language.v 5.34 KB
Newer Older
1
From iris.algebra Require Export ofe.
2
Set Default Proof Using "Type".
Ralf Jung's avatar
Ralf Jung committed
3

4
5
6
7
8
9
Structure language := Language {
  expr : Type;
  val : Type;
  state : Type;
  of_val : val  expr;
  to_val : expr  option val;
10
  prim_step : expr  state  expr  state  list expr  Prop;
Robbert Krebbers's avatar
Robbert Krebbers committed
11
12
  to_of_val v : to_val (of_val v) = Some v;
  of_to_val e v : to_val e = Some v  of_val v = e;
13
  val_stuck e σ e' σ' efs : prim_step e σ e' σ' efs  to_val e = None
Ralf Jung's avatar
Ralf Jung committed
14
}.
Janno's avatar
Janno committed
15
16
17
Delimit Scope expr_scope with E.
Delimit Scope val_scope with V.
Bind Scope expr_scope with expr.
18
Bind Scope val_scope with val.
19
20
21
22
23
Arguments of_val {_} _.
Arguments to_val {_} _.
Arguments prim_step {_} _ _ _ _ _.
Arguments to_of_val {_} _.
Arguments of_to_val {_} _ _ _.
24
Arguments val_stuck {_} _ _ _ _ _ _.
25

26
27
28
Canonical Structure stateC Λ := leibnizC (state Λ).
Canonical Structure valC Λ := leibnizC (val Λ).
Canonical Structure exprC Λ := leibnizC (expr Λ).
29
30

Definition cfg (Λ : language) := (list (expr Λ) * state Λ)%type.
Ralf Jung's avatar
Ralf Jung committed
31

32
33
34
35
36
37
38
39
40
41
42
Class LanguageCtx (Λ : language) (K : expr Λ  expr Λ) := {
  fill_not_val e :
    to_val e = None  to_val (K e) = None;
  fill_step e1 σ1 e2 σ2 efs :
    prim_step e1 σ1 e2 σ2 efs 
    prim_step (K e1) σ1 (K e2) σ2 efs;
  fill_step_inv e1' σ1 e2 σ2 efs :
    to_val e1' = None  prim_step (K e1') σ1 e2 σ2 efs 
     e2', e2 = K e2'  prim_step e1' σ1 e2' σ2 efs
}.

Robbert Krebbers's avatar
Robbert Krebbers committed
43
44
45
Instance language_ctx_id Λ : LanguageCtx Λ id.
Proof. constructor; naive_solver. Qed.

46
Section language.
47
48
  Context {Λ : language}.
  Implicit Types v : val Λ.
Ralf Jung's avatar
Ralf Jung committed
49

50
  Definition reducible (e : expr Λ) (σ : state Λ) :=
51
     e' σ' efs, prim_step e σ e' σ' efs.
52
  Definition irreducible (e : expr Λ) (σ : state Λ) :=
53
54
     e' σ' efs, ¬prim_step e σ e' σ' efs.

Ralf Jung's avatar
Ralf Jung committed
55
56
57
58
59
60
  (* This (weak) form of atomicity is enough to open invariants when WP ensures
     safety, i.e., programs never can get stuck.  We have an example in
     lambdaRust of an expression that is atomic in this sense, but not in the
     stronger sense defined below, and we have to be able to open invariants
     around that expression.  See `CasStuckS` in
     [lambdaRust](https://gitlab.mpi-sws.org/FP/LambdaRust-coq/blob/master/theories/lang/lang.v). *)
Robbert Krebbers's avatar
Robbert Krebbers committed
61
62
  Class Atomic (e : expr Λ) : Prop :=
    atomic σ e' σ' efs : prim_step e σ e' σ' efs  irreducible e' σ'.
63

Ralf Jung's avatar
Ralf Jung committed
64
65
66
67
  (* To open invariants with a WP that does not ensure safety, we need a
     stronger form of atomicity.  With the above definition, in case `e` reduces
     to a stuck non-value, there is no proof that the invariants have been
     established again. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
68
69
  Class StronglyAtomic (e : expr Λ) : Prop :=
    strongly_atomic σ e' σ' efs : prim_step e σ e' σ' efs  is_Some (to_val e').
70

71
  Inductive step (ρ1 ρ2 : cfg Λ) : Prop :=
72
    | step_atomic e1 σ1 e2 σ2 efs t1 t2 :
73
       ρ1 = (t1 ++ e1 :: t2, σ1) 
74
75
       ρ2 = (t1 ++ e2 :: t2 ++ efs, σ2) 
       prim_step e1 σ1 e2 σ2 efs 
76
77
       step ρ1 ρ2.

