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frap/EvaluationContexts_template.v
Adam Chlipala ac0a15e9f2 Rename typing relations from hasty to has_ty
2023-03-19 11:59:34 -04:00

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Require Import Frap.
(** * Evaluation Contexts for Lambda Calculus *)
Module Stlc.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp).
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
end.
(* Here's the first difference from last chapter. *)
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context.
(* Again, note how two of the rules include [value] premises. *)
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e').
(* Small-step, call-by-value evaluation, using our evaluation contexts *)
(* First: the primitive reductions *)
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2)).
(* Then: running them in context *)
Inductive step : exp -> exp -> Prop :=
| StepRule : forall C e1 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step0 e1 e2
-> step e1' e2'.
(* It's easy to wrap everything as a transition system. *)
Definition trsys_of (e : exp) := {|
Initial := {e};
Step := step
|}.
(* Typing details are the same as last chapter. *)
Inductive type :=
| Nat (* Numbers *)
| Fun (dom ran : type) (* Functions *).
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty G (Var x) t
| HtConst : forall G n,
has_ty G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty G e1 Nat
-> has_ty G e2 Nat
-> has_ty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty (G $+ (x, t1)) e1 t2
-> has_ty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty G e1 (Fun t1 t2)
-> has_ty G e2 t1
-> has_ty G (App e1 e2) t2.
Local Hint Constructors value plug step0 step has_ty : core.
(** * Now we adapt the automated proof of type soundness. *)
Ltac t0 := match goal with
| [ H : ex _ |- _ ] => invert H
| [ H : _ /\ _ |- _ ] => invert H
| [ |- context[?x ==v ?y] ] => cases (x ==v y)
| [ H : Some _ = Some _ |- _ ] => invert H
| [ H : step _ _ |- _ ] => invert H
| [ H : step0 _ _ |- _ ] => invert1 H
| [ H : has_ty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
| [ H : has_ty _ _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => invert1 H
end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
Lemma progress : forall e t,
has_ty $0 e t
-> value e
\/ (exists e' : exp, step e e').
Proof.
induct 1; t.
Qed.
Lemma weakening_override : forall (G G' : fmap var type) x t,
(forall x' t', G $? x' = Some t' -> G' $? x' = Some t')
-> (forall x' t', G $+ (x, t) $? x' = Some t'
-> G' $+ (x, t) $? x' = Some t').
Proof.
simplify.
cases (x ==v x'); simplify; eauto.
Qed.
Local Hint Resolve weakening_override : core.
Lemma weakening : forall G e t,
has_ty G e t
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
-> has_ty G' e t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve weakening : core.
Lemma has_ty_change : forall G e t,
has_ty G e t
-> forall G', G' = G
-> has_ty G' e t.
Proof.
t.
Qed.
Local Hint Resolve has_ty_change : core.
Lemma substitution : forall G x t' e t e',
has_ty (G $+ (x, t')) e t
-> has_ty $0 e' t'
-> has_ty G (subst e' x e) t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve substitution : core.
Lemma preservation0 : forall e1 e2,
step0 e1 e2
-> forall t, has_ty $0 e1 t
-> has_ty $0 e2 t.
Proof.
invert 1; t.
Qed.
Local Hint Resolve preservation0 : core.
Lemma preservation' : forall C e1 e1',
plug C e1 e1'
-> forall e2 e2' t, plug C e2 e2'
-> step0 e1 e2
-> has_ty $0 e1' t
-> has_ty $0 e2' t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve preservation' : core.
Lemma preservation : forall e1 e2,
step e1 e2
-> forall t, has_ty $0 e1 t
-> has_ty $0 e2 t.
Proof.
invert 1; t.
Qed.
Local Hint Resolve progress preservation : core.
Theorem safety : forall e t, has_ty $0 e t
-> invariantFor (trsys_of e)
(fun e' => value e'
\/ exists e'', step e' e'').
Proof.
simplify.
apply invariant_weaken with (invariant1 := fun e' => has_ty $0 e' t); eauto.
apply invariant_induction; simplify; eauto; equality.
Qed.
(* It may not be obvious that this way of defining the semantics gives us a
* unique evaluation sequence for every well-typed program. Let's prove
* it. *)
Lemma plug_not_value : forall C e v,
value v
-> plug C e v
-> C = Hole /\ e = v.
