proofmode.v 21.2 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214
(* This version provides convenient tactics that are wrappers around the
lifting lemmas. Much of this file is taking from [heap_lang/proofmode] in Iris,
but the difference is that our tactics detect whether the goal is 1.) a
weakest precondition of the basic Iron logic with explicit fractions, or 2.) a
weakest precondition of the lifted Iron++ logic. Depending on the shape of the
goal, the tactics will use the lifting lemma for either logic.

As for the lifting file, all results for total weakest preconditions have been
commented out, because we currently do not support them. These may be restored
later. *)
From iron.iron_logic Require Export weakestpre (* total_weakestpre *).
From iron.iron_logic Require Import lifting (* total_weakestpre *).
From iris.proofmode Require Import coq_tactics reduction.
From iris.proofmode Require Export tactics.
From iron.heap_lang Require Export tactics lifting.
Set Default Proof Using "Type".
Import uPred.
(** This file contains a lot of redundancy; for many tactics we have variants
for total/partial weakest preconditions, and those in Iris or in the lifted
Iron logic. *)

Lemma tac_wp_expr_eval `{heapG Σ} Δ s E (Φ : _  iProp Σ) e e' :
  ( (e'':=e'), e = e'') 
  envs_entails Δ (WP e' @ s; E {{ Φ }})  envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. by intros ->. Qed.
Lemma tac_iron_wp_expr_eval `{heapG Σ} Δ s E (Φ : _  ironProp Σ) e e' :
  ( (e'':=e'), e = e'') 
  envs_entails Δ (WP e' @ s; E {{ Φ }})  envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. by intros ->. Qed.
(*
Lemma tac_twp_expr_eval `{heapG Σ} Δ s E Φ e e' :
  (∀ (e'':=e'), e = e'') →
  envs_entails Δ (WP e' @ s; E [{ Φ }]) → envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. by intros ->. Qed.
*)

Local Notation envs_entails_wp e := (envs_entails _ (@wp _ _ _ wp' _ _ e _)).
Local Notation envs_entails_iron_wp e := (envs_entails _ (@wp _ _ _ iron_wp _ _ e _)).

Tactic Notation "wp_expr_eval" tactic(t) :=
  iStartProof;
  lazymatch goal with
  | |- envs_entails_wp ?e =>
    eapply tac_wp_expr_eval;
      [let x := fresh in intros x; t; unfold x; reflexivity|]
  | |- envs_entails_iron_wp ?e =>
    eapply tac_iron_wp_expr_eval;
      [let x := fresh in intros x; t; unfold x; reflexivity|]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    eapply tac_twp_expr_eval;
      [let x := fresh in intros x; t; unfold x; reflexivity|] *)
  | _ => fail "wp_expr_eval: not a 'wp'"
  end.

Ltac wp_expr_simpl := wp_expr_eval simpl.
Ltac wp_expr_simpl_subst := wp_expr_eval simpl_subst.

Lemma tac_wp_pure `{heapG Σ} Δ Δ' s E e1 e2 φ n (Φ : _  iProp Σ) :
  PureExec φ n e1 e2 
  φ 
  MaybeIntoLaterNEnvs n Δ Δ' 
  envs_entails Δ' (WP e2 @ s; E {{ Φ }}) 
  envs_entails Δ (WP e1 @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ??? HΔ'. rewrite into_laterN_env_sound /=.
  rewrite HΔ' -lifting.wp_pure_step_later //.
Qed.
Lemma tac_iron_wp_pure `{heapG Σ} Δ Δ' s E e1 e2 φ n (Φ : _  ironProp Σ) :
  PureExec φ n e1 e2 
  φ 
  MaybeIntoLaterNEnvs n Δ Δ' 
  envs_entails Δ' (WP e2 @ s; E {{ Φ }}) 
  envs_entails Δ (WP e1 @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ??? HΔ'. rewrite into_laterN_env_sound /=.
  rewrite HΔ' -iron_wp_pure_step_later //.
Qed.
(*
Lemma tac_twp_pure `{heapG Σ} Δ s E e1 e2 φ Φ :
  PureExec φ e1 e2 →
  φ →
  envs_entails Δ (WP e2 @ s; E [{ Φ }]) →
  envs_entails Δ (WP e1 @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq=> ?? ->. rewrite -total_lifting.twp_pure_step //.
Qed.
*)

