Fmap.v 23.4 KB
 charguer committed Mar 22, 2017 1 2 3 4 5 6 7 8 9 10 11 12 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 ``````(** This file contains a representation of finite maps that may be used for representing a store. It also provides lemmas and tactics for reasoning about operations on the store (read, write, union). Author: Arthur Charguéraud. License: MIT. *) Set Implicit Arguments. Require Import LibCore. (* ********************************************************************** *) Tactic Notation "cases" constr(E) := (* TODO For TLC *) let H := fresh "Eq" in cases E as H. (* ********************************************************************** *) (** * Maps (partial functions) *) (* ---------------------------------------------------------------------- *) (* ** Representation *) (** Type of partial functions from A to B *) Definition map (A B : Type) : Type := A -> option B. (* ---------------------------------------------------------------------- *) (* ** Operations *) (** Disjoint union of two partial functions *) Definition map_union (A B : Type) (f1 f2 : map A B) : map A B := fun (x:A) => match f1 x with | Some y => Some y | None => f2 x end. (** Finite domain of a partial function *) Definition map_finite (A B : Type) (f : map A B) := exists (L : list A), forall (x:A), f x <> None -> Mem x L. (** Disjointness of domain of two partial functions *) Definition map_disjoint (A B : Type) (f1 f2 : map A B) := forall (x:A), f1 x = None \/ f2 x = None. (** Compatibility of two partial functions on the intersection of their domains *) Definition map_agree (A B : Type) (f1 f2 : map A B) := forall x v1 v2, f1 x = Some v1 -> f2 x = Some v2 -> v1 = v2. (* ---------------------------------------------------------------------- *) (** Properties *) Section MapOps. Variables (A B : Type). Implicit Types f : map A B. (** Symmetry of disjointness *) Lemma map_disjoint_sym : sym (@map_disjoint A B). Proof using. introv H. unfolds map_disjoint. intros z. specializes H z. intuition. Qed. (** Commutativity of disjoint union *) Lemma map_union_comm : forall f1 f2, map_disjoint f1 f2 -> map_union f1 f2 = map_union f2 f1. Proof using. introv H. unfold map. extens. intros x. unfolds map_disjoint, map_union. specializes H x. cases (f1 x); cases (f2 x); auto. destruct H; false. Qed. (** Finiteness of union *) Lemma map_union_finite : forall f1 f2, map_finite f1 -> map_finite f2 -> map_finite (map_union f1 f2). Proof using. introv [L1 F1] [L2 F2]. exists (L1 ++ L2). introv M. specializes F1 x. specializes F2 x. unfold map_union in M. apply Mem_app_or. destruct~ (f1 x). Qed. End MapOps. (* ********************************************************************** *) (** * Finite maps *) (* ---------------------------------------------------------------------- *) `````` charguer committed May 16, 2017 111 ``````(** Definition of the type of finite maps *) `````` charguer committed Mar 22, 2017 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 `````` Inductive fmap (A B : Type) : Type := fmap_make { fmap_data :> map A B; fmap_finite : map_finite fmap_data }. Implicit Arguments fmap_make [A B]. (* ---------------------------------------------------------------------- *) (** Operations *) (** Empty fmap *) Program Definition fmap_empty (A B : Type) : fmap A B := fmap_make (fun l => None) _. Next Obligation. exists (@nil A). auto_false. Qed. Implicit Arguments fmap_empty [[A] [B]]. (** Singleton fmap *) Program Definition fmap_single A B (x:A) (v:B) : fmap A B := fmap_make (fun