Robbert Krebbers's avatar
Robbert Krebbers committed
78
79
  Lemma of_to_val_flip v e : of_val v = e  to_val e = Some v.
  Proof. intros <-. by rewrite to_of_val. Qed.
80
81
82

  Lemma not_reducible e σ : ¬reducible e σ  irreducible e σ.
  Proof. unfold reducible, irreducible. naive_solver. Qed.
83
  Lemma reducible_not_val e σ : reducible e σ  to_val e = None.
84
  Proof. intros (?&?&?&?); eauto using val_stuck. Qed.
85
  Global Instance of_val_inj : Inj (=) (=) (@of_val Λ).
86
  Proof. by intros v v' Hv; apply (inj Some); rewrite -!to_of_val Hv. Qed.
Ralf Jung's avatar
Ralf Jung committed
87

Robbert Krebbers's avatar
Robbert Krebbers committed
88
89
  Lemma strongly_atomic_atomic e : StronglyAtomic e  Atomic e.
  Proof. unfold StronglyAtomic, Atomic. eauto using val_irreducible. Qed.
90

91
92
93
94
95
96
97
98
99
  Lemma reducible_fill `{LanguageCtx Λ K} e σ :
    to_val e = None  reducible (K e) σ  reducible e σ.
  Proof.
    intros ? (e'&σ'&efs&Hstep); unfold reducible.
    apply fill_step_inv in Hstep as (e2' & _ & Hstep); eauto.
  Qed.
  Lemma irreducible_fill `{LanguageCtx Λ K} e σ :
    to_val e = None  irreducible e σ  irreducible (K e) σ.
  Proof. rewrite -!not_reducible. naive_solver eauto using reducible_fill. Qed.
100
101
102
103
104
105
106
107
108
109

  Lemma step_Permutation (t1 t1' t2 : list (expr Λ)) σ1 σ2 :
    t1  t1'  step (t1,σ1) (t2,σ2)   t2', t2  t2'  step (t1',σ1) (t2',σ2).
  Proof.
    intros Ht [e1 σ1' e2 σ2' efs tl tr ?? Hstep]; simplify_eq/=.
    move: Ht; rewrite -Permutation_middle (symmetry_iff ()).
    intros (tl'&tr'&->&Ht)%Permutation_cons_inv.
    exists (tl' ++ e2 :: tr' ++ efs); split; [|by econstructor].
    by rewrite -!Permutation_middle !assoc_L Ht.
  Qed.
Dan Frumin's avatar
Dan Frumin committed
110
111
112
113
114
115
116
117

  Class PureExec (P : Prop) (e1 e2 : expr Λ) := {
    pure_exec_safe σ :
      P  reducible e1 σ;
    pure_exec_puredet σ1 e2' σ2 efs :
      P  prim_step e1 σ1 e2' σ2 efs  σ1 = σ2  e2 = e2'  efs = [];
  }.

118
  Lemma hoist_pred_pure_exec (P : Prop) (e1 e2 : expr Λ) :
Dan Frumin's avatar
Dan Frumin committed
119
120
121
    (P  PureExec True e1 e2) 
    PureExec P e1 e2.
  Proof. intros HPE. split; intros; eapply HPE; eauto. Qed.
122
123
124
125

  (* This is a family of frequent assumptions for PureExec *)
  Class IntoVal (e : expr Λ) (v : val Λ) :=
    into_val : to_val e = Some v.
Robbert Krebbers's avatar
Robbert Krebbers committed
126
127
128
129
130
131
132
133
134

  Class AsVal (e : expr Λ) := as_val : is_Some (to_val e).
  (* There is no instance [IntoVal → AsVal] as often one can solve [AsVal] more
  efficiently since no witness has to be computed. *)
  Global Instance as_vals_of_val vs : TCForall AsVal (of_val <$> vs).
  Proof.
    apply TCForall_Forall, Forall_fmap, Forall_true=> v.
    rewrite /AsVal /= to_of_val; eauto.
  Qed.
135
End language.