Lemma tac_wp_value `{heapG Σ} Δ s E (Φ : _  iProp Σ) e v :
  IntoVal e v 
  envs_entails Δ (Φ v)  envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. rewrite envs_entails_eq=> ? ->. by apply wp_value. Qed.
Lemma tac_iron_wp_value `{heapG Σ} Δ s E (Φ : _  ironProp Σ) e v :
  IntoVal e v 
  envs_entails Δ (Φ v)  envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. rewrite envs_entails_eq=> ? ->. by apply iron_wp_value. Qed.
(*
Lemma tac_twp_value `{heapG Σ} Δ s E Φ e v :
  IntoVal e v →
  envs_entails Δ (Φ v) → envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. rewrite envs_entails_eq=> ? ->. by apply twp_value. Qed.
*)
Ltac wp_value_head :=
  first [eapply tac_wp_value || eapply tac_iron_wp_value];
    [iSolveTC
    |lazy beta].

Tactic Notation "wp_pure" open_constr(efoc) :=
  let fail_with e :=
    fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a redex" in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_wp ?e =>
    let e := eval simpl in e in
    reshape_expr e ltac:(fun K e' =>
      unify e' efoc;
      eapply (tac_wp_pure _ _ _ _ (fill K e'));
      [iSolveTC                       (* PureExec *)
      |try fast_done                  (* The pure condition for PureExec *)
      |iSolveTC                       (* IntoLaters *)
      |reduction.pm_prettify; wp_expr_simpl_subst; try wp_value_head (* new goal *)
      ]) || fail_with e
  | |- envs_entails_iron_wp ?e =>
    let e := eval simpl in e in
    reshape_expr e ltac:(fun K e' =>
      unify e' efoc;
      eapply (tac_iron_wp_pure _ _ _ _ (fill K e'));
      [iSolveTC                       (* PureExec *)
      |try fast_done                  (* The pure condition for PureExec *)
      |iSolveTC                       (* IntoLaters *)
      |reduction.pm_prettify; wp_expr_simpl_subst; try wp_value_head (* new goal *)
      ]) || fail_with e
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    let e := eval simpl in e in
    reshape_expr e ltac:(fun K e' =>
      unify e' efoc;
      eapply (tac_twp_pure _ _ _ (fill K e'));
      [iSolveTC                        (* PureExec *)
      |try fast_done                  (* The pure condition for PureExec *)
      |wp_expr_simpl_subst; try wp_value_head (* new goal *)
      ])
    || fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a redex" *)
  | _ => fail "wp_pure: not a 'wp'"
  end.

Tactic Notation "wp_if" := wp_pure (If _ _ _).
Tactic Notation "wp_if_true" := wp_pure (If (Lit (LitBool true)) _ _).
Tactic Notation "wp_if_false" := wp_pure (If (Lit (LitBool false)) _ _).
Tactic Notation "wp_unop" := wp_pure (UnOp _ _).
Tactic Notation "wp_binop" := wp_pure (BinOp _ _ _).
Tactic Notation "wp_op" := wp_unop || wp_binop.
Tactic Notation "wp_rec" := wp_pure (App _ _).
Tactic Notation "wp_lam" := wp_rec.
Tactic Notation "wp_let" := wp_lam.
Tactic Notation "wp_seq" := wp_lam.
Tactic Notation "wp_proj" := wp_pure (Fst _) || wp_pure (Snd _).
Tactic Notation "wp_case" := wp_pure (Case _ _ _).
Tactic Notation "wp_match" := wp_case; wp_let.

Lemma tac_wp_bind `{heapG Σ} K Δ s E (Φ : _  iProp Σ) e f :
  f = (λ e, fill K e)  (* as an eta expanded hypothesis so that we can `simpl` it *)
  envs_entails Δ (WP e @ s; E {{ v, WP f (of_val v) @ s; E {{ Φ }} }})%I 
  envs_entails Δ (WP fill K e @ s; E {{ Φ }}).
Proof. rewrite envs_entails_eq=> -> ->. by apply: wp_bind. Qed.
Lemma tac_iron_wp_bind `{heapG Σ} K Δ s E (Φ : _  ironProp Σ) e f :
  f = (λ e, fill K e)  (* as an eta expanded hypothesis so that we can `simpl` it *)
  envs_entails Δ (WP e @ s; E {{ v, WP f (of_val v) @ s; E {{ Φ }} }})%I 
  envs_entails Δ (WP fill K e @ s; E {{ Φ }}).
Proof. rewrite envs_entails_eq=> -> ->. by apply: iron_wp_bind. Qed.
(*
Lemma tac_twp_bind `{heapG Σ} K Δ s E Φ e f :
  f = (λ e, fill K e) → (* as an eta expanded hypothesis so that we can `simpl` it *)
  envs_entails Δ (WP e @ s; E [{ v, WP f (of_val v) @ s; E [{ Φ }] }])%I →
  envs_entails Δ (WP fill K e @ s; E [{ Φ }]).
Proof. rewrite envs_entails_eq=> -> ->. by apply: twp_bind. Qed.
*)
Ltac wp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
  | _ => eapply (tac_wp_bind K); [simpl; reflexivity|lazy beta]
  end.
Ltac iron_wp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
  | _ => eapply (tac_iron_wp_bind K); [simpl; reflexivity|lazy beta]
  end.
(*
Ltac twp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
  | _ => eapply (tac_twp_bind K); [simpl; reflexivity|lazy beta]
  end.
*)

