heap_lang.v 12.5 KB
Newer Older
1
Require Export program_logic.language prelude.strings.
2
Require Import prelude.gmap.
3

4
Module heap_lang.
5 6
Open Scope Z_scope.

7
(** Expressions and vals. *)
8
Definition loc := positive. (* Really, any countable type. *)
Ralf Jung's avatar
Ralf Jung committed
9

10
Inductive base_lit : Set :=
11
  | LitInt (n : Z) | LitBool (b : bool) | LitUnit.
12
Inductive un_op : Set :=
13
  | NegOp | MinusUnOp.
14 15 16
Inductive bin_op : Set :=
  | PlusOp | MinusOp | LeOp | LtOp | EqOp.

Ralf Jung's avatar
Ralf Jung committed
17
Inductive expr :=
18
  (* Base lambda calculus *)
19 20
  | Var (x : string)
  | Rec (f x : string) (e : expr)
21
  | App (e1 e2 : expr)
22 23 24 25 26
  (* Base types and their operations *)
  | Lit (l : base_lit)
  | UnOp (op : un_op) (e : expr)
  | BinOp (op : bin_op) (e1 e2 : expr)
  | If (e0 e1 e2 : expr)
27 28 29 30 31 32 33
  (* Products *)
  | Pair (e1 e2 : expr)
  | Fst (e : expr)
  | Snd (e : expr)
  (* Sums *)
  | InjL (e : expr)
  | InjR (e : expr)
34
  | Case (e0 : expr) (x1 : string) (e1 : expr) (x2 : string) (e2 : expr)
35 36 37 38 39 40 41 42
  (* Concurrency *)
  | Fork (e : expr)
  (* Heap *)
  | Loc (l : loc)
  | Alloc (e : expr)
  | Load (e : expr)
  | Store (e1 : expr) (e2 : expr)
  | Cas (e0 : expr) (e1 : expr) (e2 : expr).
Ralf Jung's avatar
Ralf Jung committed
43

44
Inductive val :=
45
  | RecV (f x : string) (e : expr) (* e should be closed *)
46
  | LitV (l : base_lit)
47 48 49 50
  | PairV (v1 v2 : val)
  | InjLV (v : val)
  | InjRV (v : val)
  | LocV (l : loc).
Ralf Jung's avatar
Ralf Jung committed
51

52 53 54
Delimit Scope lang_scope with L.
Bind Scope lang_scope with expr val.

55
Fixpoint of_val (v : val) : expr :=
Ralf Jung's avatar
Ralf Jung committed
56
  match v with
57
  | RecV f x e => Rec f x e
58
  | LitV l => Lit l
59 60 61
  | PairV v1 v2 => Pair (of_val v1) (of_val v2)
  | InjLV v => InjL (of_val v)
  | InjRV v => InjR (of_val v)
62
  | LocV l => Loc l
Ralf Jung's avatar
Ralf Jung committed
63
  end.
64
Fixpoint to_val (e : expr) : option val :=
65
  match e with
66
  | Rec f x e => Some (RecV f x e)
67
  | Lit l => Some (LitV l)
68 69 70
  | Pair e1 e2 => v1  to_val e1; v2  to_val e2; Some (PairV v1 v2)
  | InjL e => InjLV <$> to_val e
  | InjR e => InjRV <$> to_val e
71
  | Loc l => Some (LocV l)
Ralf Jung's avatar
Ralf Jung committed
72
  | _ => None
73 74
  end.