x' => If x = x' then Some v else None) _. Next Obligation. exists (x::nil). intros. case_if. subst~. Qed. (** Union of fmaps *) Program Definition fmap_union A B (h1 h2:fmap A B) : fmap A B := fmap_make (map_union h1 h2) _. Next Obligation. destruct h1. destruct h2. apply~ map_union_finite. Qed. Notation "h1 \+ h2" := (fmap_union h1 h2) (at level 51, right associativity) : fmap_scope. Open Scope fmap_scope. (** Update of a fmap *) Definition fmap_update A B (h:fmap A B) (x:A) (v:B) := fmap_union (fmap_single x v) h. (* Note: the union operation first reads in the first argument. *) (* ---------------------------------------------------------------------- *) (** Properties *) (** Inhabited type [fmap] *) Global Instance Inhab_fmap A B : Inhab (fmap A B). Proof using. intros. applys prove_Inhab (@fmap_empty A B). Qed. (** Compatible fmaps *) Definition fmap_agree A B (h1 h2:fmap A B) := map_agree h1 h2. (** Disjoint fmaps *) Definition fmap_disjoint A B (h1 h2 : fmap A B) : Prop := map_disjoint h1 h2. Notation "\# h1 h2" := (fmap_disjoint h1 h2) (at level 40, h1 at level 0, h2 at level 0, no associativity) : fmap_scope. (** Three disjoint fmaps *) Definition fmap_disjoint_3 A B (h1 h2 h3 : fmap A B) := fmap_disjoint h1 h2 /\ fmap_disjoint h2 h3 /\ fmap_disjoint h1 h3. Notation "\# h1 h2 h3" := (fmap_disjoint_3 h1 h2 h3) (at level 40, h1 at level 0, h2 at level 0, h3 at level 0, no associativity) : fmap_scope. (* ********************************************************************** *) (* * Lemmas about Fmap *) Section FmapProp. Variables (A B : Type). Implicit Types f g h : fmap A B. (* ---------------------------------------------------------------------- *) (* ** Equality *) `````` charguer committed Mar 22, 2017 201 ``````Lemma fmap_make_eq : forall (f1 f2:map A B) F1 F2, `````` charguer committed Mar 22, 2017 202 203 204 205 206 207 208 `````` (forall x, f1 x = f2 x) -> fmap_make f1 F1 = fmap_make f2 F2. Proof using. introv H. asserts: (f1 = f2). { unfold map. extens~. } subst. fequals. (* note: involves proof irrelevance *) Qed. `````` charguer committed Mar 22, 2017 209 210 211 212 213 ``````Lemma fmap_eq_inv_fmap_data_eq : forall h1 h2, h1 = h2 -> forall x, fmap_data h1 x = fmap_data h2 x. Proof using. intros. fequals. Qed. `````` charguer committed Mar 22, 2017 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 `````` (* ---------------------------------------------------------------------- *) (* ** Disjointness *) Lemma fmap_disjoint_sym : forall h1 h2, \# h1 h2 -> \# h2 h1. Proof using. intros [f1 F1] [f2 F2]. apply map_disjoint_sym. Qed. Lemma fmap_disjoint_comm : forall h1 h2, \# h1 h2 = \# h2 h1. Proof using. lets: fmap_disjoint_sym. extens*. Qed. Lemma fmap_disjoint_empty_l : forall h, \# fmap_empty h. Proof using. intros [f F] x. simple~. Qed. Lemma fmap_disjoint_empty_r : forall h, \# h fmap_empty. Proof using. intros [f F] x. simple~. Qed. Hint Resolve fmap_disjoint_sym fmap_disjoint_empty_l fmap_disjoint_empty_r. Lemma fmap_disjoint_union_eq_r : forall h1 h2 h3, \# h1 (h2 \+ h3) = (\# h1 h2 /\ \# h1 h3). Proof using. intros [f1 F1] [f2 F2] [f3 F3]. unfolds fmap_disjoint, fmap_union. simpls. unfolds map_disjoint, map_union. extens. iff M [M1 M2]. split; intros x; specializes M x; destruct (f2 x); intuition; tryfalse. intros x. specializes M1 x. specializes M2 x. destruct (f2 x); intuition. Qed. Lemma fmap_disjoint_union_eq_l : forall h1 h2 h3, \# (h2 \+ h3) h1 = (\# h1 h2 /\ \# h1 h3). Proof using. intros. rewrite fmap_disjoint_comm. apply