list_lookup_validN=> Hl i. rewrite list_lookup_fmap.
    by apply (validN_preserving (fmap f : option A  option B)).
  - intros l1 l2. rewrite !list_lookup_included=> Hl i. rewrite !list_lookup_fmap.
434
    by apply (cmra_monotone (fmap f : option A  option B)).
Robbert Krebbers's avatar
Robbert Krebbers committed
435 436
Qed.

437 438 439
Program Definition listURF (F : urFunctor) : urFunctor := {|
  urFunctor_car A B := listUR (urFunctor_car F A B);
  urFunctor_map A1 A2 B1 B2 fg := listC_map (urFunctor_map F fg)
Robbert Krebbers's avatar
Robbert Krebbers committed
440 441
|}.
Next Obligation.
442
  by intros F ???? n f g Hfg; apply listC_map_ne, urFunctor_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
443 444 445
Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(list_fmap_id x).
446
  apply list_fmap_setoid_ext=>y. apply urFunctor_id.
Robbert Krebbers's avatar
Robbert Krebbers committed
447 448 449
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose.
450
  apply list_fmap_setoid_ext=>y; apply urFunctor_compose.
Robbert Krebbers's avatar
Robbert Krebbers committed
451 452
Qed.

453 454
Instance listURF_contractive F :
  urFunctorContractive F  urFunctorContractive (listURF F).
Robbert Krebbers's avatar
Robbert Krebbers committed
455
Proof.
456
  by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, urFunctor_contractive.
Robbert Krebbers's avatar
Robbert Krebbers committed
457
Qed.