NormNormalization of STLC
Set Warnings "-notation-overridden,-parsing,-deprecated-hint-without-locality".
From Coq Require Import Lists.List.
From Coq Require Import Strings.String.
From PLF Require Import Maps.
From PLF Require Import Smallstep.
Hint Constructors multi : core.
(* Chapter written and maintained by Andrew Tolmach *)
This optional chapter is based on chapter 12 of Types and
Programming Languages (Pierce). It may be useful to look at the
two together, as that chapter includes explanations and informal
proofs that are not repeated here.
In this chapter, we consider another fundamental theoretical
property of the simply typed lambda-calculus: the fact that the
evaluation of a well-typed program is guaranteed to halt in a
finite number of steps---i.e., every well-typed term is
normalizable.
Unlike the type-safety properties we have considered so far, the
normalization property does not extend to full-blown programming
languages, because these languages nearly always extend the simply
typed lambda-calculus with constructs, such as general
recursion (see the MoreStlc chapter) or recursive types, that
can be used to write nonterminating programs. However, the issue
of normalization reappears at the level of types when we
consider the metatheory of polymorphic versions of the lambda
calculus such as System F-omega: in this system, the language of
types effectively contains a copy of the simply typed
lambda-calculus, and the termination of the typechecking algorithm
will hinge on the fact that a "normalization" operation on type
expressions is guaranteed to terminate.
Another reason for studying normalization proofs is that they are
some of the most beautiful---and mind-blowing---mathematics to be
found in the type theory literature, often (as here) involving the
fundamental proof technique of logical relations.
The calculus we shall consider here is the simply typed
lambda-calculus over a single base type bool and with
pairs. We'll give most details of the development for the basic
lambda-calculus terms treating bool as an uninterpreted base
type, and leave the extension to the boolean operators and pairs
to the reader. Even for the base calculus, normalization is not
entirely trivial to prove, since each reduction of a term can
duplicate redexes in subterms.
Exercise: 2 stars, standard (norm_fail)
Where do we fail if we attempt to prove normalization by a straightforward induction on the size of a well-typed term?(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_norm_fail : option (nat×string) := None.
☐
Exercise: 5 stars, standard, especially useful (norm)
The best ways to understand an intricate proof like this is are (1) to help fill it in and (2) to extend it. We've left out some parts of the following development, including some proofs of lemmas and the all the cases involving products and conditionals. Fill them in.(* Do not modify the following line: *)
Definition manual_grade_for_norm : option (nat×string) := None.
☐
Language
Inductive ty : Type :=
| Ty_Bool : ty
| Ty_Arrow : ty → ty → ty
| Ty_Prod : ty → ty → ty
.
Inductive tm : Type :=
(* pure STLC *)
| tm_var : string → tm
| tm_app : tm → tm → tm
| tm_abs : string → ty → tm → tm
(* booleans *)
| tm_true : tm
| tm_false : tm
| tm_if : tm → tm → tm → tm
(* pairs *)
| tm_pair : tm → tm → tm
| tm_fst : tm → tm
| tm_snd : tm → tm.
Declare Custom Entry stlc.
Notation "<{ e }>" := e (e custom stlc at level 99).
Notation "( x )" := x (in custom stlc, x at level 99).
Notation "x" := x (in custom stlc at level 0, x constr at level 0).
Notation "S -> T" := (Ty_Arrow S T) (in custom stlc at level 50, right associativity).
Notation "x y" := (tm_app x y) (in custom stlc at level 1, left associativity).
Notation "\ x : t , y" :=
(tm_abs x t y) (in custom stlc at level 90, x at level 99,
t custom stlc at level 99,
y custom stlc at level 99,
left associativity).
Coercion tm_var : string >-> tm.
Notation "{ x }" := x (in custom stlc at level 1, x constr).
Notation "'Bool'" := Ty_Bool (in custom stlc at level 0).
Notation "'if' x 'then' y 'else' z" :=
(tm_if x y z) (in custom stlc at level 89,
x custom stlc at level 99,
y custom stlc at level 99,
z custom stlc at level 99,
left associativity).
Notation "'true'" := true (at level 1).
Notation "'true'" := tm_true (in custom stlc at level 0).
Notation "'false'" := false (at level 1).
Notation "'false'" := tm_false (in custom stlc at level 0).
Notation "X * Y" :=
(Ty_Prod X Y) (in custom stlc at level 2, X custom stlc, Y custom stlc at level 0).
Notation "( x ',' y )" := (tm_pair x y) (in custom stlc at level 0,
x custom stlc at level 99,
y custom stlc at level 99).
Notation "t '.fst'" := (tm_fst t) (in custom stlc at level 0).
Notation "t '.snd'" := (tm_snd t) (in custom stlc at level 0).
Reserved Notation "'[' x ':=' s ']' t" (in custom stlc at level 20, x constr).
Fixpoint subst (x : string) (s : tm) (t : tm) : tm :=
match t with
| tm_var y ⇒
if eqb_string x y then s else t
| <{ \ y : T, t1 }> ⇒
if eqb_string x y then t else <{ \y:T, [x:=s] t1 }>
| <{t1 t2}> ⇒
<{ ([x:=s]t1) ([x:=s]t2)}>
| <{true}> ⇒
<{true}>
| <{false}> ⇒
<{false}>
| <{if t1 then t2 else t3}> ⇒
<{if ([x:=s] t1) then ([x:=s] t2) else ([x:=s] t3)}>
| <{(t1, t2)}> ⇒
<{( ([x:=s] t1), ([x:=s] t2) )}>
| <{t0.fst}> ⇒
<{ ([x:=s] t0).fst}>
| <{t0.snd}> ⇒
<{ ([x:=s] t0).snd}>
end
where "'[' x ':=' s ']' t" := (subst x s t) (in custom stlc).