75 76
(** The state: heaps of vals. *)
Definition state := gmap loc val.
Ralf Jung's avatar
Ralf Jung committed
77

78
(** Evaluation contexts *)
79 80 81
Inductive ectx_item :=
  | AppLCtx (e2 : expr)
  | AppRCtx (v1 : val)
82 83 84 85
  | UnOpCtx (op : un_op)
  | BinOpLCtx (op : bin_op) (e2 : expr)
  | BinOpRCtx (op : bin_op) (v1 : val)
  | IfCtx (e1 e2 : expr)
86 87 88 89 90 91
  | PairLCtx (e2 : expr)
  | PairRCtx (v1 : val)
  | FstCtx
  | SndCtx
  | InjLCtx
  | InjRCtx
92
  | CaseCtx (x1 : string) (e1 : expr) (x2 : string) (e2 : expr)
93 94 95 96 97 98 99
  | AllocCtx
  | LoadCtx
  | StoreLCtx (e2 : expr)
  | StoreRCtx (v1 : val)
  | CasLCtx (e1 : expr)  (e2 : expr)
  | CasMCtx (v0 : val) (e2 : expr)
  | CasRCtx (v0 : val) (v1 : val).
100

101
Notation ectx := (list ectx_item).
102

103
Definition fill_item (Ki : ectx_item) (e : expr) : expr :=
104 105 106
  match Ki with
  | AppLCtx e2 => App e e2
  | AppRCtx v1 => App (of_val v1) e
107 108 109 110
  | UnOpCtx op => UnOp op e
  | BinOpLCtx op e2 => BinOp op e e2
  | BinOpRCtx op v1 => BinOp op (of_val v1) e
  | IfCtx e1 e2 => If e e1 e2
111 112 113 114 115 116
  | PairLCtx e2 => Pair e e2
  | PairRCtx v1 => Pair (of_val v1) e
  | FstCtx => Fst e
  | SndCtx => Snd e
  | InjLCtx => InjL e
  | InjRCtx => InjR e
117
  | CaseCtx x1 e1 x2 e2 => Case e x1 e1 x2 e2
118 119 120 121 122 123 124
  | AllocCtx => Alloc e
  | LoadCtx => Load e
  | StoreLCtx e2 => Store e e2
  | StoreRCtx v1 => Store (of_val v1) e
  | CasLCtx e1 e2 => Cas e e1 e2
  | CasMCtx v0 e2 => Cas (of_val v0) e e2
  | CasRCtx v0 v1 => Cas (of_val v0) (of_val v1) e
Ralf Jung's avatar
Ralf Jung committed
125
  end.
126
Definition fill (K : ectx) (e : expr) : expr := fold_right fill_item e K.
Ralf Jung's avatar
Ralf Jung committed
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
(** Substitution *)
(** We have [subst e "" v = e] to deal with anonymous binders *)
Fixpoint subst (e : expr) (x : string) (v : val) : expr :=
  match e with
  | Var y => if decide (x = y  x  "") then of_val v else Var y
  | Rec f y e => Rec f y (if decide (x  f  x  y) then subst e x v else e)
  | App e1 e2 => App (subst e1 x v) (subst e2 x v)
  | Lit l => Lit l
  | UnOp op e => UnOp op (subst e x v)
  | BinOp op e1 e2 => BinOp op (subst e1 x v) (subst e2 x v)
  | If e0 e1 e2 => If (subst e0 x v) (subst e1 x v) (subst e2 x v)
  | Pair e1 e2 => Pair (subst e1 x v) (subst e2 x v)
  | Fst e => Fst (subst e x v)
  | Snd e => Snd (subst e x v)
  | InjL e => InjL (subst e x v)
  | InjR e => InjR (subst e x v)
  | Case e0 x1 e1 x2 e2 =>
     Case (subst e0 x v)
       x1 (if decide (x  x1) then subst e1 x v else e1)
       x2 (if decide (x  x2) then subst e2 x v else e2)
  | Fork e => Fork (subst e x v)
  | Loc l => Loc l
  | Alloc e => Alloc (subst e x v)
  | Load e => Load (subst e x v)
  | Store e1 e2 => Store (subst e1 x v) (subst e2 x v)
  | Cas e0 e1 e2 => Cas (subst e0 x v) (subst e1 x v) (subst e2 x v)
  end.

156
(** The stepping relation *)
157 158
Definition un_op_eval (op : un_op) (l : base_lit) : option base_lit :=
  match op, l with
159
  | NegOp, LitBool b => Some (LitBool (negb b))
160
  | MinusUnOp, LitInt n => Some (LitInt (- n))
161 162 163 164 165
  | _, _ => None
  end.