fmap_disjoint_union_eq_r. Qed. `````` charguer committed Mar 28, 2017 257 258 259 260 261 262 263 ``````Lemma fmap_disjoint_single_single : forall (x1 x2:A) (v1 v2:B), x1 <> x2 -> \# (fmap_single x1 v1) (fmap_single x2 v2). Proof using. introv N. intros x. simpls. do 2 case_if; auto. Qed. `````` charguer committed Mar 28, 2017 264 ``````Lemma fmap_disjoint_single_single_same_inv : forall (x:A) (v1 v2:B), `````` charguer committed Mar 22, 2017 265 266 `````` \# (fmap_single x v1) (fmap_single x v2) -> False. `````` charguer committed Mar 22, 2017 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 ``````Proof using. introv D. specializes D x. simpls. case_if. destruct D; tryfalse. Qed. Lemma fmap_disjoint_3_unfold : forall h1 h2 h3, \# h1 h2 h3 = (\# h1 h2 /\ \# h2 h3 /\ \# h1 h3). Proof using. auto. Qed. Lemma fmap_disjoint_single_set : forall h l v1 v2, \# (fmap_single l v1) h -> \# (fmap_single l v2) h. Proof using. introv M. unfolds fmap_disjoint, fmap_single, map_disjoint; simpls. intros l'. specializes M l'. case_if~. destruct M; auto_false. Qed. (* ---------------------------------------------------------------------- *) (* ** Union *) `````` charguer committed Mar 22, 2017 287 ``````Lemma fmap_union_self : forall h, `````` charguer committed Mar 22, 2017 288 289 `````` h \+ h = h. Proof using. `````` charguer committed Mar 22, 2017 290 `````` intros [f F]. apply~ fmap_make_eq. simpl. intros x. `````` charguer committed Mar 22, 2017 291 292 293 294 295 296 297 `````` unfold map_union. cases~ (f x). Qed. Lemma fmap_union_empty_l : forall h, fmap_empty \+ h = h. Proof using. intros [f F]. unfold fmap_union, map_union, fmap_empty. simpl. `````` charguer committed Mar 22, 2017 298 `````` apply~ fmap_make_eq. `````` charguer committed Mar 22, 2017 299 300 301 302 303 304 ``````Qed. Lemma fmap_union_empty_r : forall h, h \+ fmap_empty = h. Proof using. intros [f F]. unfold fmap_union, map_union, fmap_empty. simpl. `````` charguer committed Mar 22, 2017 305 `````` apply fmap_make_eq. intros x. destruct~ (f x). `````` charguer committed Mar 22, 2017 306 307 308 309 310 311 312 ``````Qed. Lemma fmap_union_eq_empty_inv_l : forall h1 h2, h1 \+ h2 = fmap_empty -> h1 = fmap_empty. Proof using. intros (f1&F1) (f2&F2) M. inverts M as M. `````` charguer committed Mar 22, 2017 313 `````` applys fmap_make_eq. intros l. `````` charguer committed Mar 22, 2017 314 315 316 317 318 319 320 321 322 323 `````` unfolds map_union. lets: func_same_1 l M. cases (f1 l); auto_false. Qed. Lemma fmap_union_eq_empty_inv_r : forall h1 h2, h1 \+ h2 = fmap_empty -> h2 = fmap_empty. Proof using. intros (f1&F1) (f2&F2) M. inverts M as M. `````` charguer committed Mar 22, 2017 324 `````` applys fmap_make_eq. intros l. `````` charguer committed Mar 22, 2017 325 326 327 328 329 `````` unfolds map_union. lets: func_same_1 l M. cases (f1 l); auto_false. Qed. `````` charguer committed Mar 22, 2017 330 ``````Lemma fmap_union_comm_of_disjoint : forall h1 h2, `````` charguer committed Mar 22, 2017 331 332 333 `````` \# h1 h2 -> h1 \+ h2 = h2 \+ h1. Proof using. `````` charguer committed Mar 22, 2017 334 `````` intros [f1 F1] [f2 F2] H. simpls. apply fmap_make_eq. simpl. `````` charguer committed Mar 22, 2017 335 336 337 `````` intros. rewrite~ map_union_comm. Qed. `````` charguer committed Mar 22, 2017 338 ``````Lemma fmap_union_comm_of_agree : forall h1 h2, `````` charguer committed Mar 22, 2017 339 340 341 `````` fmap_agree h1 h2 -> h1 \+ h2 = h2 \+ h1. Proof using. `````` charguer committed Mar 22, 2017 342 `````` intros [f1 F1] [f2 F2] H. simpls. apply fmap_make_eq. simpl. `````` charguer committed Mar 22, 2017 343 344 345 346 347 348 349 350 `````` intros l. specializes H l. unfolds map_union. simpls. cases (f1 l); cases (f2 l); auto. fequals. applys* H. Qed. Lemma fmap_union_assoc : forall h1 h2 h3, (h1 \+ h2) \+ h3 = h1 \+ (h2 \+ h3). Proof using. intros [f1 F1] [f2 F2] [f3 F3]. unfolds fmap_union. simpls. `````` charguer committed Mar 22, 2017 351 `````` apply fmap_make_eq. intros x. unfold map_union. destruct~ (f1 x). `````` charguer committed Mar 22, 2017 352 353 354 ``````Qed. (* `````` charguer committed Mar 22, 2017 355 ``````Lemma fmap_union_eq_inv_of_disjoint : forall h2 h1 h1' : fmap, `````` charguer committed Mar 22, 2017 356 357 358 359 360 361 `````` \# h1 h2 -> fmap_agree h1' h2 -> h1 \+ h2 = h1' \+ h2 -> h1 = h1'. Proof using. intros [f2 F2] [f1 F1] [f1' F1'] D D' E. `````` charguer committed Mar 22, 2017 362 363 `````` applys fmap_make_eq. intros x. specializes D x. specializes D' x. lets E': fmap_eq_inv_fmap_data_eq (rm E) x. simpls. `````` charguer committed Mar 22, 2017 364 365 366 367 368 369 370 371 372 373 374 `````` unfolds map_union. cases (f1 x); cases (f2 x); try solve [cases (f1' x); destruct D; congruence ]. destruct D; try false. rewrite H in E'. inverts E'. cases (f1' x); cases (f1 x); destruct D; try congruence. false. destruct D'; try congruence. Qed. *) `````` charguer committed Mar 22, 2017 375 ``````Lemma fmap_union_eq_inv_of_disjoint : forall h2 h1 h1', `````` charguer committed Mar 22, 2017 376 377 378 379 380 381 `````` \# h1 h2 -> \# h1' h2 -> h1 \+ h2 = h1' \+ h2 -> h1 = h1'. Proof using. intros [f2 F2] [f1' F1'] [f1 F1] D D' E. `````` charguer committed Mar 22, 2017 382 383 `````` applys fmap_make_eq. intros x. specializes D x. specializes D' x. lets E': fmap_eq_inv_fmap_data_eq (rm E) x. simpls. `````` charguer committed Mar 22, 2017 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 `````` unfolds map_union. cases (f1' x); cases (f1 x); destruct D; try congruence; destruct D'; try congruence. Qed. (* ---------------------------------------------------------------------- *) (* ** Compatibility *) Lemma fmap_agree_refl : forall h, fmap_agree h h. Proof using. intros h. introv M1 M2. congruence. Qed. Lemma fmap_agree_sym : forall f1 f2, fmap_agree f1 f2 -> fmap_agree f2 f1. Proof using. introv M. intros l v1 v2 E1 E2. specializes M l E1. Qed. `````` charguer committed Mar 22, 2017 408 ``````Lemma fmap_agree_of_disjoint : forall h1 h2, `````` charguer committed Mar 22, 2017 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 `````` fmap_disjoint h1 h2 -> fmap_agree h1 h2. Proof using. introv HD. intros l v1 v2 M1 M2. destruct (HD l); false. Qed. Lemma fmap_agree_empty_l : forall h, fmap_agree fmap_empty h. Proof using. intros h l v1 v2 E1 E2. simpls. false. Qed. Lemma fmap_agree_empty_r : forall h, fmap_agree h fmap_empty. Proof using. hint fmap_agree_sym, fmap_agree_empty_l. eauto. Qed. Lemma fmap_agree_union_l : forall f1 f2 f3, fmap_agree f1 f3 -> fmap_agree f2 f3 -> fmap_agree (f1 \+ f2) f3. Proof using. introv M1 M2. intros l v1 v2 E1 E2. specializes M1 l. specializes M2 l. simpls. unfolds map_union. cases (fmap_data f1 l). { inverts E1. applys* M1. } { applys* M2. } Qed. Lemma fmap_agree_union_r : forall f1 f2 f3, fmap_agree f1 f2 -> fmap_agree f1 f3 -> fmap_agree f1 (f2 \+ f3). Proof using. hint fmap_agree_sym, fmap_agree_union_l. eauto. Qed. Lemma fmap_agree_union_lr : forall f1 g1 f2 g2, fmap_agree g1 g2 -> \# f1 f2 (g1 \+ g2) -> fmap_agree (f1 \+ g1) (f2 \+ g2). Proof using. introv M1 (M2a&M2b&M2c). rewrite fmap_disjoint_union_eq_r in *. applys fmap_agree_union_l; applys fmap_agree_union_r; `````` charguer committed Mar 22, 2017 453 `````` try solve [ applys* fmap_agree_of_disjoint ]. `````` charguer committed Mar 22, 2017 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 `````` auto. Qed. Lemma fmap_agree_union_ll_inv : forall f1 f2 f3, fmap_agree (f1 \+ f2) f3 -> fmap_agree f1 f3. Proof using. introv M. intros l v1 v2 E1 E2. specializes M l. simpls. unfolds map_union. rewrite E1 in M. applys* M. Qed. Lemma fmap_agree_union_rl_inv : forall f1 f2 f3, fmap_agree f1 (f2 \+ f3) -> fmap_agree f1 f2. Proof using. hint fmap_agree_union_ll_inv, fmap_agree_sym. eauto. Qed. Lemma fmap_agree_union_lr_inv_agree_agree : forall f1 f2 f3, fmap_agree (f1 \+ f2) f3 -> fmap_agree f1 f2 -> fmap_agree f2 f3. Proof using. `````` charguer committed Mar 22, 2017 478 `````` introv M D. rewrite~ (@fmap_union_comm_of_agree f1 f2) in M. `````` charguer committed Mar 22, 2017 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 `````` applys* fmap_agree_union_ll_inv. Qed. Lemma fmap_agree_union_rr_inv_agree : forall f1 f2 f3, fmap_agree f1 (f2 \+ f3) -> fmap_agree f2 f3 -> fmap_agree f1 f3. Proof using. hint fmap_agree_union_lr_inv_agree_agree, fmap_agree_sym. eauto. Qed. Lemma fmap_agree_union_l_inv : forall f1 f2 f3, fmap_agree (f1 \+ f2) f3 -> fmap_agree f1 f2 -> fmap_agree f1 f3 /\ fmap_agree f2 f3. Proof using. (* TODO: proofs redundant with others above *) introv M2 M1. split. { intros l v1 v2 E1 E2. specializes M1 l v1 v2 E1. applys~ M2 l v1 v2. unfold fmap_union, map_union; simpl. rewrite~ E1. } { intros l v1 v2 E1 E2. specializes M1 l. specializes M2 l. unfolds fmap_union, map_union; simpls. cases (fmap_data f1 l). (* LATER: name b *) { applys eq_trans b. symmetry. applys~ M1. applys~ M2. } { auto. } } Qed. Lemma fmap_agree_union_r_inv : forall f1 f2 f3, fmap_agree f1 (f2 \+ f3) -> fmap_agree f2 f3 -> fmap_agree f1 f2 /\ fmap_agree f1 f3. Proof using. hint fmap_agree_sym. intros. forwards~ (M1&M2): fmap_agree_union_l_inv f2 f3 f1. Qed. (* ---------------------------------------------------------------------- *) (* ** Read and write *) `````` charguer committed Mar 22, 2017 523 ``````Lemma fmap_union_single_l_read : forall f1 f2 l v, `````` charguer committed Mar 22, 2017 524 525 526 527 528 529 `````` f1 = fmap_single l v -> fmap_data (f1 \+ f2) l = Some v. Proof using. intros. subst. simpl. unfold map_union. case_if~. Qed. `````` charguer committed Mar 22, 2017 530 ``````Lemma fmap_union_single_to_update : forall f1 f1' f2 l v v', `````` charguer committed Mar 22, 2017 531 532 533 534 535 536 `````` f1 = fmap_single l v -> f1' = fmap_single l v' -> (f1' \+ f2) = fmap_update (f1 \+ f2) l v'. Proof using. intros. subst. unfold fmap_update. rewrite <- fmap_union_assoc. fequals. `````` charguer committed Mar 22, 2017 537 `````` applys fmap_make_eq. intros l'. `````` charguer committed Mar 22, 2017 538 539 540 541 542 543 `````` unfolds map_union, fmap_single; simpl. case_if~. Qed. End FmapProp. Implicit Arguments fmap_union_assoc [A B]. `````` charguer committed Mar 22, 2017 544 545 ``````Implicit Arguments fmap_union_comm_of_disjoint [A B]. Implicit Arguments fmap_union_comm_of_agree [A B]. `````` charguer committed Mar 22, 2017 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 `````` (* ********************************************************************** *) (* * Tactics *) (* ---------------------------------------------------------------------- *) (* ** Tactic [fmap_disjoint] for proving disjointness results *) (** [fmap_disjoint] proves goals of the form [\# h1 h2] and [\# h1 h2 h3] by expanding all hypotheses into binary forms [\# h1 h2] and then exploiting symmetry and disjointness with [fmap_empty]. *) Hint Resolve fmap_disjoint_sym fmap_disjoint_empty_l fmap_disjoint_empty_r. Hint Rewrite fmap_disjoint_union_eq_l fmap_disjoint_union_eq_r fmap_disjoint_3_unfold : rew_disjoint. Tactic Notation "rew_disjoint" := autorewrite with rew_disjoint in *. Tactic Notation "rew_disjoint" "*" := rew_disjoint; auto_star. Tactic Notation "fmap_disjoint" := solve [ subst; rew_disjoint; jauto_set; auto ]. Tactic Notation "fmap_disjoint_if_needed" := match goal with | |- \# _ _ => fmap_disjoint | |- \# _ _ _ => fmap_disjoint end. Lemma fmap_disjoint_demo : forall A B (h1 h2 h3 h4 h5:fmap A B), h1 = h2 \+ h3 -> \# h2 h3 -> \# h1 h4 h5 -> \# h3 h2 h5 /\ \# h4 h5. Proof using. intros. dup 2. { subst. rew_disjoint. jauto_set. auto. auto. auto. auto. } { fmap_disjoint. } Qed. (* ---------------------------------------------------------------------- *) (* ** Tactic [fmap_eq] for proving equality of fmaps, and tactic [rew_fmap] to normalize fmap expressions. *) Section StateEq. Variables (A B : Type). Implicit Types h : fmap A B. `````` charguer committed May 16, 2017 600 601 ``````(** [fmap_eq] proves equalities between unions of fmaps, of the form [h1 \+ h2 \+ h3 \+ h4 = h1' \+ h2' \+ h3' \+ h4'] `````` charguer committed Mar 22, 2017 602 `````` It attempts to discharge the disjointness side-conditions. `````` charguer committed May 16, 2017 603 `````` Disclaimer: it cancels heaps at depth up to 4, but no more. *) `````` charguer committed Mar 22, 2017 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 `````` Lemma fmap_union_eq_cancel_1 : forall h1 h2 h2', h2 = h2' -> h1 \+ h2 = h1 \+ h2'. Proof using. intros. subst. auto. Qed. Lemma fmap_union_eq_cancel_1' : forall h1, h1 = h1. Proof using. intros. auto. Qed. Lemma fmap_union_eq_cancel_2 : forall h1 h1' h2 h2', \# h1 h1' -> h2 = h1' \+ h2' -> h1 \+ h2 = h1' \+ h1 \+ h2'. Proof using. intros. subst. rewrite <- fmap_union_assoc. `````` charguer committed Mar 22, 2017 620 `````` rewrite (fmap_union_comm_of_disjoint h1 h1'). `````` charguer committed Mar 22, 2017 621 622 623 624 625 626 627 628 `````` rewrite~ fmap_union_assoc. auto. Qed. Lemma fmap_union_eq_cancel_2' : forall h1 h1' h2, \# h1 h1' -> h2 = h1' -> h1 \+ h2 = h1' \+ h1. Proof using. `````` charguer committed Mar 22, 2017 629 `````` intros. subst. apply~ fmap_union_comm_of_disjoint. `````` charguer committed Mar 22, 2017 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 ``````Qed. Lemma fmap_union_eq_cancel_3 : forall h1 h1' h2 h2' h3', \# h1 (h1' \+ h2') -> h2 = h1' \+ h2' \+ h3' -> h1 \+ h2 = h1' \+ h2' \+ h1 \+ h3'. Proof using. intros. subst. rewrite <- (fmap_union_assoc h1' h2' h3'). rewrite <- (fmap_union_assoc h1' h2' (h1 \+ h3')). apply~ fmap_union_eq_cancel_2. Qed. Lemma fmap_union_eq_cancel_3' : forall h1 h1' h2 h2', \# h1 (h1' \+ h2') -> h2 = h1' \+ h2' -> h1 \+ h2 = h1' \+ h2' \+ h1. Proof using. intros. subst. rewrite <- (fmap_union_assoc h1' h2' h1). apply~ fmap_union_eq_cancel_2'. Qed. Lemma fmap_union_eq_cancel_4 : forall h1 h1' h2 h2' h3' h4', \# h1 ((h1' \+ h2') \+ h3') -> h2 = h1' \+ h2' \+ h3' \+ h4' -> h1 \+ h2 = h1' \+ h2' \+ h3' \+ h1 \+ h4'. Proof using. intros. subst. rewrite <- (fmap_union_assoc h1' h2' (h3' \+ h4')). rewrite <- (fmap_union_assoc