Inductive value : tm → Prop :=
| v_abs : ∀ x T2 t1,
value <{\x:T2, t1}>
| v_true :
value <{true}>
| v_false :
value <{false}>
| v_pair : ∀ v1 v2,
value v1 →
value v2 →
value <{(v1, v2)}>.
Hint Constructors value : core.
Reserved Notation "t '-->' t'" (at level 40).
Inductive step : tm → tm → Prop :=
| ST_AppAbs : ∀ x T2 t1 v2,
value v2 →
<{(\x:T2, t1) v2}> --> <{ [x:=v2]t1 }>
| ST_App1 : ∀ t1 t1' t2,
t1 --> t1' →
<{t1 t2}> --> <{t1' t2}>
| ST_App2 : ∀ v1 t2 t2',
value v1 →
t2 --> t2' →
<{v1 t2}> --> <{v1 t2'}>
| ST_IfTrue : ∀ t1 t2,
<{if true then t1 else t2}> --> t1
| ST_IfFalse : ∀ t1 t2,
<{if false then t1 else t2}> --> t2
| ST_If : ∀ t1 t1' t2 t3,
t1 --> t1' →
<{if t1 then t2 else t3}> --> <{if t1' then t2 else t3}>
| ST_Pair1 : ∀ t1 t1' t2,
t1 --> t1' →
<{ (t1,t2) }> --> <{ (t1' , t2) }>
| ST_Pair2 : ∀ v1 t2 t2',
value v1 →
t2 --> t2' →
<{ (v1, t2) }> --> <{ (v1, t2') }>
| ST_Fst1 : ∀ t0 t0',
t0 --> t0' →
<{ t0.fst }> --> <{ t0'.fst }>
| ST_FstPair : ∀ v1 v2,
value v1 →
value v2 →
<{ (v1,v2).fst }> --> v1
| ST_Snd1 : ∀ t0 t0',
t0 --> t0' →
<{ t0.snd }> --> <{ t0'.snd }>
| ST_SndPair : ∀ v1 v2,
value v1 →
value v2 →
<{ (v1,v2).snd }> --> v2
where "t '-->' t'" := (step t t').
Hint Constructors step : core.
Notation multistep := (multi step).
Notation "t1 '-->*' t2" := (multistep t1 t2) (at level 40).
Notation step_normal_form := (normal_form step).
Lemma value__normal : ∀ t, value t → step_normal_form t.
Proof with eauto.
intros t H; induction H; intros [t' ST]; inversion ST...
Qed.
intros t H; induction H; intros [t' ST]; inversion ST...
Qed.
Definition context := partial_map ty.
Reserved Notation "Gamma '⊢' t '∈' T" (at level 40,
t custom stlc, T custom stlc at level 0).
Inductive has_type : context → tm → ty → Prop :=
(* Same as before: *)
(* pure STLC *)
| T_Var : ∀ Gamma x T1,
Gamma x = Some T1 →
Gamma ⊢ x \in T1
| T_Abs : ∀ Gamma x T1 T2 t1,
(x ⊢> T2 ; Gamma) ⊢ t1 \in T1 →
Gamma ⊢ \x:T2, t1 \in (T2 → T1)
| T_App : ∀ T1 T2 Gamma t1 t2,
Gamma ⊢ t1 \in (T2 → T1) →
Gamma ⊢ t2 \in T2 →
Gamma ⊢ t1 t2 \in T1
| T_True : ∀ Gamma,
Gamma ⊢ true \in Bool
| T_False : ∀ Gamma,
Gamma ⊢ false \in Bool
| T_If : ∀ t1 t2 t3 T1 Gamma,
Gamma ⊢ t1 \in Bool →
Gamma ⊢ t2 \in T1 →
Gamma ⊢ t3 \in T1 →
Gamma ⊢ if t1 then t2 else t3 \in T1
| T_Pair : ∀ Gamma t1 t2 T1 T2,
Gamma ⊢ t1 \in T1 →
Gamma ⊢ t2 \in T2 →
Gamma ⊢ (t1, t2) \in (T1 × T2)
| T_Fst : ∀ Gamma t0 T1 T2,
Gamma ⊢ t0 \in (T1 × T2) →
Gamma ⊢ t0.fst \in T1
| T_Snd : ∀ Gamma t0 T1 T2,
Gamma ⊢ t0 \in (T1 × T2) →
Gamma ⊢ t0.snd \in T2
where "Gamma '⊢' t '∈' T" := (has_type Gamma t T).
Hint Constructors has_type : core.
Hint Extern 2 (has_type _ (app _ _) _) ⇒ eapply T_App; auto : core.
Hint Extern 2 (_ = _) ⇒ compute; reflexivity : core.
Lemma weakening : ∀ Gamma Gamma' t T,
inclusion Gamma Gamma' →
Gamma ⊢ t \in T →
Gamma' ⊢ t \in T.
Proof.
intros Gamma Gamma' t T H Ht.
generalize dependent Gamma'.
induction Ht; eauto using inclusion_update.
Qed.