Proof.
invert 1; invert 1; auto.
Qed.
Lemma step0_value : forall v e,
value v
-> step0 v e
-> False.
Proof.
invert 1; invert 1.
Qed.
Lemma plug_det : forall C e1 e2 e1' f1 f1',
step0 e1 e1'
-> step0 f1 f1'
-> plug C e1 e2
-> forall C', plug C' f1 e2
-> C = C' /\ e1 = f1.
Proof.
induct 3; invert 1;
repeat match goal with
| [ H : step0 _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => eapply plug_not_value in H; [ | solve [ eauto ] ];
propositional; subst
| [ IH : step0 _ _ -> _, H : plug _ _ _ |- _ ] => eapply IH in H; [ | solve [ auto ] ];
equality
| [ _ : value ?v, _ : step0 ?v _ |- _ ] => exfalso; eapply step0_value; eauto
end; equality.
Qed.
Lemma step0_det : forall e e', step0 e e'
-> forall e'', step0 e e''
-> e' = e''.
Proof.
invert 1; invert 1; auto.
Qed.
Lemma plug_func : forall C e e1,
plug C e e1
-> forall e2, plug C e e2
-> e1 = e2.
Proof.
induct 1; invert 1; auto; f_equal; auto.
Qed.
Theorem deterministic : forall e e', step e e'
-> forall e'', step e e''
-> e' = e''.
Proof.
invert 1; invert 1.
assert (C = C0 /\ e1 = e0) by (eapply plug_det; eassumption).
propositional; subst.
assert (e2 = e3) by (eapply step0_det; eassumption).
subst.
eapply plug_func; eassumption.
Qed.
End Stlc.
(** * Pairs *)
Module StlcPairs.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
| Pair (e1 e2 : exp)
| Fst (e1 : exp)
| Snd (e2 : exp).
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1)
(* A pair of values is a value. (Now this relation finally becomes
* recursive.) *)
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
(* Some bureaucratic work here to add predictable cases *)
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
| Fst e2' => Fst (subst e1 x e2')
| Snd e2' => Snd (subst e1 x e2')
end.
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context
(* Two new context kinds, indicating left-to-right evaluation order for
* pairs *)
| Pair1 : context -> exp -> context
| Pair2 : exp -> context -> context
(* And similar for projections *)
| Fst1 : context -> context
| Snd1 : context -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e')
(* Our new plugging rules *)
| PlugPair1 : forall e e' C e2,
plug C e e'
-> plug (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall e e' C,
plug C e e'
-> plug (Fst1 C) e (Fst e')
| PlugSnd1 : forall e e' C,
plug C e e'
-> plug (Snd1 C) e (Snd e').
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
(* Reducing projections *)
| FstPair : forall v1 v2,
value v1
-> value v2
-> step0 (Fst (Pair v1 v2)) v1
| SndPair : forall v1 v2,
value v1
-> value v2
-> step0 (Snd (Pair v1 v2)) v2.
Inductive step : exp -> exp -> Prop :=
| StepRule : forall C e1 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step0 e1 e2
-> step e1' e2'.
Definition trsys_of (e : exp) := {|
Initial := {e};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type) (* "Prod" for "product," as in Cartesian product *).
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty G (Var x) t
| HtConst : forall G n,
has_ty G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty G e1 Nat
-> has_ty G e2 Nat
-> has_ty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty (G $+ (x, t1)) e1 t2
-> has_ty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty G e1 (Fun t1 t2)
-> has_ty G e2 t1
-> has_ty G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
has_ty G e1 t1
-> has_ty G e2 t2
-> has_ty G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Snd e1) t2.
(* Let's copy and paste the type-soundness proof from above and adapt it. *)
End StlcPairs.
(** * Variants *)
Print sum.
Module StlcSums.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
| Pair (e1 e2 : exp)
| Fst (e1 : exp)
| Snd (e2 : exp)
(* New cases: *)
| Inl (e1 : exp)
| Inr (e2 : exp)
| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp).