h1' h2' (h3' \+ h1 \+ h4')). apply~ fmap_union_eq_cancel_3. Qed. Lemma fmap_union_eq_cancel_4' : forall h1 h1' h2 h2' h3', \# h1 (h1' \+ h2' \+ h3') -> h2 = h1' \+ h2' \+ h3' -> h1 \+ h2 = h1' \+ h2' \+ h3' \+ h1. Proof using. intros. subst. rewrite <- (fmap_union_assoc h2' h3' h1). apply~ fmap_union_eq_cancel_3'. Qed. End StateEq. Hint Rewrite fmap_union_assoc fmap_union_empty_l fmap_union_empty_r : rew_fmap. Tactic Notation "rew_fmap" := autorewrite with rew_fmap in *. Tactic Notation "rew_fmap" "~" := rew_fmap; auto_tilde. Tactic Notation "rew_fmap" "*" := rew_fmap; auto_star. Ltac fmap_eq_step tt := match goal with | |- ?G => match G with | ?h1 = ?h1 => apply fmap_union_eq_cancel_1' | ?h1 \+ ?h2 = ?h1 \+ ?h2' => apply fmap_union_eq_cancel_1 | ?h1 \+ ?h2 = ?h1' \+ ?h1 => apply fmap_union_eq_cancel_2' | ?h1 \+ ?h2 = ?h1' \+ ?h1 \+ ?h2' => apply fmap_union_eq_cancel_2 | ?h1 \+ ?h2 = ?h1' \+ ?h2' \+ ?h1 => apply fmap_union_eq_cancel_3' | ?h1 \+ ?h2 = ?h1' \+ ?h2' \+ ?h1 \+ ?h3' => apply fmap_union_eq_cancel_3 | ?h1 \+ ?h2 = ?h1' \+ ?h2' \+ ?h3' \+ ?h1 => apply fmap_union_eq_cancel_4' | ?h1 \+ ?h2 = ?h1' \+ ?h2' \+ ?h3' \+ ?h1 \+ ?h4' => apply fmap_union_eq_cancel_4 end end. Tactic Notation "fmap_eq" := subst; rew_fmap; repeat (fmap_eq_step tt); try fmap_disjoint_if_needed. Tactic Notation "fmap_eq" "~" := fmap_eq; auto_tilde. Tactic Notation "fmap_eq" "*" := fmap_eq; auto_star. Lemma fmap_eq_demo : forall A B (h1 h2 h3 h4 h5:fmap A B), \# h1 h2 h3 -> \# (h1 \+ h2 \+ h3) h4 h5 -> h1 = h2 \+ h3 -> h4 \+ h1 \+ h5 = h2 \+ h5 \+ h4 \+ h3. Proof using. intros. dup 2. { subst. rew_fmap. fmap_eq_step tt. fmap_disjoint. fmap_eq_step tt. fmap_eq_step tt. fmap_disjoint. auto. } { fmap_eq. } Qed. (* ---------------------------------------------------------------------- *) (* ** Tactic [fmap_red] for proving [red] goals (reduction according to a big-step semantics) modulo equalities between fmaps *) (** [fmap_red] proves a goal of the form [red h1 t h2 v] using an hypothesis of the shape [red h1' t h2' v], generating [h1 = h1'] and [h2 = h2'] as subgoals, and `````` charguer committed Mar 22, 2017 737 738 739 740 741 742 743 744 745 746 `````` attempting to solve them using the tactic [fmap_eq]. The tactic should be configured depending on [red]. For example: Ltac fmap_red_base tt := match goal with H: red _ ?t _ _ |- red _ ?t _ _ => applys_eq H 2 4; try fmap_eq end. The default implementation is a dummy one. *) `````` charguer committed Mar 22, 2017 747 748 749 750 751 752 753 754 755 756 757 `````` Ltac fmap_red_base tt := fail. Tactic Notation "fmap_red" := fmap_red_base tt. Tactic Notation "fmap_red" "~" := fmap_red; auto_tilde. Tactic Notation "fmap_red" "*" := fmap_red; auto_star. `````` charguer committed Mar 28, 2017 758 759 760 761 762 763 764 765 `````` (* ********************************************************************** *) (** * Consecutive locations and fresh locations *) (* ---------------------------------------------------------------------- *) (** ** Existence of fresh locations *) `````` charguer committed Mar 31, 2017 766 ``````Fixpoint fmap_conseq (B:Type) (l:nat) (k:nat) (v:B) : fmap nat B := `````` charguer committed Mar 28, 2017 767 `````` match k with `````` charguer committed Mar 31, 2017 768 769 `````` | O => fmap_empty | S k' => (fmap_single l v) \+ (fmap_conseq (S l) k' v) `````` charguer committed Mar 28, 2017 770 771 772 773 774 775 776 `````` end. Lemma fmap_conseq_zero : forall B (l:nat) (v:B), fmap_conseq l O v = fmap_empty. Proof using. auto. Qed. Lemma