Lemma weakening_empty : ∀ Gamma t T,
empty ⊢ t \in T →
Gamma ⊢ t \in T.
Proof.
intros Gamma t T.
eapply weakening.
discriminate.
Qed.
Lemma substitution_preserves_typing : ∀ Gamma x U t v T,
(x ⊢> U ; Gamma) ⊢ t \in T →
empty ⊢ v \in U →
Gamma ⊢ [x:=v]t \in T.
Proof with eauto.
intros Gamma x U t v T Ht Hv.
generalize dependent Gamma. generalize dependent T.
induction t; intros T Gamma H;
(* in each case, we'll want to get at the derivation of H *)
inversion H; clear H; subst; simpl; eauto.
- (* var *)
rename s into y. destruct (eqb_stringP x y); subst.
+ (* x=y *)
rewrite update_eq in H2.
injection H2 as H2; subst.
apply weakening_empty. assumption.
+ (* x<>y *)
apply T_Var. rewrite update_neq in H2; auto.
- (* abs *)
rename s into y, t into S.
destruct (eqb_stringP x y); subst; apply T_Abs.
+ (* x=y *)
rewrite update_shadow in H5. assumption.
+ (* x<>y *)
apply IHt.
rewrite update_permute; auto.
Qed.
Theorem preservation : ∀ t t' T,
empty ⊢ t \in T →
t --> t' →
empty ⊢ t' \in T.
Proof with eauto.
intros t t' T HT. generalize dependent t'.
remember empty as Gamma.
induction HT;
intros t' HE; subst; inversion HE; subst...
- (* T_App *)
inversion HE; subst...
+ (* ST_AppAbs *)
apply substitution_preserves_typing with T2...
inversion HT1...
- (* T_Fst *)
inversion HT...
- (* T_Snd *)
inversion HT...
Qed.
Inductive appears_free_in : string → tm → Prop :=
| afi_var : ∀ (x : string),
appears_free_in x <{ x }>
| afi_app1 : ∀ x t1 t2,
appears_free_in x t1 → appears_free_in x <{ t1 t2 }>
| afi_app2 : ∀ x t1 t2,
appears_free_in x t2 → appears_free_in x <{ t1 t2 }>
| afi_abs : ∀ x y T11 t12,
y ≠ x →
appears_free_in x t12 →
appears_free_in x <{ \y : T11, t12 }>
(* booleans *)
| afi_test0 : ∀ x t0 t1 t2,
appears_free_in x t0 →
appears_free_in x <{ if t0 then t1 else t2 }>
| afi_test1 : ∀ x t0 t1 t2,
appears_free_in x t1 →
appears_free_in x <{ if t0 then t1 else t2 }>
| afi_test2 : ∀ x t0 t1 t2,
appears_free_in x t2 →
appears_free_in x <{ if t0 then t1 else t2 }>
(* pairs *)
| afi_pair1 : ∀ x t1 t2,
appears_free_in x t1 →
appears_free_in x <{ (t1, t2) }>
| afi_pair2 : ∀ x t1 t2,
appears_free_in x t2 →
appears_free_in x <{ (t1 , t2) }>
| afi_fst : ∀ x t,
appears_free_in x t →
appears_free_in x <{ t.fst }>
| afi_snd : ∀ x t,
appears_free_in x t →
appears_free_in x <{ t.snd }>
.
Hint Constructors appears_free_in : core.
Definition closed (t:tm) :=
∀ x, ¬ appears_free_in x t.
Lemma context_invariance : ∀ Gamma Gamma' t S,
Gamma ⊢ t \in S →
(∀ x, appears_free_in x t → Gamma x = Gamma' x) →
Gamma' ⊢ t \in S.
Proof.
intros.
generalize dependent Gamma'.
induction H; intros; eauto 12.
- (* T_Var *)
apply T_Var. rewrite <- H0; auto.
- (* T_Abs *)
apply T_Abs.
apply IHhas_type. intros x1 Hafi.
(* the only tricky step... *)
destruct (eqb_stringP x x1); subst.
+ rewrite update_eq.
rewrite update_eq.
reflexivity.
+ rewrite update_neq; [| assumption].
rewrite update_neq; [| assumption].
auto.
Qed.
intros.
generalize dependent Gamma'.
induction H; intros; eauto 12.
- (* T_Var *)
apply T_Var. rewrite <- H0; auto.
- (* T_Abs *)
apply T_Abs.
apply IHhas_type. intros x1 Hafi.
(* the only tricky step... *)
destruct (eqb_stringP x x1); subst.
+ rewrite update_eq.
rewrite update_eq.
reflexivity.
+ rewrite update_neq; [| assumption].
rewrite update_neq; [| assumption].
auto.
Qed.
Lemma free_in_context : ∀ x t T Gamma,
appears_free_in x t →
Gamma ⊢ t \in T →
∃ T', Gamma x = Some T'.
Proof with eauto.
intros x t T Gamma Hafi Htyp.
induction Htyp; inversion Hafi; subst...
- (* T_Abs *)
destruct IHHtyp as [T' Hctx]... ∃ T'.
unfold update, t_update in Hctx.
rewrite false_eqb_string in Hctx...
Qed.
intros x t T Gamma Hafi Htyp.
induction Htyp; inversion Hafi; subst...
- (* T_Abs *)
destruct IHHtyp as [T' Hctx]... ∃ T'.
unfold update, t_update in Hctx.
rewrite false_eqb_string in Hctx...
