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EvaluationContexts.v
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EvaluationContexts.v
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 12: More on Evaluation Contexts
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Require Import Frap.
(** * Evaluation Contexts for Lambda Calculus *)
(* Let's revisit the typed language from the end of the previous chapter, this
* time casting its small-step semantics using evaluation contexts. *)
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. This is our grammar of
* contexts. Note a difference from the book: we don't enforce here that
* the first argument of a [Plus1] or [App1] is a value, but rather that
* constraint comes in the next relation definition. *)
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 hasty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> hasty G (Var x) t
| HtConst : forall G n,
hasty G (Const n) Nat
| HtPlus : forall G e1 e2,
hasty G e1 Nat
-> hasty G e2 Nat
-> hasty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
hasty (G $+ (x, t1)) e1 t2
-> hasty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
hasty G e1 (Fun t1 t2)
-> hasty G e2 t1
-> hasty G (App e1 e2) t2.
Local Hint Constructors value plug step0 step hasty : 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 : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H
| [ H : hasty _ _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => invert1 H
end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
Lemma progress : forall e t,
hasty $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,
hasty G e t
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
-> hasty G' e t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve weakening : core.
(* Replacing a typing context with an equal one has no effect (useful to guide
* proof search as a hint). *)
Lemma hasty_change : forall G e t,
hasty G e t
-> forall G', G' = G
-> hasty G' e t.
Proof.
t.
Qed.
Local Hint Resolve hasty_change : core.
Lemma substitution : forall G x t' e t e',
hasty (G $+ (x, t')) e t
-> hasty $0 e' t'
-> hasty 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, hasty $0 e1 t
-> hasty $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
-> hasty $0 e1' t
-> hasty $0 e2' t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve preservation' : core.
Lemma preservation : forall e1 e2,
step e1 e2
-> forall t, hasty $0 e1 t
-> hasty $0 e2 t.
Proof.
invert 1; t.
Qed.
Local Hint Resolve progress preservation : core.
Theorem safety : forall e t, hasty $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' => hasty $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.
(** * Some More Classic Features *)
(* Here's how easy it is to extend those definitions and proofs to two other
* common features of functional-programming languages. We'll use comments to
* mark the only places where code is added. Very little old code needs to be
* changed! The version in the book PDF shows even more clearly how evaluation
* contexts make for compact descriptions of features, since here we are
* manually writing [plug] relations, following clear conventions in
* evaluation-context grammars. *)
Module StlcPairs.
Inductive exp : Set :=
| Var (x : var)
| Const (n : nat)
| Plus (e1 e2 : exp)
| Abs (x : var) (e1 : exp)
| App (e1 e2 : exp)
(* We can combine two values together into a pair, and then we can use
* projection functions to retrieve the first and second components,
* respectively. *)
| 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 hasty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> hasty G (Var x) t
| HtConst : forall G n,
hasty G (Const n) Nat
| HtPlus : forall G e1 e2,
hasty G e1 Nat
-> hasty G e2 Nat
-> hasty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
hasty (G $+ (x, t1)) e1 t2
-> hasty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
hasty G e1 (Fun t1 t2)
-> hasty G e2 t1
-> hasty G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
hasty G e1 t1
-> hasty G e2 t2
-> hasty G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
hasty G e1 (Prod t1 t2)
-> hasty G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
hasty G e1 (Prod t1 t2)
-> hasty G (Snd e1) t2.
Local Hint Constructors value plug step0 step hasty : core.
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 : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; []
(* Change here! We need to enforce there is at most one
* remaining subgoal, or we'll keep doing useless [value]
* inversions ad infinitum. *)
| [ H : hasty _ _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => invert1 H
end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 6.
Lemma progress : forall e t,
hasty $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,
hasty G e t
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
-> hasty G' e t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve weakening : core.
(* Replacing a typing context with an equal one has no effect (useful to guide
* proof search as a hint). *)
Lemma hasty_change : forall G e t,
hasty G e t
-> forall G', G' = G
-> hasty G' e t.
Proof.
t.
Qed.
Local Hint Resolve hasty_change : core.
Lemma substitution : forall G x t' e t e',
hasty (G $+ (x, t')) e t
-> hasty $0 e' t'
-> hasty 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, hasty $0 e1 t
-> hasty $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
-> hasty $0 e1' t
-> hasty $0 e2' t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve preservation' : core.
