(** * Stlc: The Simply Typed Lambda-Calculus *)
(* $Date: 2012-07-25 16:43:16 -0400 (Wed, 25 Jul 2012) $ *)
Require Export Ch11_Types.
(* ###################################################################### *)
(** * The Simply Typed Lambda-Calculus *)
(** The simply typed lambda-calculus (STLC) is a tiny core calculus
embodying the key concept of _functional abstraction_, which shows
up in pretty much every real-world programming language in some
form (functions, procedures, methods, etc.).
We will follow exactly the same pattern as in the previous
chapter when formalizing this calculus (syntax, small-step
semantics, typing rules) and its main properties (progress and
preservation). The new technical challenges (which will take some
work to deal with) all arise from the mechanisms of _variable
binding_ and _substitution_. *)
(* ###################################################################### *)
(** ** Overview *)
(** The STLC is built on some collection of _base types_ -- booleans,
numbers, strings, etc. The exact choice of base types doesn't
matter -- the construction of the language and its theoretical
properties work out pretty much the same -- so for the sake of
brevity let's take just [Bool] for the moment. At the end of the
chapter we'll see how to add more base types, and in later
chapters we'll enrich the pure STLC with other useful constructs
like pairs, records, subtyping, and mutable state.
Starting from the booleans, we add three things:
- variables
- function abstractions
- application
This gives us the following collection of abstract syntax
constructors (written out here in informal BNF notation -- we'll
formalize it below).
*)
(** Informal concrete syntax:
<<
t ::= x variable
| \x:T.t1 abstraction
| t1 t2 application
| true constant true
| false constant false
| if t1 then t2 else t3 conditional
>>
*)
(** The [\] symbol in a function abstraction [\x:T.t1] is often
written as a greek "lambda" (hence the name of the calculus). The
variable [x] is called the _parameter_ to the function; the term
[t1] is its _body_. The annotation [:T] specifies the type of
arguments that the function can be applied to.
Some examples:
- [\x:Bool. x]
The identity function for booleans.
- [(\x:Bool. x) true]
The identity function for booleans, applied to the boolean [true].
- [\x:Bool. if x then false else true]
The boolean "not" function.
- [\x:Bool. true]
The constant function that takes every (boolean) argument to
[true].
- [\x:Bool. \y:Bool. x]
A two-argument function that takes two booleans and returns
the first one. (Note that, as in Coq, a two-argument function
is really a one-argument function whose body is also a
one-argument function.)
- [(\x:Bool. \y:Bool. x) false true]
A two-argument function that takes two booleans and returns
the first one, applied to the booleans [false] and [true].
Note that, as in Coq, application associates to the left --
i.e., this expression is parsed as [((\x:Bool. \y:Bool. x)
false) true].
- [\f:Bool->Bool. f (f true)]
A higher-order function that takes a _function_ [f] (from
booleans to booleans) as an argument, applies [f] to [true],
and applies [f] again to the result.
- [(\f:Bool->Bool. f (f true)) (\x:Bool. false)]
The same higher-order function, applied to the constantly
[false] function. *)
(** As the last several examples show, the STLC is a language of
_higher-order_ functions: we can write down functions that take
other functions as arguments and/or return other functions as
results.
Another point to note is that the STLC doesn't provide any
primitive syntax for defining _named_ functions -- all functions
are "anonymous." We'll see in chapter [MoreStlc] that it is easy
to add named functions to what we've got -- indeed, the
fundamental naming and binding mechanisms are exactly the same.
The _types_ of the STLC include [Bool], which classifies the
boolean constants [true] and [false] as well as more complex
computations that yield booleans, plus _arrow types_ that classify
functions. *)
(**
<<
T ::= Bool
| T1 -> T2
>>
For example:
- [\x:Bool. false] has type [Bool->Bool]
- [\x:Bool. x] has type [Bool->Bool]
- [(\x:Bool. x) true] has type [Bool]
- [\x:Bool. \y:Bool. x] has type [Bool->Bool->Bool] (i.e. [Bool -> (Bool->Bool)])
- [(\x:Bool. \y:Bool. x) false] has type [Bool->Bool]
- [(\x:Bool. \y:Bool. x) false true] has type [Bool]
- [\f:Bool->Bool. f (f true)] has type [(Bool->Bool) -> Bool]
- [(\f:Bool->Bool. f (f true)) (\x:Bool. false)] has type [Bool]
*)
(* ###################################################################### *)
(** ** Syntax *)
Module STLC.
(* ################################### *)
(** *** Types *)
Inductive ty : Type :=
| TBool : ty
| TArrow : ty -> ty -> ty.
(* ################################### *)
(** *** Terms *)
Inductive tm : Type :=
| tvar : id -> tm
| tapp : tm -> tm -> tm
| tabs : id -> ty -> tm -> tm
| ttrue : tm
| tfalse : tm
| tif : tm -> tm -> tm -> tm.