Qed.
Corollary typable_empty__closed : ∀ t T,
empty ⊢ t \in T →
closed t.
Proof.
intros. unfold closed. intros x H1.
destruct (free_in_context _ _ _ _ H1 H) as [T' C].
discriminate C. Qed.
intros. unfold closed. intros x H1.
destruct (free_in_context _ _ _ _ H1 H) as [T' C].
discriminate C. Qed.
Lemma step_deterministic :
deterministic step.
Proof with eauto.
unfold deterministic.
intros t t' t'' E1 E2.
generalize dependent t''.
induction E1; intros t'' E2; inversion E2; subst; clear E2...
(* ST_AppAbs *)
- inversion H3.
- exfalso; apply value__normal in H...
(* ST_App1 *)
- inversion E1.
- f_equal...
- exfalso; apply value__normal in H1...
(* ST_App2 *)
- exfalso; apply value__normal in H3...
- exfalso; apply value__normal in H...
- f_equal...
- (* ST_IfTrue *)
inversion H3.
- (* ST_IfFalse *)
inversion H3.
(* ST_If *)
- inversion E1.
- inversion E1.
- f_equal...
(* ST_Pair1 *)
- f_equal...
- exfalso; apply value__normal in H1...
(* ST_Pair2 *)
- exfalso; apply value__normal in H...
- f_equal...
(* ST_Fst1 *)
- f_equal...
- exfalso.
inversion E1; subst.
+ apply value__normal in H0...
+ apply value__normal in H1...
(* ST_FstPair *)
- exfalso.
inversion H2; subst.
+ apply value__normal in H...
+ apply value__normal in H0...
(* ST_Snd1 *)
- f_equal...
- exfalso.
inversion E1; subst.
+ apply value__normal in H0...
+ apply value__normal in H1...
(* ST_SndPair *)
- exfalso.
inversion H2; subst.
+ apply value__normal in H...
+ apply value__normal in H0...
Qed.
unfold deterministic.
intros t t' t'' E1 E2.
generalize dependent t''.
induction E1; intros t'' E2; inversion E2; subst; clear E2...
(* ST_AppAbs *)
- inversion H3.
- exfalso; apply value__normal in H...
(* ST_App1 *)
- inversion E1.
- f_equal...
- exfalso; apply value__normal in H1...
(* ST_App2 *)
- exfalso; apply value__normal in H3...
- exfalso; apply value__normal in H...
- f_equal...
- (* ST_IfTrue *)
inversion H3.
- (* ST_IfFalse *)
inversion H3.
(* ST_If *)
- inversion E1.
- inversion E1.
- f_equal...
(* ST_Pair1 *)
- f_equal...
- exfalso; apply value__normal in H1...
(* ST_Pair2 *)
- exfalso; apply value__normal in H...
- f_equal...
(* ST_Fst1 *)
- f_equal...
- exfalso.
inversion E1; subst.
+ apply value__normal in H0...
+ apply value__normal in H1...
(* ST_FstPair *)
- exfalso.
inversion H2; subst.
+ apply value__normal in H...
+ apply value__normal in H0...
(* ST_Snd1 *)
- f_equal...
- exfalso.
inversion E1; subst.
+ apply value__normal in H0...
+ apply value__normal in H1...
(* ST_SndPair *)
- exfalso.
inversion H2; subst.
+ apply value__normal in H...
+ apply value__normal in H0...
Qed.
Normalization
A trivial fact:
The key issue in the normalization proof (as in many proofs by
induction) is finding a strong enough induction hypothesis. To
this end, we begin by defining, for each type T, a set R_T of
closed terms of type T. We will specify these sets using a
relation R and write R T t when t is in R_T. (The sets
R_T are sometimes called saturated sets or reducibility
candidates.)
Here is the definition of R for the base language:
This definition gives us the strengthened induction hypothesis that we
need. Our primary goal is to show that all programs ---i.e., all
closed terms of base type---halt. But closed terms of base type can
contain subterms of functional type, so we need to know something
about these as well. Moreover, it is not enough to know that these
subterms halt, because the application of a normalized function to a
normalized argument involves a substitution, which may enable more
reduction steps. So we need a stronger condition for terms of
functional type: not only should they halt themselves, but, when
applied to halting arguments, they should yield halting results.
The form of R is characteristic of the logical relations proof
technique. (Since we are just dealing with unary relations here, we
could perhaps more properly say logical properties.) If we want to
prove some property P of all closed terms of type A, we proceed by
proving, by induction on types, that all terms of type A possess
property P, all terms of type A→A preserve property P, all
terms of type (A→A)->(A→A) preserve the property of preserving
property P, and so on. We do this by defining a family of
properties, indexed by types. For the base type A, the property is
just P. For functional types, it says that the function should map
values satisfying the property at the input type to values satisfying
the property at the output type.
When we come to formalize the definition of R in Coq, we hit a
problem. The most obvious formulation would be as a parameterized
Inductive proposition like this:
Inductive R : ty → tm → Prop :=
| R_bool : ∀ b t, empty ⊢ t \in Bool →
halts t →
R Bool t
| R_arrow : ∀ T1 T2 t, empty ⊢ t \in (Arrow T1 T2) →
halts t →
(∀ s, R T1 s → R T2 (app t s)) →
R (Arrow T1 T2) t.