(* The last one roughly means "match e' with inl x1 => e1 | inr x2 => e2". *)
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1)
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
| VInl : forall v, value v -> value (Inl v)
| VInr : forall v, value v -> value (Inr v).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
| Fst e2' => Fst (subst e1 x e2')
| Snd e2' => Snd (subst e1 x e2')
(* Some bureaucratic work here to add predictable cases *)
| Inl e2' => Inl (subst e1 x e2')
| Inr e2' => Inr (subst e1 x e2')
| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
x1 (if x1 ==v x then e21 else subst e1 x e21)
x2 (if x2 ==v x then e22 else subst e1 x e22)
end.
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context
| Pair1 : context -> exp -> context
| Pair2 : exp -> context -> context
| Fst1 : context -> context
| Snd1 : context -> context
(* New cases: *)
| Inl1 : context -> context
| Inr1 : context -> context
| Match1 : context -> var -> exp -> var -> exp -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e')
| PlugPair1 : forall e e' C e2,
plug C e e'
-> plug (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall e e' C,
plug C e e'
-> plug (Fst1 C) e (Fst e')
| PlugSnd1 : forall e e' C,
plug C e e'
-> plug (Snd1 C) e (Snd e')
(* Our new plugging rules *)
| PlugInl1 : forall e e' C,
plug C e e'
-> plug (Inl1 C) e (Inl e')
| PlugInr1 : forall e e' C,
plug C e e'
-> plug (Inr1 C) e (Inr e')
| PluMatch1 : forall e e' C x1 e1 x2 e2,
plug C e e'
-> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2).
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
| FstPair : forall v1 v2,
value v1
-> value v2
-> step0 (Fst (Pair v1 v2)) v1
| SndPair : forall v1 v2,
value v1
-> value v2
-> step0 (Snd (Pair v1 v2)) v2
(* Reducing a [Match] *)
| MatchInl : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
| MatchInr : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2).
Inductive step : exp -> exp -> Prop :=
| StepRule : forall C e1 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step0 e1 e2
-> step e1' e2'.
Definition trsys_of (e : exp) := {|
Initial := {e};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type)
(* New case: *)
| Sum (t1 t2 : type).
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty G (Var x) t
| HtConst : forall G n,
has_ty G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty G e1 Nat
-> has_ty G e2 Nat
-> has_ty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty (G $+ (x, t1)) e1 t2
-> has_ty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty G e1 (Fun t1 t2)
-> has_ty G e2 t1
-> has_ty G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
has_ty G e1 t1
-> has_ty G e2 t2
-> has_ty G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Snd e1) t2
(* New cases: *)
| HtInl : forall G e1 t1 t2,
has_ty G e1 t1
-> has_ty G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
has_ty G e1 t2
-> has_ty G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
has_ty G e (Sum t1 t2)
-> has_ty (G $+ (x1, t1)) e1 t
-> has_ty (G $+ (x2, t2)) e2 t
-> has_ty G (Match e x1 e1 x2 e2) t.
(* Type-soundness proof here *)
End StlcSums.
(** * Exceptions *)
Module StlcExceptions.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
| Pair (e1 e2 : exp)
| Fst (e1 : exp)
| Snd (e2 : exp)
| Inl (e1 : exp)
| Inr (e2 : exp)
| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp)
| Throw (e1 : exp)
| Catch (e1 : exp) (x : var) (e2 : exp).
(* The last one roughly means "try e1 catch x => e2". *)
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1)
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
| VInl : forall v, value v -> value (Inl v)
| VInr : forall v, value v -> value (Inr v).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
| Fst e2' => Fst (subst e1 x e2')
| Snd e2' => Snd (subst e1 x e2')
| Inl e2' => Inl (subst e1 x e2')
| Inr e2' => Inr (subst e1 x e2')
| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
x1 (if x1 ==v x then e21 else subst e1 x e21)
x2 (if x2 ==v x then e22 else subst e1 x e22)
(* New cases: *)
| Throw e2' => Throw (subst e1 x e2')
| Catch e2' x1 e2'' => Catch (subst e1 x e2')
x1 (if x1 ==v x then e2'' else subst e1 x e2'')
end.
