list.v 15.6 KB
 Robbert Krebbers committed May 25, 2016 1 2 ``````From iris.algebra Require Export cmra. From iris.prelude Require Export list. `````` Robbert Krebbers committed May 23, 2016 3 ``````From iris.algebra Require Import upred. `````` Robbert Krebbers committed Mar 21, 2016 4 5 6 7 8 9 `````` Section cofe. Context {A : cofeT}. Instance list_dist : Dist (list A) := λ n, Forall2 (dist n). `````` Robbert Krebbers committed May 23, 2016 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 committed Mar 21, 2016 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ``````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 : Proper (dist n ==> dist n) (lookup (M:=list A) i). Proof. intros ???. by apply dist_option_Forall2, Forall2_lookup. Qed. Global Instance list_alter_ne n f i : Proper (dist n ==> dist n) f → Proper (dist n ==> dist n) (alter (M:=list A) f i) := _. Global Instance list_insert_ne n i : Proper (dist n ==> dist n ==> dist n) (insert (M:=list A) i) := _. 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 : Proper (dist n ==> dist n) (delete (M:=list A) i) := _. 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 := `````` Robbert Krebbers committed May 27, 2016 46 `````` {| chain_car n := from_option id x (c n !! k) |}. `````` Robbert Krebbers committed Mar 21, 2016 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. `````` Robbert Krebbers committed May 27, 2016 65 66 `````` apply Forall2_lookup=> i. rewrite -dist_option_Forall2 list_lookup_fmap. destruct (decide (i < length (c n))); last first. `````` Robbert Krebbers committed Mar 21, 2016 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. `````` Robbert Krebbers committed May 25, 2016 71 ``````Canonical Structure listC := CofeT (list A) list_cofe_mixin. `````` Robbert Krebbers committed Mar 21, 2016 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 ``````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 *) 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). Proof. intros Hf l k ?; by eapply Forall2_fmap, Forall2_impl; eauto. Qed. 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). Proof. intros f f' ? l; by apply Forall2_fmap, Forall_Forall2, Forall_true. Qed. 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 committed May 23, 2016 113 114 115 `````` (* CMRA *) Section cmra. `````` Robbert Krebbers committed May 27, 2016 116 `````` Context {A : ucmraT}. `````` Robbert Krebbers committed May 23, 2016 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 `````` 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. Instance list_core : Core (list A) := fmap core. Instance list_valid : Valid (list A) := Forall (λ x, ✓ x). Instance list_validN : ValidN (list A) := λ n, Forall (λ x, ✓{n} x). 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). Proof. revert i; induction l; intros [|i]; simpl; auto. Qed. 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. split. - intros n l l1 l2; rewrite !list_dist_lookup=> Hl i. by rewrite !list_lookup_op Hl. - apply _. - 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. rewrite !list_lookup_core. by apply cmra_core_preserving. - intros n l1 l2. rewrite !list_lookup_validN. setoid_rewrite list_lookup_op. eauto using cmra_validN_op_l. - intros n l. induction l as [|x l IH]=> -[|y1 l1] [|y2 l2] Hl Hl'; try (by exfalso; inversion_clear Hl'). + by exists ([], []). + by exists ([], x :: l). + by exists (x :: l, []). + destruct (IH l1 l2) as ([l1' l2']&?&?&?), (cmra_extend n x y1 y2) as ([y1' y2']&?&?&?); [inversion_clear Hl; inversion_clear Hl'; auto ..