tactics.v 11.1 KB
Newer Older
1
From iris.heap_lang Require Export lang.
2
Set Default Proof Using "Type".
3 4
Import heap_lang.

Robbert Krebbers's avatar
Robbert Krebbers committed
5 6 7 8
(** We define an alternative representation of expressions in which the
embedding of values and closed expressions is explicit. By reification of
expressions into this type we can implementation substitution, closedness
checking, atomic checking, and conversion into values, by computation. *)
9 10
Module W.
Inductive expr :=
11 12
  (* Value together with the original expression *)
  | Val (v : val) (e : heap_lang.expr) (H : to_val e = Some v)
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
  | ClosedExpr (e : heap_lang.expr) `{!Closed [] e}
  (* Base lambda calculus *)
  | Var (x : string)
  | Rec (f x : binder) (e : expr)
  | App (e1 e2 : expr)
  (* 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)
  (* Products *)
  | Pair (e1 e2 : expr)
  | Fst (e : expr)
  | Snd (e : expr)
  (* Sums *)
  | InjL (e : expr)
  | InjR (e : expr)
  | Case (e0 : expr) (e1 : expr) (e2 : expr)
  (* Concurrency *)
  | Fork (e : expr)
  (* Heap *)
  | Alloc (e : expr)
  | Load (e : expr)
  | Store (e1 : expr) (e2 : expr)
  | CAS (e0 : expr) (e1 : expr) (e2 : expr).

Fixpoint to_expr (e : expr) : heap_lang.expr :=
  match e with
41
  | Val v e' _ => e'
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
  | ClosedExpr e _ => e
  | Var x => heap_lang.Var x
  | Rec f x e => heap_lang.Rec f x (to_expr e)
  | App e1 e2 => heap_lang.App (to_expr e1) (to_expr e2)
  | Lit l => heap_lang.Lit l
  | UnOp op e => heap_lang.UnOp op (to_expr e)
  | BinOp op e1 e2 => heap_lang.BinOp op (to_expr e1) (to_expr e2)
  | If e0 e1 e2 => heap_lang.If (to_expr e0) (to_expr e1) (to_expr e2)
  | Pair e1 e2 => heap_lang.Pair (to_expr e1) (to_expr e2)
  | Fst e => heap_lang.Fst (to_expr e)
  | Snd e => heap_lang.Snd (to_expr e)
  | InjL e => heap_lang.InjL (to_expr e)
  | InjR e => heap_lang.InjR (to_expr e)
  | Case e0 e1 e2 => heap_lang.Case (to_expr e0) (to_expr e1) (to_expr e2)
  | Fork e => heap_lang.Fork (to_expr e)
  | Alloc e => heap_lang.Alloc (to_expr e)
  | Load e => heap_lang.Load (to_expr e)
  | Store e1 e2 => heap_lang.Store (to_expr e1) (to_expr e2)
  | CAS e0 e1 e2 => heap_lang.CAS (to_expr e0) (to_expr e1) (to_expr e2)
  end.

Ltac of_expr e :=
  lazymatch e with
  | heap_lang.Var ?x => constr:(Var x)
  | heap_lang.Rec ?f ?x ?e => let e := of_expr e in constr:(Rec f x e)
  | heap_lang.App ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(App e1 e2)
  | heap_lang.Lit ?l => constr:(Lit l)
  | heap_lang.UnOp ?op ?e => let e := of_expr e in constr:(UnOp op e)
  | heap_lang.BinOp ?op ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(BinOp op e1 e2)
  | heap_lang.If ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(If e0 e1 e2)
  | heap_lang.Pair ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(Pair e1 e2)
  | heap_lang.Fst ?e => let e := of_expr e in constr:(Fst e)
  | heap_lang.Snd ?e => let e := of_expr e in constr:(Snd e)
  | heap_lang.InjL ?e => let e := of_expr e in constr:(InjL e)
  | heap_lang.InjR ?e => let e := of_expr e in constr:(InjR e)
  | heap_lang.Case ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(Case e0 e1 e2)
  | heap_lang.Fork ?e => let e := of_expr e in constr:(Fork e)
  | heap_lang.Alloc ?e => let e := of_expr e in constr:(Alloc e)
  | heap_lang.Load ?e => let e := of_expr e in constr:(Load e)
  | heap_lang.Store ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(Store e1 e2)
  | heap_lang.CAS ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(CAS e0 e1 e2)
  | to_expr ?e => e
94 95 96 97 98
  | of_val ?v => constr:(Val v (of_val v) (to_of_val v))
  | _ => match goal with
         | H : to_val e = Some ?v |- _ => constr:(Val v e H)
         | H : Closed [] e |- _ => constr:(@ClosedExpr e H)
         end
99 100 101 102
  end.