Unfortunately, Coq rejects this definition because it violates the
strict positivity requirement for inductive definitions, which says
that the type being defined must not occur to the left of an arrow in
the type of a constructor argument. Here, it is the third argument to
R_arrow, namely (∀ s, R T1 s → R TS (app t s)), and
specifically the R T1 s part, that violates this rule. (The
outermost arrows separating the constructor arguments don't count when
applying this rule; otherwise we could never have genuinely inductive
properties at all!) The reason for the rule is that types defined
with non-positive recursion can be used to build non-terminating
functions, which as we know would be a disaster for Coq's logical
soundness. Even though the relation we want in this case might be
perfectly innocent, Coq still rejects it because it fails the
positivity test.
Fortunately, it turns out that we can define R using a
Fixpoint:
- R bool t iff t is a closed term of type bool and t halts
in a value
- R (T1 → T2) t iff t is a closed term of type T1 → T2 and t halts in a value and for any term s such that R T1 s, we have R T2 (t s).
Inductive R : ty → tm → Prop :=
| R_bool : ∀ b t, empty ⊢ t \in Bool →
halts t →
R Bool t
| R_arrow : ∀ T1 T2 t, empty ⊢ t \in (Arrow T1 T2) →
halts t →
(∀ s, R T1 s → R T2 (app t s)) →
R (Arrow T1 T2) t.
Fixpoint R (T:ty) (t:tm) : Prop :=
empty ⊢ t \in T ∧ halts t ∧
(match T with
| <{ Bool }> ⇒ True
| <{ T1 → T2 }> ⇒ (∀ s, R T1 s → R T2 <{t s}> )
(* ... edit the next line when dealing with products *)
| <{ T1 × T2 }> ⇒ False (* FILL IN HERE *)
end).
As immediate consequences of this definition, we have that every
element of every set R_T halts in a value and is closed with type
t :
Lemma R_halts : ∀ {T} {t}, R T t → halts t.
Lemma R_typable_empty : ∀ {T} {t}, R T t → empty ⊢ t \in T.
Now we proceed to show the main result, which is that every
well-typed term of type T is an element of R_T. Together with
R_halts, that will show that every well-typed term halts in a
value.
Membership in R_T Is Invariant Under Reduction
Lemma step_preserves_halting :
∀ t t', (t --> t') → (halts t ↔ halts t').
Proof.
intros t t' ST. unfold halts.
split.
- (* -> *)
intros [t'' [STM V]].
destruct STM.
+ exfalso; apply value__normal in V; eauto.
+ rewrite (step_deterministic _ _ _ ST H). ∃ z. split; assumption.
- (* <- *)
intros [t'0 [STM V]].
∃ t'0. split; eauto.
Qed.
intros t t' ST. unfold halts.
split.
- (* -> *)
intros [t'' [STM V]].
destruct STM.
+ exfalso; apply value__normal in V; eauto.
+ rewrite (step_deterministic _ _ _ ST H). ∃ z. split; assumption.
- (* <- *)
intros [t'0 [STM V]].
∃ t'0. split; eauto.
Qed.
Now the main lemma, which comes in two parts, one for each
direction. Each proceeds by induction on the structure of the type
T. In fact, this is where we make fundamental use of the
structure of types.
One requirement for staying in R_T is to stay in type T. In the
forward direction, we get this from ordinary type Preservation.
Lemma step_preserves_R : ∀ T t t', (t --> t') → R T t → R T t'.
Proof.
induction T; intros t t' E Rt; unfold R; fold R; unfold R in Rt; fold R in Rt;
destruct Rt as [typable_empty_t [halts_t RRt]].
(* Bool *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
auto.
(* Arrow *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
intros.
eapply IHT2.
apply ST_App1. apply E.
apply RRt; auto.
(* FILL IN HERE *) Admitted.
induction T; intros t t' E Rt; unfold R; fold R; unfold R in Rt; fold R in Rt;
destruct Rt as [typable_empty_t [halts_t RRt]].
(* Bool *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
auto.
(* Arrow *)
split. eapply preservation; eauto.
split. apply (step_preserves_halting _ _ E); eauto.
intros.
eapply IHT2.
apply ST_App1. apply E.
apply RRt; auto.
(* FILL IN HERE *) Admitted.
The generalization to multiple steps is trivial:
Lemma multistep_preserves_R : ∀ T t t',
(t -->* t') → R T t → R T t'.
Proof.
intros T t t' STM; induction STM; intros.
assumption.
apply IHSTM. eapply step_preserves_R. apply H. assumption.
Qed.
intros T t t' STM; induction STM; intros.
assumption.
apply IHSTM. eapply step_preserves_R. apply H. assumption.
Qed.
In the reverse direction, we must add the fact that t has type
T before stepping as an additional hypothesis.
Lemma step_preserves_R' : ∀ T t t',
empty ⊢ t \in T → (t --> t') → R T t' → R T t.
Proof.
(* FILL IN HERE *) Admitted.
(* FILL IN HERE *) Admitted.
Lemma multistep_preserves_R' : ∀ T t t',
empty ⊢ t \in T → (t -->* t') → R T t' → R T t.
Proof.
intros T t t' HT STM.
induction STM; intros.
assumption.
eapply step_preserves_R'. assumption. apply H. apply IHSTM.
eapply preservation; eauto. auto.
Qed.
intros T t t' HT STM.
induction STM; intros.
assumption.
eapply step_preserves_R'. assumption. apply H. apply IHSTM.
eapply preservation; eauto. auto.
Qed.