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context
| Pair1 : context -> exp -> context
| Pair2 : exp -> context -> context
| Fst1 : context -> context
| Snd1 : context -> context
| Inl1 : context -> context
| Inr1 : context -> context
| Match1 : context -> var -> exp -> var -> exp -> context
(* New cases: *)
| Throw1 : context -> context
| Catch1 : context -> var -> exp -> context.
(* We modify [plug] with a new Boolean argument, to control whether [Catch1]
* context kinds are allowed. *)
Inductive plug : bool -> context -> exp -> exp -> Prop :=
| PlugHole : forall ac e, plug ac Hole e e
| PlugPlus1 : forall ac e e' C e2,
plug ac C e e'
-> plug ac (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall ac e e' v1 C,
value v1
-> plug ac C e e'
-> plug ac (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall ac e e' C e2,
plug ac C e e'
-> plug ac (App1 C e2) e (App e' e2)
| PlugApp2 : forall ac e e' v1 C,
value v1
-> plug ac C e e'
-> plug ac (App2 v1 C) e (App v1 e')
| PlugPair1 : forall ac e e' C e2,
plug ac C e e'
-> plug ac (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall ac e e' v1 C,
value v1
-> plug ac C e e'
-> plug ac (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall ac e e' C,
plug ac C e e'
-> plug ac (Fst1 C) e (Fst e')
| PlugSnd1 : forall ac e e' C,
plug ac C e e'
-> plug ac (Snd1 C) e (Snd e')
| PlugInl1 : forall ac e e' C,
plug ac C e e'
-> plug ac (Inl1 C) e (Inl e')
| PlugInr1 : forall ac e e' C,
plug ac C e e'
-> plug ac (Inr1 C) e (Inr e')
| PluMatch1 : forall ac e e' C x1 e1 x2 e2,
plug ac C e e'
-> plug ac (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
| PlugThrow1 : forall ac e e' C,
plug ac C e e'
-> plug ac (Throw1 C) e (Throw e')
| PlugCatch1 : forall e e' C x1 e1,
plug true C e e'
-> plug true (Catch1 C x1 e1) e (Catch e' x1 e1).
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
| FstPair : forall v1 v2,
value v1
-> value v2
-> step0 (Fst (Pair v1 v2)) v1
| SndPair : forall v1 v2,
value v1
-> value v2
-> step0 (Snd (Pair v1 v2)) v2
| MatchInl : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
| MatchInr : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2)
| ThrowBubble : forall v C e,
plug false C (Throw v) e
-> value v
-> C <> Hole
-> step0 e (Throw v)
| CatchValue : forall v x1 e1,
value v
-> step0 (Catch v x1 e1) v
| CatchThrow : forall v x1 e1,
value v
-> step0 (Catch (Throw v) x1 e1) (subst v x1 e1).
Inductive step : exp -> exp -> Prop :=
| StepRule : forall C e1 e2 e1' e2',
plug true C e1 e1'
-> plug true C e2 e2'
-> step0 e1 e2
-> step e1' e2'.
Definition trsys_of (e : exp) := {|
Initial := {e};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type)
| Sum (t1 t2 : type).
Inductive has_ty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty G (Var x) t
| HtConst : forall G n,
has_ty G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty G e1 Nat
-> has_ty G e2 Nat
-> has_ty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty (G $+ (x, t1)) e1 t2
-> has_ty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty G e1 (Fun t1 t2)
-> has_ty G e2 t1
-> has_ty G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
has_ty G e1 t1
-> has_ty G e2 t2
-> has_ty G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
has_ty G e1 (Prod t1 t2)
-> has_ty G (Snd e1) t2
| HtInl : forall G e1 t1 t2,
has_ty G e1 t1
-> has_ty G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
has_ty G e1 t2
-> has_ty G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
has_ty G e (Sum t1 t2)
-> has_ty (G $+ (x1, t1)) e1 t
-> has_ty (G $+ (x2, t2)) e2 t
-> has_ty G (Match e x1 e1 x2 e2) t
(* New cases: *)
| HtThrow : forall G e1 t,
has_ty G e1 Nat
-> has_ty G (Throw e1) t
| HtCatch : forall G e x1 e1 t,
has_ty G e t
-> has_ty (G $+ (x1, Nat)) e1 t
-> has_ty G (Catch e x1 e1) t.
End StlcExceptions.