Definition bin_op_eval (op : bin_op) (l1 l2 : base_lit) : option base_lit :=
  match op, l1, l2 with
166 167 168 169 170
  | PlusOp, LitInt n1, LitInt n2 => Some $ LitInt (n1 + n2)
  | MinusOp, LitInt n1, LitInt n2 => Some $ LitInt (n1 - n2)
  | LeOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1  n2)
  | LtOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1 < n2)
  | EqOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1 = n2)
171 172 173
  | _, _, _ => None
  end.

174
Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
175
  | BetaS f x e1 e2 v2 σ :
176
     to_val e2 = Some v2 
177 178 179
     head_step (App (Rec f x e1) e2) σ
       (subst (subst e1 f (RecV f x e1)) x v2) σ None
  | UnOpS op l l' σ :
180 181
     un_op_eval op l = Some l'  
     head_step (UnOp op (Lit l)) σ (Lit l') σ None
182
  | BinOpS op l1 l2 l' σ :
183 184 185
     bin_op_eval op l1 l2 = Some l'  
     head_step (BinOp op (Lit l1) (Lit l2)) σ (Lit l') σ None
  | IfTrueS e1 e2 σ :
Ralf Jung's avatar
Ralf Jung committed
186
     head_step (If (Lit $ LitBool true) e1 e2) σ e1 σ None
187
  | IfFalseS e1 e2 σ :
Ralf Jung's avatar
Ralf Jung committed
188
     head_step (If (Lit $ LitBool false) e1 e2) σ e2 σ None
189 190 191 192 193 194
  | FstS e1 v1 e2 v2 σ :
     to_val e1 = Some v1  to_val e2 = Some v2 
     head_step (Fst (Pair e1 e2)) σ e1 σ None
  | SndS e1 v1 e2 v2 σ :
     to_val e1 = Some v1  to_val e2 = Some v2 
     head_step (Snd (Pair e1 e2)) σ e2 σ None
195
  | CaseLS e0 v0 x1 e1 x2 e2 σ :
196
     to_val e0 = Some v0 
197 198
     head_step (Case (InjL e0) x1 e1 x2 e2) σ (subst e1 x1 v0) σ None
  | CaseRS e0 v0 x1 e1 x2 e2 σ :
199
     to_val e0 = Some v0 
200
     head_step (Case (InjR e0) x1 e1 x2 e2) σ (subst e2 x2 v0) σ None
201
  | ForkS e σ:
202
     head_step (Fork e) σ (Lit LitUnit) σ (Some e)
203 204 205 206 207 208 209 210
  | AllocS e v σ l :
     to_val e = Some v  σ !! l = None 
     head_step (Alloc e) σ (Loc l) (<[l:=v]>σ) None
  | LoadS l v σ :
     σ !! l = Some v 
     head_step (Load (Loc l)) σ (of_val v) σ None
  | StoreS l e v σ :
     to_val e = Some v  is_Some (σ !! l) 
211
     head_step (Store (Loc l) e) σ (Lit LitUnit) (<[l:=v]>σ) None
212 213 214
  | CasFailS l e1 v1 e2 v2 vl σ :
     to_val e1 = Some v1  to_val e2 = Some v2 
     σ !! l = Some vl  vl  v1 
Ralf Jung's avatar
Ralf Jung committed
215
     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool false) σ None
216 217 218
  | CasSucS l e1 v1 e2 v2 σ :
     to_val e1 = Some v1  to_val e2 = Some v2 
     σ !! l = Some v1 
Ralf Jung's avatar
Ralf Jung committed
219
     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool true) (<[l:=v2]>σ) None.
Ralf Jung's avatar
Ralf Jung committed
220

221
(** Atomic expressions *)
222
Definition atomic (e: expr) : Prop :=
223 224 225 226 227 228 229
  match e with
  | Alloc e => is_Some (to_val e)
  | Load e => is_Some (to_val e)
  | Store e1 e2 => is_Some (to_val e1)  is_Some (to_val e2)
  | Cas e0 e1 e2 => is_Some (to_val e0)  is_Some (to_val e1)  is_Some (to_val e2)
  | _ => False
  end.
230

231 232 233 234
(** Close reduction under evaluation contexts.
We could potentially make this a generic construction. *)
Inductive prim_step
    (e1 : expr) (σ1 : state) (e2 : expr) (σ2: state) (ef: option expr) : Prop :=
235
  Ectx_step K e1' e2' :
236 237 238 239 240 241
    e1 = fill K e1'  e2 = fill K e2' 
    head_step e1' σ1 e2' σ2 ef  prim_step e1 σ1 e2 σ2 ef.