Closed Instances of Terms of Type t Belong to R_T
Multisubstitutions, Multi-Extensions, and Instantiations
Definition env := list (string × tm).
Fixpoint msubst (ss:env) (t:tm) : tm :=
match ss with
| nil ⇒ t
| ((x,s)::ss') ⇒ msubst ss' <{ [x:=s]t }>
end.
We need similar machinery to talk about repeated extension of a
typing context using a list of (identifier, type) pairs, which we
call a type assignment.
Definition tass := list (string × ty).
Fixpoint mupdate (Gamma : context) (xts : tass) :=
match xts with
| nil ⇒ Gamma
| ((x,v)::xts') ⇒ update (mupdate Gamma xts') x v
end.
We will need some simple operations that work uniformly on
environments and type assigments
Fixpoint lookup {X:Set} (k : string) (l : list (string × X))
: option X :=
match l with
| nil ⇒ None
| (j,x) :: l' ⇒
if eqb_string j k then Some x else lookup k l'
end.
Fixpoint drop {X:Set} (n:string) (nxs:list (string × X))
: list (string × X) :=
match nxs with
| nil ⇒ nil
| ((n',x)::nxs') ⇒
if eqb_string n' n then drop n nxs'
else (n',x)::(drop n nxs')
end.
An instantiation combines a type assignment and a value
environment with the same domains, where corresponding elements are
in R.
Inductive instantiation : tass → env → Prop :=
| V_nil :
instantiation nil nil
| V_cons : ∀ x T v c e,
value v → R T v →
instantiation c e →
instantiation ((x,T)::c) ((x,v)::e).
We now proceed to prove various properties of these definitions.
Lemma vacuous_substitution : ∀ t x,
¬ appears_free_in x t →
∀ t', <{ [x:=t']t }> = t.
Proof with eauto.
(* FILL IN HERE *) Admitted.
(* FILL IN HERE *) Admitted.
Lemma subst_closed: ∀ t,
closed t →
∀ x t', <{ [x:=t']t }> = t.
Lemma subst_not_afi : ∀ t x v,
closed v → ¬ appears_free_in x <{ [x:=v]t }>.
Proof with eauto. (* rather slow this way *)
unfold closed, not.
induction t; intros x v P A; simpl in A.
- (* var *)
destruct (eqb_stringP x s)...
inversion A; subst. auto.
- (* app *)
inversion A; subst...
- (* abs *)
destruct (eqb_stringP x s)...
+ inversion A; subst...
+ inversion A; subst...
- (* tru *)
inversion A.
- (* fls *)
inversion A.
- (* test *)
inversion A; subst...
- (* pair *)
inversion A; subst...
- (* fst *)
inversion A; subst...
- (* snd *)
inversion A; subst...
Qed.
unfold closed, not.
induction t; intros x v P A; simpl in A.
- (* var *)
destruct (eqb_stringP x s)...
inversion A; subst. auto.
- (* app *)
inversion A; subst...
- (* abs *)
destruct (eqb_stringP x s)...
+ inversion A; subst...
+ inversion A; subst...
- (* tru *)
inversion A.
- (* fls *)
inversion A.
- (* test *)
inversion A; subst...
- (* pair *)
inversion A; subst...
- (* fst *)
inversion A; subst...
- (* snd *)
inversion A; subst...
Qed.
Lemma duplicate_subst : ∀ t' x t v,
closed v → <{ [x:=t]([x:=v]t') }> = <{ [x:=v]t' }>.
Lemma swap_subst : ∀ t x x1 v v1,
x ≠ x1 →
closed v → closed v1 →
<{ [x1:=v1]([x:=v]t) }> = <{ [x:=v]([x1:=v1]t) }>.
Proof with eauto.
induction t; intros; simpl.
- (* var *)
destruct (eqb_stringP x s); destruct (eqb_stringP x1 s).
+ subst. exfalso...
+ subst. simpl. rewrite <- eqb_string_refl. apply subst_closed...
+ subst. simpl. rewrite <- eqb_string_refl. rewrite subst_closed...
+ simpl. rewrite false_eqb_string... rewrite false_eqb_string...
(* FILL IN HERE *) Admitted.
induction t; intros; simpl.
- (* var *)
destruct (eqb_stringP x s); destruct (eqb_stringP x1 s).
+ subst. exfalso...
+ subst. simpl. rewrite <- eqb_string_refl. apply subst_closed...
+ subst. simpl. rewrite <- eqb_string_refl. rewrite subst_closed...
+ simpl. rewrite false_eqb_string... rewrite false_eqb_string...
(* FILL IN HERE *) Admitted.
Closed environments are those that contain only closed terms.
Fixpoint closed_env (env:env) :=
match env with
| nil ⇒ True
| (x,t)::env' ⇒ closed t ∧ closed_env env'
end.
Next come a series of lemmas charcterizing how msubst of closed terms
distributes over subst and over each term form
Lemma subst_msubst: ∀ env x v t, closed v → closed_env env →
msubst env <{ [x:=v]t }> = <{ [x:=v] { msubst (drop x env) t } }> .
Proof.
induction env0; intros; auto.
destruct a. simpl.
inversion H0.
destruct (eqb_stringP s x).
- subst. rewrite duplicate_subst; auto.
- simpl. rewrite swap_subst; eauto.
Qed.
induction env0; intros; auto.
destruct a. simpl.
inversion H0.
destruct (eqb_stringP s x).