(** * Mutable Variables *)
Module StlcMutable.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
| Pair (e1 e2 : exp)
| Fst (e1 : exp)
| Snd (e2 : exp)
| Inl (e1 : exp)
| Inr (e2 : exp)
| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp)
(* New cases: *)
| GetVar (x : var)
| SetVar (x : var) (e : exp).
(* Note that we are distinguishing between mutable and immutable variables,
* keeping the latter to bind in [Abs] and [Match]. *)
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1)
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
| VInl : forall v, value v -> value (Inl v)
| VInr : forall v, value v -> value (Inr v).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
| Fst e2' => Fst (subst e1 x e2')
| Snd e2' => Snd (subst e1 x e2')
| Inl e2' => Inl (subst e1 x e2')
| Inr e2' => Inr (subst e1 x e2')
| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
x1 (if x1 ==v x then e21 else subst e1 x e21)
x2 (if x2 ==v x then e22 else subst e1 x e22)
| GetVar y => GetVar y
| SetVar y e2' => SetVar y (subst e1 x e2')
end.
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context
| Pair1 : context -> exp -> context
| Pair2 : exp -> context -> context
| Fst1 : context -> context
| Snd1 : context -> context
| Inl1 : context -> context
| Inr1 : context -> context
| Match1 : context -> var -> exp -> var -> exp -> context
(* New case: *)
| SetVar1 : var -> context -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e')
| PlugPair1 : forall e e' C e2,
plug C e e'
-> plug (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall e e' C,
plug C e e'
-> plug (Fst1 C) e (Fst e')
| PlugSnd1 : forall e e' C,
plug C e e'
-> plug (Snd1 C) e (Snd e')
| PlugInl1 : forall e e' C,
plug C e e'
-> plug (Inl1 C) e (Inl e')
| PlugInr1 : forall e e' C,
plug C e e'
-> plug (Inr1 C) e (Inr e')
| PluMatch1 : forall e e' C x1 e1 x2 e2,
plug C e e'
-> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
(* Our new plugging rules *)
| PluSetVar1 : forall x e e' C,
plug C e e'
-> plug (SetVar1 x C) e (SetVar x e').
Definition valuation := fmap var exp.
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
| FstPair : forall v1 v2,
value v1
-> value v2
-> step0 (Fst (Pair v1 v2)) v1
| SndPair : forall v1 v2,
value v1
-> value v2
-> step0 (Snd (Pair v1 v2)) v2
| MatchInl : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
| MatchInr : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2).
(* We build an intermediate relation on top of [step0], allowing for
* consultation of the mutable-variable valuation. *)
Inductive step1 : valuation * exp -> valuation * exp -> Prop :=
| Lift : forall env e1 e2,
step0 e1 e2
-> step1 (env, e1) (env, e2)
| Read : forall env x v,
env $? x = Some v
-> step1 (env, GetVar x) (env, v)
| Overwrite : forall env x v,
value v
-> step1 (env, SetVar x v) (env $+ (x, v), v).
Inductive step : valuation * exp -> valuation * exp -> Prop :=
| StepRule : forall C env1 e1 env2 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step1 (env1, e1) (env2, e2)
-> step (env1, e1') (env2, e2').
Definition trsys_of (env : valuation) (e : exp) := {|
Initial := {(env, e)};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type)
| Sum (t1 t2 : type).
(* Now there will be two typing contexts, one for mutable variables and one
* for immutable. *)
Inductive has_ty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty M G (Var x) t
| HtConst : forall G n,
has_ty M G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty M G e1 Nat
-> has_ty M G e2 Nat
-> has_ty M G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty M (G $+ (x, t1)) e1 t2
-> has_ty M G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty M G e1 (Fun t1 t2)
-> has_ty M G e2 t1
-> has_ty M G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
has_ty M G e1 t1
-> has_ty M G e2 t2
-> has_ty M G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
has_ty M G e1 (Prod t1 t2)
-> has_ty M G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
has_ty M G e1 (Prod t1 t2)
-> has_ty M G (Snd e1) t2
| HtInl : forall G e1 t1 t2,
has_ty M G e1 t1
-> has_ty M G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
has_ty M G e1 t2
-> has_ty M G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
has_ty M G e (Sum t1 t2)
-> has_ty M (G $+ (x1, t1)) e1 t
-> has_ty M (G $+ (x2, t2)) e2 t
-> has_ty M G (Match e x1 e1 x2 e2) t
(* New cases: *)
| HtGetVar : forall G x t,
M $? x = Some t
-> has_ty M G (GetVar x) t
| HtSetVar : forall G x e t,
M $? x = Some t
-> has_ty M G e t
-> has_ty M G (SetVar x e) t.