Fixpoint is_closed (X : list string) (e : expr) : bool :=
  match e with
103
  | Val _ _ _ | ClosedExpr _ _ => true
104 105 106 107 108 109 110 111 112 113 114 115 116
  | Var x => bool_decide (x  X)
  | Rec f x e => is_closed (f :b: x :b: X) e
  | Lit _ => true
  | UnOp _ e | Fst e | Snd e | InjL e | InjR e | Fork e | Alloc e | Load e =>
     is_closed X e
  | App e1 e2 | BinOp _ e1 e2 | Pair e1 e2 | Store e1 e2 =>
     is_closed X e1 && is_closed X e2
  | If e0 e1 e2 | Case e0 e1 e2 | CAS e0 e1 e2 =>
     is_closed X e0 && is_closed X e1 && is_closed X e2
  end.
Lemma is_closed_correct X e : is_closed X e  heap_lang.is_closed X (to_expr e).
Proof.
  revert X.
117
  induction e; naive_solver eauto using is_closed_to_val, is_closed_weaken_nil.
118 119
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
120 121 122 123
(* We define [to_val (ClosedExpr _)] to be [None] since [ClosedExpr]
constructors are only generated for closed expressions of which we know nothing
about apart from being closed. Notice that the reverse implication of
[to_val_Some] thus does not hold. *)
124 125
Fixpoint to_val (e : expr) : option val :=
  match e with
126
  | Val v _ _ => Some v
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
  | Rec f x e =>
     if decide (is_closed (f :b: x :b: []) e) is left H
     then Some (@RecV f x (to_expr e) (is_closed_correct _ _ H)) else None
  | Lit l => Some (LitV l)
  | 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
  | _ => None
  end.
Lemma to_val_Some e v :
  to_val e = Some v  heap_lang.to_val (to_expr e) = Some v.
Proof.
  revert v. induction e; intros; simplify_option_eq; rewrite ?to_of_val; auto.
  - do 2 f_equal. apply proof_irrel.
  - exfalso. unfold Closed in *; eauto using is_closed_correct.
Qed.
143 144 145
Lemma to_val_is_Some e :
  is_Some (to_val e)  is_Some (heap_lang.to_val (to_expr e)).
Proof. intros [v ?]; exists v; eauto using to_val_Some. Qed.
146 147 148

Fixpoint subst (x : string) (es : expr) (e : expr)  : expr :=
  match e with
149
  | Val v e H => Val v e H
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
  | ClosedExpr e H => @ClosedExpr e H
  | Var y => if decide (x = y) then es else Var y
  | Rec f y e =>
     Rec f y $ if decide (BNamed x  f  BNamed x  y) then subst x es e else e
  | App e1 e2 => App (subst x es e1) (subst x es e2)
  | Lit l => Lit l
  | UnOp op e => UnOp op (subst x es e)
  | BinOp op e1 e2 => BinOp op (subst x es e1) (subst x es e2)
  | If e0 e1 e2 => If (subst x es e0) (subst x es e1) (subst x es e2)
  | Pair e1 e2 => Pair (subst x es e1) (subst x es e2)
  | Fst e => Fst (subst x es e)
  | Snd e => Snd (subst x es e)
  | InjL e => InjL (subst x es e)
  | InjR e => InjR (subst x es e)
  | Case e0 e1 e2 => Case (subst x es e0) (subst x es e1) (subst x es e2)
  | Fork e => Fork (subst x es e)
  | Alloc e => Alloc (subst x es e)
  | Load e => Load (subst x es e)
  | Store e1 e2 => Store (subst x es e1) (subst x es e2)
  | CAS e0 e1 e2 => CAS (subst x es e0) (subst x es e1) (subst x es e2)
  end.
Lemma to_expr_subst x er e :
  to_expr (subst x er e) = heap_lang.subst x (to_expr er) (to_expr e).
Proof.
  induction e; simpl; repeat case_decide;
175
    f_equal; eauto using subst_is_closed_nil, is_closed_to_val, eq_sym.
176
Qed.
177

Robbert Krebbers's avatar
Robbert Krebbers committed
178
Definition is_atomic (e : expr) :=
179 180 181 182 183 184
  match e with
  | Alloc e => bool_decide (is_Some (to_val e))
  | Load e => bool_decide (is_Some (to_val e))
  | Store e1 e2 => bool_decide (is_Some (to_val e1)  is_Some (to_val e2))
  | CAS e0 e1 e2 =>
     bool_decide (is_Some (to_val e0)  is_Some (to_val e1)  is_Some (to_val e2))
Ralf Jung's avatar
Ralf Jung committed
185
  | Fork _ => true
186 187 188 189
  (* Make "skip" atomic *)
  | App (Rec _ _ (Lit _)) (Lit _) => true
  | _ => false
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
190
Lemma is_atomic_correct e : is_atomic e  Atomic (to_expr e).
191
Proof.
192
  intros He. apply ectx_language_atomic.
193 194 195 196 197
  - intros σ e' σ' ef Hstep; simpl in *.
    apply language.val_irreducible; revert Hstep.
    destruct e=> //=; repeat (simplify_eq/=; case_match=>//);
      inversion 1; simplify_eq/=; rewrite ?to_of_val; eauto.
    unfold subst'; repeat (simplify_eq/=; case_match=>//); eauto.
198
  - apply ectxi_language_sub_values=> /= Ki e' Hfill.
199 200
    destruct e=> //; destruct Ki; repeat (simplify_eq/=; case_match=>//);
      naive_solver eauto using to_val_is_Some.
201
Qed.
202 203 204 205 206 207 208 209 210 211 212
End W.