(** Basic properties about the language *)
Lemma to_of_val v : to_val (of_val v) = Some v.
Proof. by induction v; simplify_option_equality. Qed.
242

243
Lemma of_to_val e v : to_val e = Some v  of_val v = e.
244
Proof.
245
  revert v; induction e; intros; simplify_option_equality; auto with f_equal.
246
Qed.
247

248 249
Instance: Inj (=) (=) of_val.
Proof. by intros ?? Hv; apply (inj Some); rewrite -!to_of_val Hv. Qed.
250

251
Instance fill_item_inj Ki : Inj (=) (=) (fill_item Ki).
252
Proof. destruct Ki; intros ???; simplify_equality'; auto with f_equal. Qed.
253

254
Instance ectx_fill_inj K : Inj (=) (=) (fill K).
255
Proof. red; induction K as [|Ki K IH]; naive_solver. Qed.
256

257 258
Lemma fill_app K1 K2 e : fill (K1 ++ K2) e = fill K1 (fill K2 e).
Proof. revert e; induction K1; simpl; auto with f_equal. Qed.
259

260
Lemma fill_val K e : is_Some (to_val (fill K e))  is_Some (to_val e).
261
Proof.
262 263
  intros [v' Hv']; revert v' Hv'.
  induction K as [|[]]; intros; simplify_option_equality; eauto.
264
Qed.
265

266 267
Lemma fill_not_val K e : to_val e = None  to_val (fill K e) = None.
Proof. rewrite !eq_None_not_Some; eauto using fill_val. Qed.
268

269 270 271
Lemma values_head_stuck e1 σ1 e2 σ2 ef :
  head_step e1 σ1 e2 σ2 ef  to_val e1 = None.
Proof. destruct 1; naive_solver. Qed.
272

273 274
Lemma values_stuck e1 σ1 e2 σ2 ef : prim_step e1 σ1 e2 σ2 ef  to_val e1 = None.
Proof. intros [??? -> -> ?]; eauto using fill_not_val, values_head_stuck. Qed.
275

276 277
Lemma atomic_not_val e : atomic e  to_val e = None.
Proof. destruct e; naive_solver. Qed.
278

279
Lemma atomic_fill K e : atomic (fill K e)  to_val e = None  K = [].
280
Proof.
281 282
  rewrite eq_None_not_Some.
  destruct K as [|[]]; naive_solver eauto using fill_val.
283
Qed.
284

285 286 287
Lemma atomic_head_step e1 σ1 e2 σ2 ef :
  atomic e1  head_step e1 σ1 e2 σ2 ef  is_Some (to_val e2).
Proof. destruct 2; simpl; rewrite ?to_of_val; naive_solver. Qed.
288

289 290
Lemma atomic_step e1 σ1 e2 σ2 ef :
  atomic e1  prim_step e1 σ1 e2 σ2 ef  is_Some (to_val e2).
291
Proof.
292 293 294
  intros Hatomic [K e1' e2' -> -> Hstep].
  assert (K = []) as -> by eauto 10 using atomic_fill, values_head_stuck.
  naive_solver eauto using atomic_head_step.
Ralf Jung's avatar
Ralf Jung committed
295
Qed.
296

297
Lemma head_ctx_step_val Ki e σ1 e2 σ2 ef :
298
  head_step (fill_item Ki e) σ1 e2 σ2 ef  is_Some (to_val e).
299
Proof. destruct Ki; inversion_clear 1; simplify_option_equality; eauto. Qed.
300