End StlcMutable.
(** * Concurrency *)
Module StlcConcur.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
| Pair (e1 e2 : exp)
| Fst (e1 : exp)
| Snd (e2 : exp)
| Inl (e1 : exp)
| Inr (e2 : exp)
| Match (e' : exp) (x1 : var) (e1 : exp) (x2 : var) (e2 : exp)
| GetVar (x : var)
| SetVar (x : var) (e : exp)
(* New cases: *)
| Par (e1 e2 : exp).
(* This form evaluates both expressions and forms a pair of their
* results, if they terminate. *)
Inductive value : exp -> Prop :=
| VConst : forall n, value (Const n)
| VAbs : forall x e1, value (Abs x e1)
| VPair : forall v1 v2, value v1 -> value v2 -> value (Pair v1 v2)
| VInl : forall v, value v -> value (Inl v)
| VInr : forall v, value v -> value (Inr v).
Fixpoint subst (e1 : exp) (x : string) (e2 : exp) : exp :=
match e2 with
| Var y => if y ==v x then e1 else Var y
| Const n => Const n
| Plus e2' e2'' => Plus (subst e1 x e2') (subst e1 x e2'')
| Abs y e2' => Abs y (if y ==v x then e2' else subst e1 x e2')
| App e2' e2'' => App (subst e1 x e2') (subst e1 x e2'')
| Pair e2' e2'' => Pair (subst e1 x e2') (subst e1 x e2'')
| Fst e2' => Fst (subst e1 x e2')
| Snd e2' => Snd (subst e1 x e2')
| Inl e2' => Inl (subst e1 x e2')
| Inr e2' => Inr (subst e1 x e2')
| Match e2' x1 e21 x2 e22 => Match (subst e1 x e2')
x1 (if x1 ==v x then e21 else subst e1 x e21)
x2 (if x2 ==v x then e22 else subst e1 x e22)
| GetVar y => GetVar y
| SetVar y e2' => SetVar y (subst e1 x e2')
| Par e2' e2'' => Par (subst e1 x e2') (subst e1 x e2'')
end.
Inductive context : Set :=
| Hole : context
| Plus1 : context -> exp -> context
| Plus2 : exp -> context -> context
| App1 : context -> exp -> context
| App2 : exp -> context -> context
| Pair1 : context -> exp -> context
| Pair2 : exp -> context -> context
| Fst1 : context -> context
| Snd1 : context -> context
| Inl1 : context -> context
| Inr1 : context -> context
| Match1 : context -> var -> exp -> var -> exp -> context
| SetVar1 : var -> context -> context
(* New cases: *)
| Par1 : context -> exp -> context
| Par2 : exp -> context -> context.
Inductive plug : context -> exp -> exp -> Prop :=
| PlugHole : forall e, plug Hole e e
| PlugPlus1 : forall e e' C e2,
plug C e e'
-> plug (Plus1 C e2) e (Plus e' e2)
| PlugPlus2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Plus2 v1 C) e (Plus v1 e')
| PlugApp1 : forall e e' C e2,
plug C e e'
-> plug (App1 C e2) e (App e' e2)
| PlugApp2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (App2 v1 C) e (App v1 e')
| PlugPair1 : forall e e' C e2,
plug C e e'
-> plug (Pair1 C e2) e (Pair e' e2)
| PlugPair2 : forall e e' v1 C,
value v1
-> plug C e e'
-> plug (Pair2 v1 C) e (Pair v1 e')
| PlugFst1 : forall e e' C,
plug C e e'
-> plug (Fst1 C) e (Fst e')
| PlugSnd1 : forall e e' C,
plug C e e'
-> plug (Snd1 C) e (Snd e')
| PlugInl1 : forall e e' C,
plug C e e'
-> plug (Inl1 C) e (Inl e')
| PlugInr1 : forall e e' C,
plug C e e'
-> plug (Inr1 C) e (Inr e')
| PluMatch1 : forall e e' C x1 e1 x2 e2,
plug C e e'
-> plug (Match1 C x1 e1 x2 e2) e (Match e' x1 e1 x2 e2)
| PluSetVar1 : forall x e e' C,
plug C e e'
-> plug (SetVar1 x C) e (SetVar x e')
(* Our new plugging rules: *)
| PlugPar1 : forall e e' C e2,
plug C e e'
-> plug (Par1 C e2) e (Par e' e2)
| PlugPar2 : forall e e' e1 C,
plug C e e'
-> plug (Par2 e1 C) e (Par e1 e').