|]; simplify_eq/=. exists (y1' :: l1', y2' :: l2'); repeat constructor; auto. Qed. `````` Robbert Krebbers committed May 27, 2016 199 `````` Canonical Structure listR := `````` Robbert Krebbers committed May 25, 2016 200 `````` CMRAT (list A) list_cofe_mixin list_cmra_mixin. `````` Robbert Krebbers committed May 23, 2016 201 202 `````` Global Instance empty_list : Empty (list A) := []. `````` Robbert Krebbers committed May 27, 2016 203 `````` Definition list_ucmra_mixin : UCMRAMixin (list A). `````` Robbert Krebbers committed May 23, 2016 204 205 206 207 208 209 `````` Proof. split. - constructor. - by intros l. - by inversion_clear 1. Qed. `````` Robbert Krebbers committed May 27, 2016 210 211 `````` Canonical Structure listUR := UCMRAT (list A) list_cofe_mixin list_cmra_mixin list_ucmra_mixin. `````` Robbert Krebbers committed May 23, 2016 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 `````` 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. intros ?; apply list_equiv_lookup=> i. by rewrite list_lookup_core (persistent (l !! i)). Qed. (** Internalized properties *) Lemma list_equivI {M} l1 l2 : (l1 ≡ l2) ⊣⊢ (∀ i, l1 !! i ≡ l2 !! i : uPred M). Proof. uPred.unseal; constructor=> n x ?. apply list_dist_lookup. Qed. Lemma list_validI {M} l : (✓ l) ⊣⊢ (∀ i, ✓ (l !! i) : uPred M). Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed. End cmra. Arguments listR : clear implicits. `````` Robbert Krebbers committed May 27, 2016 233 ``````Arguments listUR : clear implicits. `````` Robbert Krebbers committed May 23, 2016 234 `````` `````` Robbert Krebbers committed May 27, 2016 235 ``````Instance list_singletonM {A : ucmraT} : SingletonM nat A (list A) := λ n x, `````` Robbert Krebbers committed May 23, 2016 236 237 238 `````` replicate n ∅ ++ [x]. Section properties. `````` Robbert Krebbers committed May 27, 2016 239 `````` Context {A : ucmraT}. `````` Robbert Krebbers committed May 23, 2016 240 `````` Implicit Types l : list A. `````` Robbert Krebbers committed May 27, 2016 241 `````` Implicit Types x y z : A. `````` Robbert Krebbers committed May 23, 2016 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 `````` Local Arguments op _ _ !_ !_ / : simpl nomatch. Local Arguments cmra_op _ !_ !_ / : simpl nomatch. Lemma list_op_app l1 l2 l3 : length l2 ≤ length l1 → (l1 ++ l3) ⋅ l2 = (l1 ⋅ l2) ++ l3. Proof. revert l2 l3. induction l1 as [|x1 l1]=> -[|x2 l2] [|x3 l3] ?; f_equal/=; auto with lia. Qed. 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. `````` Robbert Krebbers committed May 27, 2016 262 263 264 265 266 `````` Global Instance list_singletonM_ne n i : Proper (dist n ==> dist n) (@list_singletonM A i). Proof. intros l1 l2 ?. apply Forall2_app; by repeat constructor. Qed. Global Instance list_singletonM_proper i : Proper ((≡) ==> (≡)) (list_singletonM i) := ne_proper _. `````` Robbert Krebbers committed May 23, 2016 267 `````` `````` Robbert Krebbers committed May 27, 2016 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 `````` Lemma elem_of_list_singletonM i z x : z ∈ {[i := x]} → z = ∅ ∨ z = x. Proof. rewrite elem_of_app elem_of_list_singleton elem_of_replicate. naive_solver. Qed. Lemma list_lookup_singletonM i x : {[ i := x ]} !! i = Some x. Proof. induction i; by f_equal/=. Qed. Lemma list_lookup_singletonM_ne i j x : i ≠ j → {[ i := x ]} !! j = None ∨ {[ i := x ]} !! j = Some ∅. Proof. revert j; induction i; intros [|j]; naive_solver auto with omega. Qed. Lemma list_singletonM_validN n i x : ✓{n} {[ i := x ]} ↔ ✓{n} x. Proof. rewrite list_lookup_validN. split. { move=> /(_ i). by rewrite list_lookup_singletonM. } intros Hx j; destruct (decide (i = j)); subst. - by rewrite list_lookup_singletonM. - destruct (list_lookup_singletonM_ne i j x) as [Hi|Hi]; first done; rewrite Hi; by try apply (ucmra_unit_validN (A:=A)). Qed. 