Lemma preservation : forall e1 e2,
step e1 e2
-> forall t, hasty $0 e1 t
-> hasty $0 e2 t.
Proof.
invert 1; t.
Qed.
Local Hint Resolve progress preservation : core.
Theorem safety : forall e t, hasty $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' => hasty $0 e' t); eauto.
apply invariant_induction; simplify; eauto; equality.
Qed.
End StlcPairs.
(* Next, the dual feature of *variants*, corresponding to the following type
* family from Coq's standard library. *)
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 hasty : fmap var type -> exp -> type -> Prop :=
| HtVar : forall G x t,
G $? x = Some t
-> hasty G (Var x) t
| HtConst : forall G n,
hasty G (Const n) Nat
| HtPlus : forall G e1 e2,
hasty G e1 Nat
-> hasty G e2 Nat
-> hasty G (Plus e1 e2) Nat
| HtAbs : forall G x e1 t1 t2,
hasty (G $+ (x, t1)) e1 t2
-> hasty G (Abs x e1) (Fun t1 t2)
| HtApp : forall G e1 e2 t1 t2,
hasty G e1 (Fun t1 t2)
-> hasty G e2 t1
-> hasty G (App e1 e2) t2
| HtPair : forall G e1 e2 t1 t2,
hasty G e1 t1
-> hasty G e2 t2
-> hasty G (Pair e1 e2) (Prod t1 t2)
| HtFst : forall G e1 t1 t2,
hasty G e1 (Prod t1 t2)
-> hasty G (Fst e1) t1
| HtSnd : forall G e1 t1 t2,
hasty G e1 (Prod t1 t2)
-> hasty G (Snd e1) t2
(* New cases: *)
| HtInl : forall G e1 t1 t2,
hasty G e1 t1
-> hasty G (Inl e1) (Sum t1 t2)
| HtInr : forall G e1 t1 t2,
hasty G e1 t2
-> hasty G (Inr e1) (Sum t1 t2)
| HtMatch : forall G e t1 t2 x1 e1 x2 e2 t,
hasty G e (Sum t1 t2)
-> hasty (G $+ (x1, t1)) e1 t
-> hasty (G $+ (x2, t2)) e2 t
-> hasty G (Match e x1 e1 x2 e2) t.
Local Hint Constructors value plug step0 step hasty : core.
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 : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; []
(* New case! For sums, we sometimes need to consider two rules for
* one [value] inversion. *)
| [ H : hasty _ ?e _, H' : value ?e |- _ ] => invert H'; invert H; [|]
| [ H : hasty _ _ _ |- _ ] => invert1 H
| [ H : plug _ _ _ |- _ ] => invert1 H
end; subst.
Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
(* change! --^ *)
Lemma progress : forall e t,
hasty $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,
hasty G e t
-> forall G', (forall x t, G $? x = Some t -> G' $? x = Some t)
-> hasty G' e t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve weakening : core.
(* Replacing a typing context with an equal one has no effect (useful to guide
* proof search as a hint). *)
Lemma hasty_change : forall G e t,
hasty G e t
-> forall G', G' = G
-> hasty G' e t.
Proof.
t.
Qed.
Local Hint Resolve hasty_change : core.
Lemma substitution : forall G x t' e t e',
hasty (G $+ (x, t')) e t
-> hasty $0 e' t'
-> hasty 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, hasty $0 e1 t
-> hasty $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
-> hasty $0 e1' t
-> hasty $0 e2' t.
Proof.
induct 1; t.
Qed.
Local Hint Resolve preservation' : core.
Lemma preservation : forall e1 e2,
step e1 e2
-> forall t, hasty $0 e1 t
-> hasty $0 e2 t.
Proof.
invert 1; t.
Qed.
Local Hint Resolve progress preservation : core.
Theorem safety : forall e t, hasty $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' => hasty $0 e' t); eauto.
apply invariant_induction; simplify; eauto; equality.
Qed.
End StlcSums.
(** * Exceptions *)
(* Evaluation contexts are very helpful for concise modeling of control-flow
* constructs like exceptions. Let's look at an example where exceptions are
* just numbers, for simplicity. *)
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