Definition valuation := fmap var exp.
Inductive step0 : exp -> exp -> Prop :=
| Beta : forall x e v,
value v
-> step0 (App (Abs x e) v) (subst v x e)
| Add : forall n1 n2,
step0 (Plus (Const n1) (Const n2)) (Const (n1 + n2))
| FstPair : forall v1 v2,
value v1
-> value v2
-> step0 (Fst (Pair v1 v2)) v1
| SndPair : forall v1 v2,
value v1
-> value v2
-> step0 (Snd (Pair v1 v2)) v2
| MatchInl : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inl v) x1 e1 x2 e2) (subst v x1 e1)
| MatchInr : forall v x1 e1 x2 e2,
value v
-> step0 (Match (Inr v) x1 e1 x2 e2) (subst v x2 e2)
(* New case: *)
| ParDone : forall v1 v2,
value v1
-> value v2
-> step0 (Par v1 v2) (Pair v1 v2).
Inductive step1 : valuation * exp -> valuation * exp -> Prop :=
| Lift : forall env e1 e2,
step0 e1 e2
-> step1 (env, e1) (env, e2)
| Read : forall env x v,
env $? x = Some v
-> step1 (env, GetVar x) (env, v)
| Overwrite : forall env x v,
value v
-> step1 (env, SetVar x v) (env $+ (x, v), v).
Inductive step : valuation * exp -> valuation * exp -> Prop :=
| StepRule : forall C env1 e1 env2 e2 e1' e2',
plug C e1 e1'
-> plug C e2 e2'
-> step1 (env1, e1) (env2, e2)
-> step (env1, e1') (env2, e2').
Definition trsys_of (env : valuation) (e : exp) := {|
Initial := {(env, e)};
Step := step
|}.
Inductive type :=
| Nat
| Fun (dom ran : type)
| Prod (t1 t2 : type)
| Sum (t1 t2 : type).
Inductive has_ty (M : fmap var type) : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> has_ty M G (Var x) t
| HtConst : forall G n,
has_ty M G (Const n) Nat
| HtPlus : forall G e1 e2,
has_ty M G e1 Nat
-> has_ty M G e2 Nat
-> has_ty M G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
has_ty M (G $+ (x, t1)) e1 t2
-> has_ty M G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
has_ty M G e1 (Fun t1 t2)
-> has_ty M G e2 t1
-> has_ty M G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
has_ty M G e1 t1
-> has_ty M G e2 t2
-> has_ty M G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
has_ty M G e1 (Prod t1 t2)
-> has_ty M G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
has_ty M G e1 (Prod t1 t2)
-> has_ty M G (Snd e1) t2
| HtInl : forall G e1 t1 t2,
has_ty M G e1 t1
-> has_ty M G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
has_ty M G e1 t2
-> has_ty M G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
has_ty M G e (Sum t1 t2)
-> has_ty M (G $+ (x1, t1)) e1 t
-> has_ty M (G $+ (x2, t2)) e2 t
-> has_ty M G (Match e x1 e1 x2 e2) t
| HtGetVar : forall G x t,
M $? x = Some t
-> has_ty M G (GetVar x) t
| HtSetVar : forall G x e t,
M $? x = Some t
-> has_ty M G e t
-> has_ty M G (SetVar x e) t
(* New cases: *)
| HtPar : forall G e1 t1 e2 t2,
has_ty M G e1 t1
-> has_ty M G e2 t2
-> has_ty M G (Par e1 e2) (Prod t1 t2).
(* Type-soundness proof *)
End StlcConcur.
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