Ltac solve_closed :=
  match goal with
  | |- Closed ?X ?e =>
     let e' := W.of_expr e in change (Closed X (W.to_expr e'));
     apply W.is_closed_correct; vm_compute; exact I
  end.
Hint Extern 0 (Closed _ _) => solve_closed : typeclass_instances.

Ltac solve_to_val :=
Robbert Krebbers's avatar
Robbert Krebbers committed
213
  rewrite /AsVal /IntoVal;
214
  try match goal with
215 216
  | |- context E [language.to_val ?e] =>
     let X := context E [to_val e] in change X
217 218 219 220
  end;
  match goal with
  | |- to_val ?e = Some ?v =>
     let e' := W.of_expr e in change (to_val (W.to_expr e') = Some v);
221
     apply W.to_val_Some; simpl; unfold W.to_expr; reflexivity
222 223 224
  | |- is_Some (to_val ?e) =>
     let e' := W.of_expr e in change (is_Some (to_val (W.to_expr e')));
     apply W.to_val_is_Some, (bool_decide_unpack _); vm_compute; exact I
225
  end.
226
Hint Extern 10 (IntoVal _ _) => solve_to_val : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
227
Hint Extern 10 (AsVal _) => solve_to_val : typeclass_instances.
228

229 230
Ltac solve_atomic :=
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
231 232 233
  | |- Atomic ?e =>
     let e' := W.of_expr e in change (Atomic (W.to_expr e'));
     apply W.is_atomic_correct; vm_compute; exact I
234
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
235
Hint Extern 10 (Atomic _) => solve_atomic : typeclass_instances.
236

237 238
(** Substitution *)
Ltac simpl_subst :=
Robbert Krebbers's avatar
Robbert Krebbers committed
239
  simpl;
240 241 242 243 244 245 246 247 248 249
  repeat match goal with
  | |- context [subst ?x ?er ?e] =>
      let er' := W.of_expr er in let e' := W.of_expr e in
      change (subst x er e) with (subst x (W.to_expr er') (W.to_expr e'));
      rewrite <-(W.to_expr_subst x); simpl (* ssr rewrite is slower *)
  end;
  unfold W.to_expr.
Arguments W.to_expr : simpl never.
Arguments subst : simpl never.

250 251 252
(** The tactic [reshape_expr e tac] decomposes the expression [e] into an
evaluation context [K] and a subexpression [e']. It calls the tactic [tac K e']
for each possible decomposition until [tac] succeeds. *)
253 254
Ltac reshape_val e tac :=
  let rec go e :=
255
  lazymatch e with
256 257 258 259
  | of_val ?v => v
  | Rec ?f ?x ?e => constr:(RecV f x e)
  | Lit ?l => constr:(LitV l)
  | Pair ?e1 ?e2 =>
260 261 262
    let v1 := go e1 in let v2 := go e2 in constr:(PairV v1 v2)
  | InjL ?e => let v := go e in constr:(InjLV v)
  | InjR ?e => let v := go e in constr:(InjRV v)
263
  end in let v := go e in tac v.
264

265 266 267
Ltac reshape_expr e tac :=
  let rec go K e :=
  match e with
268
  | _ => tac K e
269 270 271
  | App ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (AppRCtx v1 :: K) e2)
  | App ?e1 ?e2 => go (AppLCtx e2 :: K) e1
  | UnOp ?op ?e => go (UnOpCtx op :: K) e
272
  | BinOp ?op ?e1 ?e2 =>
273 274
     reshape_val e1 ltac:(fun v1 => go (BinOpRCtx op v1 :: K) e2)
  | BinOp ?op ?e1 ?e2 => go (BinOpLCtx op e2 :: K) e1
275
  | If ?e0 ?e1 ?e2 => go (IfCtx e1 e2 :: K) e0
276 277
  | Pair ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (PairRCtx v1 :: K) e2)
  | Pair ?e1 ?e2 => go (PairLCtx e2 :: K) e1
278 279 280 281 282 283 284
  | Fst ?e => go (FstCtx :: K) e
  | Snd ?e => go (SndCtx :: K) e
  | InjL ?e => go (InjLCtx :: K) e
  | InjR ?e => go (InjRCtx :: K) e
  | Case ?e0 ?e1 ?e2 => go (CaseCtx e1 e2 :: K) e0
  | Alloc ?e => go (AllocCtx :: K) e
  | Load ?e => go (LoadCtx :: K) e
285 286
  | Store ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (StoreRCtx v1 :: K) e2)
  | Store ?e1 ?e2 => go (StoreLCtx e2 :: K) e1
287
  | CAS ?e0 ?e1 ?e2 => reshape_val e0 ltac:(fun v0 => first
288 289
     [ reshape_val e1 ltac:(fun v1 => go (CasRCtx v0 v1 :: K) e2)
     | go (CasMCtx v0 e2 :: K) e1 ])
290
  | CAS ?e0 ?e1 ?e2 => go (CasLCtx e1 e2 :: K) e0
291
  end in go (@nil ectx_item) e.