301
Lemma fill_item_no_val_inj Ki1 Ki2 e1 e2 :
302
  to_val e1 = None  to_val e2 = None 
303
  fill_item Ki1 e1 = fill_item Ki2 e2  Ki1 = Ki2.
304
Proof.
305
  destruct Ki1, Ki2; intros; try discriminate; simplify_equality';
306
    repeat match goal with
307 308
    | H : to_val (of_val _) = None |- _ => by rewrite to_of_val in H
    end; auto.
Ralf Jung's avatar
Ralf Jung committed
309
Qed.
310

311 312 313 314 315 316
(* When something does a step, and another decomposition of the same expression
has a non-val [e] in the hole, then [K] is a left sub-context of [K'] - in
other words, [e] also contains the reducible expression *)
Lemma step_by_val K K' e1 e1' σ1 e2 σ2 ef :
  fill K e1 = fill K' e1'  to_val e1 = None  head_step e1' σ1 e2 σ2 ef 
  K `prefix_of` K'.
317
Proof.
318 319 320
  intros Hfill Hred Hnval; revert K' Hfill.
  induction K as [|Ki K IH]; simpl; intros K' Hfill; auto using prefix_of_nil.
  destruct K' as [|Ki' K']; simplify_equality'.
Ralf Jung's avatar
Ralf Jung committed
321
  { exfalso; apply (eq_None_not_Some (to_val (fill K e1)));
322 323
      eauto using fill_not_val, head_ctx_step_val. }
  cut (Ki = Ki'); [naive_solver eauto using prefix_of_cons|].
324
  eauto using fill_item_no_val_inj, values_head_stuck, fill_not_val.
325
Qed.
326

327 328 329
Lemma alloc_fresh e v σ :
  let l := fresh (dom _ σ) in
  to_val e = Some v  head_step (Alloc e) σ (Loc l) (<[l:=v]>σ) None.
330
Proof. by intros; apply AllocS, (not_elem_of_dom (D:=gset _)), is_fresh. Qed.
331

332 333
Lemma subst_empty e v : subst e "" v = e.
Proof. induction e; simpl; repeat case_decide; intuition auto with f_equal. Qed.
334 335 336 337 338 339 340 341 342 343
End heap_lang.

(** Language *)
Program Canonical Structure heap_lang : language := {|
  expr := heap_lang.expr; val := heap_lang.val; state := heap_lang.state;
  of_val := heap_lang.of_val; to_val := heap_lang.to_val;
  atomic := heap_lang.atomic; prim_step := heap_lang.prim_step;
|}.
Solve Obligations with eauto using heap_lang.to_of_val, heap_lang.of_to_val,
  heap_lang.values_stuck, heap_lang.atomic_not_val, heap_lang.atomic_step.
344

345
Global Instance heap_lang_ctx K : LanguageCtx heap_lang (heap_lang.fill K).
346
Proof.
347 348
  split.
  * eauto using heap_lang.fill_not_val.
349
  * intros ????? [K' e1' e2' Heq1 Heq2 Hstep].
350
    by exists (K ++ K') e1' e2'; rewrite ?heap_lang.fill_app ?Heq1 ?Heq2.
351
  * intros e1 σ1 e2 σ2 ? Hnval [K'' e1'' e2'' Heq1 -> Hstep].
352 353
    destruct (heap_lang.step_by_val
      K K'' e1 e1'' σ1 e2'' σ2 ef) as [K' ->]; eauto.
354
    rewrite heap_lang.fill_app in Heq1; apply (inj _) in Heq1.
Ralf Jung's avatar
Ralf Jung committed
355
    exists (heap_lang.fill K' e2''); rewrite heap_lang.fill_app; split; auto.
356
    econstructor; eauto.
357
Qed.
358 359 360 361 362 363 364

Global Instance heap_lang_ctx_item Ki :
  LanguageCtx heap_lang (heap_lang.fill_item Ki).
Proof.
  change (LanguageCtx heap_lang (heap_lang.fill [Ki])).
  by apply _.
Qed.