- subst. rewrite duplicate_subst; auto.
- simpl. rewrite swap_subst; eauto.
Qed.
Lemma msubst_var: ∀ ss x, closed_env ss →
msubst ss (tm_var x) =
match lookup x ss with
| Some t ⇒ t
| None ⇒ tm_var x
end.
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eqb_string s x).
apply msubst_closed. inversion H; auto.
apply IHss. inversion H; auto.
Qed.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eqb_string s x).
apply msubst_closed. inversion H; auto.
apply IHss. inversion H; auto.
Qed.
Lemma msubst_abs: ∀ ss x T t,
msubst ss <{ \ x : T, t }> = <{ \x : T, {msubst (drop x ss) t} }>.
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eqb_string s x); simpl; auto.
Qed.
induction ss; intros.
reflexivity.
destruct a.
simpl. destruct (eqb_string s x); simpl; auto.
Qed.
Lemma msubst_app : ∀ ss t1 t2,
msubst ss <{ t1 t2 }> = <{ {msubst ss t1} ({msubst ss t2}) }>.
Proof.
induction ss; intros.
reflexivity.
destruct a.
simpl. rewrite <- IHss. auto.
Qed.
induction ss; intros.
reflexivity.
destruct a.
simpl. rewrite <- IHss. auto.
Qed.
You'll need similar functions for the other term constructors.
(* FILL IN HERE *)
Properties of Multi-Extensions
Lemma mupdate_lookup : ∀ (c : tass) (x:string),
lookup x c = (mupdate empty c) x.
Proof.
induction c; intros.
auto.
destruct a. unfold lookup, mupdate, update, t_update. destruct (eqb_string s x); auto.
Qed.
induction c; intros.
auto.
destruct a. unfold lookup, mupdate, update, t_update. destruct (eqb_string s x); auto.
Qed.
Lemma mupdate_drop : ∀ (c: tass) Gamma x x',
mupdate Gamma (drop x c) x'
= if eqb_string x x' then Gamma x' else mupdate Gamma c x'.
Proof.
induction c; intros.
- destruct (eqb_stringP x x'); auto.
- destruct a. simpl.
destruct (eqb_stringP s x).
+ subst. rewrite IHc.
unfold update, t_update. destruct (eqb_stringP x x'); auto.
+ simpl. unfold update, t_update. destruct (eqb_stringP s x'); auto.
subst. rewrite false_eqb_string; congruence.
Qed.
induction c; intros.
- destruct (eqb_stringP x x'); auto.
- destruct a. simpl.
destruct (eqb_stringP s x).
+ subst. rewrite IHc.
unfold update, t_update. destruct (eqb_stringP x x'); auto.
+ simpl. unfold update, t_update. destruct (eqb_stringP s x'); auto.
subst. rewrite false_eqb_string; congruence.
Qed.
Lemma instantiation_domains_match: ∀ {c} {e},
instantiation c e →
∀ {x} {T},
lookup x c = Some T → ∃ t, lookup x e = Some t.
Proof.
intros c e V. induction V; intros x0 T0 C.
solve_by_invert.
simpl in ×.
destruct (eqb_string x x0); eauto.
Qed.
intros c e V. induction V; intros x0 T0 C.
solve_by_invert.
simpl in ×.
destruct (eqb_string x x0); eauto.
Qed.
Lemma instantiation_env_closed : ∀ c e,
instantiation c e → closed_env e.
Proof.
intros c e V; induction V; intros.
econstructor.
unfold closed_env. fold closed_env.
split; [|assumption].
eapply typable_empty__closed. eapply R_typable_empty. eauto.
Qed.
intros c e V; induction V; intros.
econstructor.
unfold closed_env. fold closed_env.
split; [|assumption].
eapply typable_empty__closed. eapply R_typable_empty. eauto.
Qed.
Lemma instantiation_R : ∀ c e,
instantiation c e →
∀ x t T,
lookup x c = Some T →
lookup x e = Some t → R T t.
Proof.
intros c e V. induction V; intros x' t' T' G E.
solve_by_invert.
unfold lookup in ×. destruct (eqb_string x x').
inversion G; inversion E; subst. auto.
eauto.
Qed.
intros c e V. induction V; intros x' t' T' G E.
solve_by_invert.
unfold lookup in ×. destruct (eqb_string x x').
inversion G; inversion E; subst. auto.
eauto.
Qed.
Lemma instantiation_drop : ∀ c env,
instantiation c env →
∀ x, instantiation (drop x c) (drop x env).
Proof.
intros c e V. induction V.
intros. simpl. constructor.
intros. unfold drop.
destruct (eqb_string x x0); auto. constructor; eauto.
Qed.
intros c e V. induction V.
intros. simpl. constructor.
intros. unfold drop.
destruct (eqb_string x x0); auto. constructor; eauto.
Qed.
Lemma multistep_App2 : ∀ v t t',
value v → (t -->* t') → <{ v t }> -->* <{ v t' }>.
Proof.
intros v t t' V STM. induction STM.
apply multi_refl.
eapply multi_step.
apply ST_App2; eauto. auto.
Qed.
intros v t t' V STM. induction STM.
apply multi_refl.
eapply multi_step.
apply ST_App2; eauto. auto.
Qed.
(* FILL IN HERE *)
The R Lemma
Lemma msubst_preserves_typing : ∀ c e,
instantiation c e →
∀ Gamma t S, (mupdate Gamma c) ⊢ t \in S →
Gamma ⊢ { (msubst e t) } \in S.