Tactic Notation "wp_bind" open_constr(efoc) :=
  iStartProof;
  lazymatch goal with
  | |- envs_entails_wp ?e =>
    reshape_expr e ltac:(fun K e' => unify e' efoc; wp_bind_core K)
    || fail "wp_bind: cannot find" efoc "in" e
  | |- envs_entails_iron_wp ?e =>
    reshape_expr e ltac:(fun K e' => unify e' efoc; iron_wp_bind_core K)
    || fail "wp_bind: cannot find" efoc "in" e
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    reshape_expr e ltac:(fun K e' => unify e' efoc; twp_bind_core K)
    || fail "wp_bind: cannot find" efoc "in" e *)
  | _ => fail "wp_bind: not a 'wp'"
  end.

Tactic Notation "wp_apply" open_constr(lem) :=
  let fail_with H :=
    lazymatch iTypeOf H with
    | Some (_,?P) => fail "wp_apply: cannot apply" P
    end in
215
  iPoseProofCore lem as false (fun H =>
Robbert Krebbers's avatar
Robbert Krebbers committed
216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554
    lazymatch goal with
    | |- envs_entails_wp ?e =>
      reshape_expr e ltac:(fun K e' =>
        wp_bind_core K; iApplyHyp H; try iNext; try wp_expr_simpl) || fail_with H
    | |- envs_entails_iron_wp ?e =>
      reshape_expr e ltac:(fun K e' =>
        iron_wp_bind_core K; iApplyHyp H; try iNext; try wp_expr_simpl) || fail_with H
(*    | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
      reshape_expr e ltac:(fun K e' =>
        twp_bind_core K; iApplyHyp H; try wp_expr_simpl) || fail H *)
    | _ => fail "wp_apply: not a 'wp'"
    end).

(** Heap tactics *)
Section heap.
Context `{heapG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val  ironProp Σ.
Implicit Types Δ : envs (ironPropI Σ).

Lemma tac_wp_alloc Δ Δ' s E j K e v Φ :
  IntoVal e v 
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  ( l,  Δ'',
    envs_app false (Esnoc Enil j (l  v)) Δ' = Some Δ'' 
    envs_entails Δ'' (WP fill K (Lit (LitLoc l)) @ s; E {{ Φ }})) 
  envs_entails Δ (WP fill K (Alloc e) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ?? HΔ.
  rewrite -iron_wp_bind. eapply wand_apply; first exact: iron_wp_alloc.
  rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
  destruct (HΔ l) as (Δ''&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.
(*
Lemma tac_twp_alloc Δ s E j K e v Φ :
  IntoVal e v →
  (∀ l, ∃ Δ',
    envs_app false (Esnoc Enil j (l ↦ v)) Δ = Some Δ' ∧
    envs_entails Δ' (WP fill K (Lit (LitLoc l)) @ s; E [{ Φ }])) →
  envs_entails Δ (WP fill K (Alloc e) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq=> ? HΔ.
  rewrite -twp_bind. eapply wand_apply; first exact: twp_alloc.
  rewrite left_id. apply forall_intro=> l.
  destruct (HΔ l) as (Δ'&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.
*)

Lemma tac_wp_load Δ Δ' s E i K l q v Φ :
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l {q} v)%I 
  envs_entails Δ' (WP fill K (of_val v) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (Load (Lit (LitLoc l))) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ???.
  rewrite -iron_wp_bind. eapply wand_apply; first exact: iron_wp_load.
  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
  by apply later_mono, sep_mono_r, wand_mono.
Qed.
(*
Lemma tac_twp_load Δ s E i K l q v Φ :
  envs_lookup i Δ = Some (false, l ↦{q} v)%I →
  envs_entails Δ (WP fill K (of_val v) @ s; E [{ Φ }]) →
  envs_entails Δ (WP fill K (Load (Lit (LitLoc l))) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq=> ??.
  rewrite -twp_bind. eapply wand_apply; first exact: twp_load.
  rewrite envs_lookup_split //; simpl.
  by apply sep_mono_r, wand_mono.
Qed.
*)