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. Proof. rewrite /singletonM /list_singletonM app_length replicate_length /=; lia. Qed. Lemma list_core_singletonM i (x : A) : core {[ i := x ]} ≡ {[ i := core x ]}. Proof. rewrite /singletonM /list_singletonM /=. induction i; constructor; auto using ucmra_core_unit. Qed. 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. intros. by rewrite /Persistent list_core_singletonM persistent. Qed. `````` Robbert Krebbers committed May 23, 2016 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 `````` (* Update *) Lemma list_update_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 Hx%option_updateP' HP n mf; rewrite list_lookup_validN=> Hm. destruct (Hx n (mf !! length l1)) as ([y|]&H1&H2); simpl in *; try done. { move: (Hm (length l1)). by rewrite list_lookup_op list_lookup_middle. } exists (l1 ++ y :: l2); split; auto. apply list_lookup_validN=> i. destruct (lt_eq_lt_dec i (length l1)) as [[?|?]|?]; subst. - move: (Hm i); by rewrite !list_lookup_op !lookup_app_l. - by rewrite list_lookup_op list_lookup_middle. - move: (Hm i). rewrite !(cons_middle _ l1 l2) !assoc. rewrite !list_lookup_op !lookup_app_r !app_length //=; lia. Qed. Lemma list_update_update l1 l2 x y : x ~~> y → l1 ++ x :: l2 ~~> l1 ++ y :: l2. Proof. rewrite !cmra_update_updateP => H; eauto using list_update_updateP with subst. Qed. (* Applying a local update at a position we own is a local update. *) Global Instance list_alter_update `{LocalUpdate A Lv L} i : LocalUpdate (λ L, ∃ x, L !! i = Some x ∧ Lv x) (alter L i). Proof. split; [apply _|]; intros n l1 l2 (x&Hi1&?) Hm; apply list_dist_lookup=> j. destruct (decide (j = i)); subst; last first. { by rewrite list_lookup_op !list_lookup_alter_ne // list_lookup_op. } rewrite list_lookup_op !list_lookup_alter list_lookup_op Hi1. destruct (l2 !! i) as [y|] eqn:Hi2; rewrite Hi2; constructor; auto. eapply (local_updateN L), (list_lookup_validN_Some _ _ i); eauto. by rewrite list_lookup_op Hi1 Hi2. Qed. End properties. (** Functor *) `````` Robbert Krebbers committed May 27, 2016 351 ``````Instance list_fmap_cmra_monotone {A B : ucmraT} (f : A → B) `````` Robbert Krebbers committed May 23, 2016 352 353 354 355 356 357 358 359 360 `````` `{!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. by apply (included_preserving (fmap f : option A → option B)). Qed. `````` Robbert Krebbers committed May 27, 2016 361 362 363 ``````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 committed May 23, 2016 364 365 ``````|}. Next Obligation. `````` Robbert Krebbers committed May 27, 2016 366 `````` by intros F ???? n f g Hfg; apply listC_map_ne, urFunctor_ne. `````` Robbert Krebbers committed May 23, 2016 367 368 369 ``````Qed. Next Obligation. intros F A B x. rewrite /= -{2}(list_fmap_id x). `````` Robbert Krebbers committed May 27, 2016 370 `````` apply list_fmap_setoid_ext=>y. apply urFunctor_id. `````` Robbert Krebbers committed May 23, 2016 371 372 373 ``````Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -list_fmap_compose. `````` Robbert Krebbers committed May 27, 2016 374 `````` apply list_fmap_setoid_ext=>y; apply urFunctor_compose. `````` Robbert Krebbers committed May 23, 2016 375 376 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 377 378 ``````Instance listURF_contractive F : urFunctorContractive F → urFunctorContractive (listURF F). `````` Robbert Krebbers committed May 23, 2016 379 ``````Proof. `````` Robbert Krebbers committed May 27, 2016 380 `````` by intros ? A1 A2 B1 B2 n f g Hfg; apply listC_map_ne, urFunctor_contractive. `````` Robbert Krebbers committed May 23, 2016 381 ``Qed.``