Proof.
induction 1; intros.
simpl in H. simpl. auto.
simpl in H2. simpl.
apply IHinstantiation.
eapply substitution_preserves_typing; eauto.
apply (R_typable_empty H0).
Qed.
induction 1; intros.
simpl in H. simpl. auto.
simpl in H2. simpl.
apply IHinstantiation.
eapply substitution_preserves_typing; eauto.
apply (R_typable_empty H0).
Qed.
And at long last, the main lemma.
Lemma msubst_R : ∀ c env t T,
(mupdate empty c) ⊢ t \in T →
instantiation c env →
R T (msubst env t).
Proof.
intros c env0 t T HT V.
generalize dependent env0.
(* We need to generalize the hypothesis a bit before setting up the induction. *)
remember (mupdate empty c) as Gamma.
assert (∀ x, Gamma x = lookup x c).
intros. rewrite HeqGamma. rewrite mupdate_lookup. auto.
clear HeqGamma.
generalize dependent c.
induction HT; intros.
- (* T_Var *)
rewrite H0 in H. destruct (instantiation_domains_match V H) as [t P].
eapply instantiation_R; eauto.
rewrite msubst_var. rewrite P. auto. eapply instantiation_env_closed; eauto.
- (* T_Abs *)
rewrite msubst_abs.
(* We'll need variants of the following fact several times, so its simplest to
establish it just once. *)
assert (WT : empty ⊢ \x : T2, {msubst (drop x env0) t1 } \in (T2 → T1) ).
{ eapply T_Abs. eapply msubst_preserves_typing.
{ eapply instantiation_drop; eauto. }
eapply context_invariance.
{ apply HT. }
intros.
unfold update, t_update. rewrite mupdate_drop. destruct (eqb_stringP x x0).
+ auto.
+ rewrite H.
clear - c n. induction c.
simpl. rewrite false_eqb_string; auto.
simpl. destruct a. unfold update, t_update.
destruct (eqb_string s x0); auto. }
unfold R. fold R. split.
auto.
split. apply value_halts. apply v_abs.
intros.
destruct (R_halts H0) as [v [P Q]].
pose proof (multistep_preserves_R _ _ _ P H0).
apply multistep_preserves_R' with (msubst ((x,v)::env0) t1).
eapply T_App. eauto.
apply R_typable_empty; auto.
eapply multi_trans. eapply multistep_App2; eauto.
eapply multi_R.
simpl. rewrite subst_msubst.
eapply ST_AppAbs; eauto.
eapply typable_empty__closed.
apply (R_typable_empty H1).
eapply instantiation_env_closed; eauto.
eapply (IHHT ((x,T2)::c)).
intros. unfold update, t_update, lookup. destruct (eqb_string x x0); auto.
constructor; auto.
- (* T_App *)
rewrite msubst_app.
destruct (IHHT1 c H env0 V) as [_ [_ P1]].
pose proof (IHHT2 c H env0 V) as P2. fold R in P1. auto.
(* FILL IN HERE *) Admitted.
intros c env0 t T HT V.
generalize dependent env0.
(* We need to generalize the hypothesis a bit before setting up the induction. *)
remember (mupdate empty c) as Gamma.
assert (∀ x, Gamma x = lookup x c).
intros. rewrite HeqGamma. rewrite mupdate_lookup. auto.
clear HeqGamma.
generalize dependent c.
induction HT; intros.
- (* T_Var *)
rewrite H0 in H. destruct (instantiation_domains_match V H) as [t P].
eapply instantiation_R; eauto.
rewrite msubst_var. rewrite P. auto. eapply instantiation_env_closed; eauto.
- (* T_Abs *)
rewrite msubst_abs.
(* We'll need variants of the following fact several times, so its simplest to
establish it just once. *)
assert (WT : empty ⊢ \x : T2, {msubst (drop x env0) t1 } \in (T2 → T1) ).
{ eapply T_Abs. eapply msubst_preserves_typing.
{ eapply instantiation_drop; eauto. }
eapply context_invariance.
{ apply HT. }
intros.
unfold update, t_update. rewrite mupdate_drop. destruct (eqb_stringP x x0).
+ auto.
+ rewrite H.
clear - c n. induction c.
simpl. rewrite false_eqb_string; auto.
simpl. destruct a. unfold update, t_update.
destruct (eqb_string s x0); auto. }
unfold R. fold R. split.
auto.
split. apply value_halts. apply v_abs.
intros.
destruct (R_halts H0) as [v [P Q]].
pose proof (multistep_preserves_R _ _ _ P H0).
apply multistep_preserves_R' with (msubst ((x,v)::env0) t1).
eapply T_App. eauto.
apply R_typable_empty; auto.
eapply multi_trans. eapply multistep_App2; eauto.
eapply multi_R.
simpl. rewrite subst_msubst.
eapply ST_AppAbs; eauto.
eapply typable_empty__closed.
apply (R_typable_empty H1).
eapply instantiation_env_closed; eauto.
eapply (IHHT ((x,T2)::c)).
intros. unfold update, t_update, lookup. destruct (eqb_string x x0); auto.
constructor; auto.
- (* T_App *)
rewrite msubst_app.
destruct (IHHT1 c H env0 V) as [_ [_ P1]].
pose proof (IHHT2 c H env0 V) as P2. fold R in P1. auto.
(* FILL IN HERE *) Admitted.