tactics.v 11.4 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
  | 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)
37 38
  | CAS (e0 : expr) (e1 : expr) (e2 : expr)
  | FAA (e1 : expr) (e2 : expr).
39 40 41

Fixpoint to_expr (e : expr) : heap_lang.expr :=
  match e with
42
  | Val v e' _ => e'
43
  | ClosedExpr e => e
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
  | 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)
62
  | FAA e1 e2 => heap_lang.FAA (to_expr e1) (to_expr e2)
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
  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)
95 96
  | heap_lang.FAA ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(FAA e1 e2)
97
  | to_expr ?e => e
98 99 100 101 102
  | 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
103 104 105 106
  end.

Fixpoint is_closed (X : list string) (e : expr) : bool :=
  match e with
107
  | Val _ _ _ | ClosedExpr _ => true
108 109 110 111 112
  | 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
113
  | App e1 e2 | BinOp _ e1 e2 | Pair e1 e2 | Store e1 e2 | FAA e1 e2 =>
114 115 116 117 118 119 120
     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.
121
  induction e; naive_solver eauto using is_closed_to_val, is_closed_weaken_nil.
122 123
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
124 125 126 127
(* 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. *)
128 129
Fixpoint to_val (e : expr) : option val :=
  match e with
130
  | Val v _ _ => Some v
131 132 133 134 135 136 137 138 139 140
  | 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 :
Ralf Jung's avatar
Ralf Jung committed
141
  to_val e = Some v  heap_lang.of_val v = W.to_expr e.
142
Proof.
143
  revert v. induction e; intros; simplify_option_eq; try f_equal; auto using of_to_val.
144
Qed.
145
Lemma to_val_is_Some e :
Ralf Jung's avatar
Ralf Jung committed
146
  is_Some (to_val e)   v, heap_lang.of_val v = to_expr e.
147
Proof. intros [v ?]; exists v; eauto using to_val_Some. Qed.
148 149 150

Fixpoint subst (x : string) (es : expr) (e : expr)  : expr :=
  match e with
151
  | Val v e H => Val v e H
152
  | ClosedExpr e => ClosedExpr e
153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171
  | 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)
172
  | FAA e1 e2 => FAA (subst x es e1) (subst x es e2)
173 174 175 176 177
  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;
178
    f_equal; eauto using subst_is_closed_nil, is_closed_to_val, eq_sym.
179
Qed.
180

Robbert Krebbers's avatar
Robbert Krebbers committed
181
Definition is_atomic (e : expr) :=
182 183 184 185 186 187
  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))
188
  | FAA e1 e2 => bool_decide (is_Some (to_val e1)  is_Some (to_val e2))
Ralf Jung's avatar
Ralf Jung committed
189
  | Fork _ => true
190 191 192 193
  (* Make "skip" atomic *)
  | App (Rec _ _ (Lit _)) (Lit _) => true
  | _ => false
  end.
194
Lemma is_atomic_correct s e : is_atomic e  Atomic s (to_expr e).
195
Proof.
196
  intros He. apply strongly_atomic_atomic, ectx_language_atomic.
197
  - intros σ e' σ' ef Hstep; simpl in *. revert Hstep.
198 199 200
    destruct e=> //=; repeat (simplify_eq/=; case_match=>//);
      inversion 1; simplify_eq/=; rewrite ?to_of_val; eauto.
    unfold subst'; repeat (simplify_eq/=; case_match=>//); eauto.
201
  - apply ectxi_language_sub_redexes_are_values=> /= Ki e' Hfill.
Ralf Jung's avatar
Ralf Jung committed
202 203
    destruct e=> //; destruct Ki; repeat (simplify_eq/=; case_match=>//); try
      naive_solver eauto using as_val_is_Some, to_val_is_Some.
204
Qed.
205 206 207 208 209 210 211 212 213 214
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.

Robbert Krebbers's avatar
Robbert Krebbers committed
215
Ltac solve_into_val :=
216
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
217
  | |- IntoVal ?e ?v =>
218
     let e' := W.of_expr e in change (of_val v = W.to_expr e');
219
     apply W.to_val_Some; simpl; unfold W.to_expr; reflexivity
Robbert Krebbers's avatar
Robbert Krebbers committed
220 221 222 223 224 225
  end.
Hint Extern 10 (IntoVal _ _) => solve_into_val : typeclass_instances.

Ltac solve_as_val :=
  match goal with
  | |- AsVal ?e =>
226
     let e' := W.of_expr e in change ( v, of_val v = W.to_expr e');
227
     apply W.to_val_is_Some, (bool_decide_unpack _); vm_compute; exact I
228
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
229
Hint Extern 10 (AsVal _) => solve_as_val : typeclass_instances.
230

231 232
Ltac solve_atomic :=
  match goal with
233 234
  | |- Atomic ?s ?e =>
     let e' := W.of_expr e in change (Atomic s (W.to_expr e'));
235
     apply W.is_atomic_correct; vm_compute; exact I
236
  end.
237
Hint Extern 10 (Atomic _ _) => solve_atomic : typeclass_instances.
238

239 240
(** Substitution *)
Ltac simpl_subst :=
Robbert Krebbers's avatar
Robbert Krebbers committed
241
  simpl;
242 243 244 245 246 247 248 249 250 251
  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.

252 253 254
(** 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. *)
255 256
Ltac reshape_val e tac :=
  let rec go e :=
257
  lazymatch e with
258 259 260 261
  | of_val ?v => v
  | Rec ?f ?x ?e => constr:(RecV f x e)
  | Lit ?l => constr:(LitV l)
  | Pair ?e1 ?e2 =>
262 263 264
    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)
265
  end in let v := go e in tac v.
266

267 268 269
Ltac reshape_expr e tac :=
  let rec go K e :=
  match e with
270
  | _ => tac K e
271 272 273
  | 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
274
  | BinOp ?op ?e1 ?e2 =>
275 276
     reshape_val e1 ltac:(fun v1 => go (BinOpRCtx op v1 :: K) e2)
  | BinOp ?op ?e1 ?e2 => go (BinOpLCtx op e2 :: K) e1
277
  | If ?e0 ?e1 ?e2 => go (IfCtx e1 e2 :: K) e0
278 279
  | Pair ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (PairRCtx v1 :: K) e2)
  | Pair ?e1 ?e2 => go (PairLCtx e2 :: K) e1
280 281 282 283 284 285 286
  | 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
287 288
  | Store ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (StoreRCtx v1 :: K) e2)
  | Store ?e1 ?e2 => go (StoreLCtx e2 :: K) e1
289
  | CAS ?e0 ?e1 ?e2 => reshape_val e0 ltac:(fun v0 => first
290 291
     [ reshape_val e1 ltac:(fun v1 => go (CasRCtx v0 v1 :: K) e2)
     | go (CasMCtx v0 e2 :: K) e1 ])
292
  | CAS ?e0 ?e1 ?e2 => go (CasLCtx e1 e2 :: K) e0
293 294
  | FAA ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (FaaRCtx v1 :: K) e2)
  | FAA ?e1 ?e2 => go (FaaLCtx e2 :: K) e1
295
  end in go (@nil ectx_item) e.