fmap_conseq_succ : forall B (l:nat) (k:nat) (v:B), `````` charguer committed Mar 31, 2017 777 `````` fmap_conseq l (S k) v = (fmap_single l v) \+ (fmap_conseq (S l) k v). `````` charguer committed Mar 28, 2017 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 ``````Proof using. auto. Qed. Opaque fmap_conseq. (* ---------------------------------------------------------------------- *) (** ** Existence of fresh locations *) (** These lemmas are useful to prove: [forall h v, exists l, fmap_disjoint (fmap_single l v) h]. *) Definition loc_fresh_gen (L : list nat) := (1 + fold_right plus O L)%nat. Lemma loc_fresh_ind : forall l L, Mem l L -> (l < loc_fresh_gen L)%nat. Proof using. intros l L. unfold loc_fresh_gen. induction L; introv M; inverts M; rew_list. { math. } { forwards~: IHL. math. } Qed. Lemma loc_fresh_nat_ge : forall (L:list nat), exists (l:nat), forall (i:nat), ~ Mem (l+i)%nat L. Proof using. intros L. exists (loc_fresh_gen L). intros i M. lets: loc_fresh_ind M. math. Qed. Lemma loc_fresh_nat : forall (L:list nat), exists (l:nat), ~ Mem l L. Proof using. intros L. forwards (l&P): loc_fresh_nat_ge L. exists l. intros M. applys (P 0%nat). applys_eq M 2. math. Qed. (* ---------------------------------------------------------------------- *) `````` charguer committed May 16, 2017 818 ``````(** ** Extension of a number of consecutive fresh locations *) `````` charguer committed Mar 28, 2017 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 `````` Section FmapFresh. Variables (B : Type). Implicit Types h : fmap nat B. Lemma fmap_single_fresh : forall null h v, exists l, \# (fmap_single l v) h /\ l <> null. Proof using. intros null (m&(L&M)) v. unfold fmap_disjoint, map_disjoint. simpl. lets (l&F): (loc_fresh_nat (null::L)). exists l. split. { intros l'. case_if~. { right. applys not_not_elim. intros H. applys F. constructor. applys~ M. } } { intro_subst. applys~ F. } Qed. Lemma fmap_conseq_fresh : forall null h k v, exists l, \# (fmap_conseq l k v) h /\ l <> null. Proof using. intros null (m&(L&M)) k v. unfold fmap_disjoint, map_disjoint. simpl. lets (l&F): (loc_fresh_nat_ge (null::L)). exists l. split. `````` charguer committed Mar 31, 2017 844 845 846 847 848 `````` { intros l'. gen l. induction k; intros. { simple~. } { rewrite fmap_conseq_succ. destruct (IHk (S l)%nat) as [E|?]. { intros i N. applys F (S i). applys_eq N 2. math. } `````` charguer committed Mar 28, 2017 849 `````` { simpl. unfold map_union. case_if~. `````` charguer committed Mar 31, 2017 850 851 `````` { right. applys not_not_elim. intros H. applys F 0%nat. constructor. math_rewrite (l'+0 = l')%nat. applys~ M. } } `````` charguer committed Mar 28, 2017 852 853 854 855 856 `````` { auto. } } } { intro_subst. applys~ F 0%nat. rew_nat. auto. } Qed. Lemma fmap_disjoint_single_conseq : forall B l l' k (v:B), `````` charguer committed Mar 31, 2017 857 858 `````` (l < l')%nat \/ (l >= l'+k)%nat -> \# (fmap_single l v) (fmap_conseq l' k v). `````` charguer committed Mar 28, 2017 859 ``````Proof using. `````` charguer committed Mar 31, 2017 860 `````` introv N. gen l'. induction k; intros. `````` charguer committed Mar 28, 2017 861 862 `````` { rewrite~ fmap_conseq_zero. } { rewrite fmap_conseq_succ. rew_disjoint. split. `````` charguer committed Mar 31, 2017 863 864 `````` { applys fmap_disjoint_single_single. destruct N; math. } { applys IHk. destruct N. { left; math. } { right; math. } } } `````` charguer committed Mar 28, 2017 865 866 867 868 869 870 871 ``````Qed. End FmapFresh. ``````