Lemma tac_wp_store Δ Δ' Δ'' s E i K l v e v' Φ :
  IntoVal e v' 
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l  v)%I 
  envs_simple_replace i false (Esnoc Enil i (l  v')) Δ' = Some Δ'' 
  envs_entails Δ'' (WP fill K (Lit LitUnit) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (Store (Lit (LitLoc l)) e) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ?????.
  rewrite -iron_wp_bind. eapply wand_apply; first by eapply iron_wp_store.
  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
(*
Lemma tac_twp_store Δ Δ' s E i K l v e v' Φ :
  IntoVal e v' →
  envs_lookup i Δ = Some (false, l ↦ v)%I →
  envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ = Some Δ' →
  envs_entails Δ' (WP fill K (Lit LitUnit) @ s; E [{ Φ }]) →
  envs_entails Δ (WP fill K (Store (Lit (LitLoc l)) e) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq. intros. rewrite -twp_bind.
  eapply wand_apply; first by eapply twp_store.
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
*)

Lemma tac_wp_cas_fail Δ Δ' s E i K l q v e1 v1 e2 Φ :
  IntoVal e1 v1  AsVal e2 
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l {q} v)%I  v  v1 
  envs_entails Δ' (WP fill K (Lit (LitBool false)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ??????.
  rewrite -iron_wp_bind. eapply wand_apply; first exact: iron_wp_cas_fail.
  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
  by apply later_mono, sep_mono_r, wand_mono.
Qed.
(*
Lemma tac_twp_cas_fail Δ s E i K l q v e1 v1 e2 Φ :
  IntoVal e1 v1 → AsVal e2 →
  envs_lookup i Δ = Some (false, l ↦{q} v)%I → v ≠ v1 →
  envs_entails Δ (WP fill K (Lit (LitBool false)) @ s; E [{ Φ }]) →
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq. intros. rewrite -twp_bind.
  eapply wand_apply; first exact: twp_cas_fail.
  rewrite envs_lookup_split //=. by do 2 f_equiv.
Qed. *)

Lemma tac_wp_cas_suc Δ Δ' Δ'' s E i K l v e1 v1 e2 v2 Φ :
  IntoVal e1 v1  IntoVal e2 v2 
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l  v)%I  v = v1 
  envs_simple_replace i false (Esnoc Enil i (l  v2)) Δ' = Some Δ'' 
  envs_entails Δ'' (WP fill K (Lit (LitBool true)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ???????; subst.
  rewrite -iron_wp_bind. eapply wand_apply; first exact: iron_wp_cas_suc.
  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
(*
Lemma tac_twp_cas_suc Δ Δ' s E i K l v e1 v1 e2 v2 Φ :
  IntoVal e1 v1 → IntoVal e2 v2 →
  envs_lookup i Δ = Some (false, l ↦ v)%I → v = v1 →
  envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ = Some Δ' →
  envs_entails Δ' (WP fill K (Lit (LitBool true)) @ s; E [{ Φ }]) →
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq. intros; subst.
  rewrite -twp_bind. eapply wand_apply; first exact: twp_cas_suc.
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
*)

Lemma tac_wp_faa Δ Δ' Δ'' s E i K l i1 e2 i2 Φ :
  IntoVal e2 (LitV (LitInt i2)) 
  MaybeIntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l  LitV (LitInt i1))%I 
  envs_simple_replace i false (Esnoc Enil i (l  LitV (LitInt (i1 + i2)))) Δ' = Some Δ'' 
  envs_entails Δ'' (WP fill K (Lit (LitInt i1)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (FAA (Lit (LitLoc l)) e2) @ s; E {{ Φ }}).
Proof.
  rewrite envs_entails_eq=> ?????; subst.
  rewrite -iron_wp_bind. eapply wand_apply; first exact: (iron_wp_faa _ _ _ i1 _ i2).
  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
(*
Lemma tac_twp_faa Δ Δ' s E i K l i1 e2 i2 Φ :
  IntoVal e2 (LitV (LitInt i2)) →
  envs_lookup i Δ = Some (false, l ↦ LitV (LitInt i1))%I →
  envs_simple_replace i false (Esnoc Enil i (l ↦ LitV (LitInt (i1 + i2)))) Δ = Some Δ' →
  envs_entails Δ' (WP fill K (Lit (LitInt i1)) @ s; E [{ Φ }]) →
  envs_entails Δ (WP fill K (FAA (Lit (LitLoc l)) e2) @ s; E [{ Φ }]).
Proof.
  rewrite envs_entails_eq=> ????; subst.
  rewrite -twp_bind. eapply wand_apply; first exact: (twp_faa _ _ _ i1 _ i2).
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
*)
End heap.

Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
  let finish _ :=
    first [intros l | fail 1 "wp_alloc:" l "not fresh"];
      eexists; split;
        [pm_reflexivity || fail "wp_alloc:" H "not fresh"
        |wp_expr_simpl; try wp_value_head] in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_wp_alloc _ _ _ _ H K); [iSolveTC|..])
      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
    [iSolveTC
    |finish ()]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_alloc _ _ _ H K); [iSolveTC|..])
      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
    finish () *)
  | _ => fail "wp_alloc: not a 'wp'"
  end.

Tactic Notation "wp_alloc" ident(l) :=
  let H := iFresh in wp_alloc l as H.

Tactic Notation "wp_load" :=
  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_load: cannot find" l "↦ ?" in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_load _ _ _ _ _ K))
      |fail 1 "wp_load: cannot find 'Load' in" e];
    [iSolveTC
    |solve_mapsto ()
    |wp_expr_simpl; try wp_value_head]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' => eapply (tac_twp_load _ _ _ _ K))
      |fail 1 "wp_load: cannot find 'Load' in" e];
    [solve_mapsto ()
    |wp_expr_simpl; try wp_value_head] *)
  | _ => fail "wp_load: not a 'wp'"
  end.

Tactic Notation "wp_store" :=
  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_store: cannot find" l "↦ ?" in
  let finish _ :=
    wp_expr_simpl; try first [wp_pure (Seq (Lit LitUnit) _)|wp_value_head] in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_wp_store _ _ _ _ _ _ K); [iSolveTC|..])
      |fail 1 "wp_store: cannot find 'Store' in" e];
    [iSolveTC
    |solve_mapsto ()
    |pm_reflexivity
    |finish ()]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_store _ _ _ _ _ K); [iSolveTC|..])
      |fail 1 "wp_store: cannot find 'Store' in" e];
    [solve_mapsto ()
    |env_cbv; reflexivity
    |finish ()] *)
  | _ => fail "wp_store: not a 'wp'"
  end.

Tactic Notation "wp_cas_fail" :=
  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?" in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_wp_cas_fail _ _ _ _ _ K); [iSolveTC|iSolveTC|..])
      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
    [iSolveTC
    |solve_mapsto ()
    |try congruence
    |simpl; try wp_value_head]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_cas_fail _ _ _ _ K); [iSolveTC|iSolveTC|..])
      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
    [solve_mapsto ()
    |try congruence
    |wp_expr_simpl; try wp_value_head] *)
  | _ => fail "wp_cas_fail: not a 'wp'"
  end.

Tactic Notation "wp_cas_suc" :=
  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_suc: cannot find" l "↦ ?" in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_wp_cas_suc _ _ _ _ _ _ K); [iSolveTC|iSolveTC|..])
      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
    [iSolveTC
    |solve_mapsto ()
    |try congruence
    |pm_reflexivity
    |simpl; try wp_value_head]
(*  | |- envs_entails _ (twp ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_cas_suc _ _ _ _ _ K); [iSolveTC|iSolveTC|..])
      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
    [solve_mapsto ()
    |try congruence
    |env_cbv; reflexivity
    |wp_expr_simpl; try wp_value_head] *)
  | _ => fail "wp_cas_suc: not a 'wp'"
  end.

Tactic Notation "wp_faa" :=
  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_suc: cannot find" l "↦ ?" in
  iStartProof;
  lazymatch goal with
  | |- envs_entails_iron_wp ?e =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_wp_faa _ _ _ _ _ _ K); [iSolveTC|..])
      |fail 1 "wp_faa: cannot find 'CAS' in" e];
    [iSolveTC
    |solve_mapsto ()
    |pm_reflexivity
    |wp_expr_simpl; try wp_value_head]
(*  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_faa _ _ _ _ _ K); [iSolveTC|..])
      |fail 1 "wp_faa: cannot find 'CAS' in" e];
    [solve_mapsto ()
    |env_cbv; reflexivity
    |wp_expr_simpl; try wp_value_head] *)
  | _ => fail "wp_faa: not a 'wp'"
  end.