Tactic Notation "t_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "tvar" | Case_aux c "tapp"
| Case_aux c "tabs" | Case_aux c "ttrue"
| Case_aux c "tfalse" | Case_aux c "tif" ].
(** Note that an abstraction [\x:T.t] (formally, [tabs x T t]) is
always annotated with the type [T] of its parameter, in contrast
to Coq (and other functional languages like ML, Haskell, etc.),
which use _type inference_ to fill in missing annotations. We're
not considering type inference here, to keep things simple. *)
(** Some examples... *)
Definition x := (Id 0).
Definition y := (Id 1).
Definition z := (Id 2).
Hint Unfold x.
Hint Unfold y.
Hint Unfold z.
(** [idB = \x:Bool. x] *)
Notation idB :=
(tabs x TBool (tvar x)).
(** [idBB = \x:Bool->Bool. x] *)
Notation idBB :=
(tabs x (TArrow TBool TBool) (tvar x)).
(** [idBBBB = \x:(Bool->Bool)->(Bool->Bool). x] *)
Notation idBBBB :=
(tabs x (TArrow (TArrow TBool TBool)
(TArrow TBool TBool))
(tvar x)).
(** [k = \x:Bool. \y:Bool. x] *)
Notation k := (tabs x TBool (tabs y TBool (tvar x))).
(** (We write these as [Notation]s rather than [Definition]s to make
things easier for [auto].) *)
(* ###################################################################### *)
(** ** Operational Semantics *)
(** To define the small-step semantics of STLC terms, we begin -- as
always -- by defining the set of values. Next, we define the
critical notions of _free variables_ and _substitution_, which are
used in the reduction rule for application expressions. And
finally we give the small-step relation itself. *)
(* ################################### *)
(** *** Values *)
(** To define the values of the STLC, we have a few cases to consider.
First, for the boolean part of the language, the situation is
clear: [true] and [false] are the only values. (An [if]
expression is never a value.)
Second, an application is clearly not a value: It represents a
function being invoked on some argument, which clearly still has
work left to do.
Third, for abstractions, we have a choice:
- We can say that [\x:T.t1] is a value only when [t1] is a
value -- i.e., only if the function's body has been
reduced (as much as it can be without knowing what argument it
is going to be applied to).
- Or we can say that [\x:T.t1] is always a value, no matter
whether [t1] is one or not -- in other words, we can say that
reduction stops at abstractions.
Coq (in its built-in functional programming langauge) makes the
first choice -- for example,
Eval simpl in (fun x:bool => 3 + 4)
yields [fun x:bool => 7]. But most real-world functional
programming languages make the second choice -- reduction of a
function's body only begins when the function is actually applied
to an argument. We also make the second choice here.
Finally, having made the choice not to reduce under abstractions,
we don't need to worry about whether variables are values, since
we'll always be reducing programs "from the outside in," and that
means the [step] relation will always be working with closed
terms (ones with no free variables). *)
Inductive value : tm -> Prop :=
| v_abs : forall x T t,
value (tabs x T t)
| t_true :
value ttrue
| t_false :
value tfalse.
Hint Constructors value.
(* ###################################################################### *)
(** *** Free Variables and Substitution *)
(** Now we come to the heart of the matter: the operation of
substituting one term for a variable in another term.
This operation will be used below to define the operational
semantics of function application, where we will need to
substitute the argument term for the function parameter in the
function's body. For example, we reduce
(\x:Bool. if x then true else x) false
to [false] by substituting [false] for the parameter [x] in the
body of the function. In general, we need to be able to
substitute some given term [s] for occurrences of some variable
[x] in another term [t]. In informal discussions, this is usually
written [ [x:=s]t ] and pronounced "substitute [x] with [s] in [t]."
Here are some examples:
- [[x:=true] (if x then x else false)] yields [if true then true else false]
- [[x:=true] x] yields [true]
- [[x:=true] (if x then x else y)] yields [if true then true else y]
- [[x:=true] y] yields [y]
- [[x:=true] false] yields [false] (vacuous substitution)
- [[x:=true] (\y:Bool. if y then x else false)] yields [\y:Bool. if y then true else false]
- [[x:=true] (\y:Bool. x)] yields [\y:Bool. true]
- [[x:=true] (\y:Bool. y)] yields [\y:Bool. y]
- [[x:=true] (\x:Bool. x)] yields [\x:Bool. x]
The last example is very important: substituting [x] with [true] in
[\x:Bool. x] does _not_ yield [\x:Bool. true]! The reason for
this is that the [x] in the body of [\x:Bool. x] is _bound_ by the
abstraction: it is a new, local name that just happens to be
spelled the same as some global name [x]. *)
(** Here is the definition, informally...
[x:=s]x = s
[x:=s]y = y if x <> y
[x:=s](\x:T11.t12) = \x:T11. t12
[x:=s](\y:T11.t12) = \y:T11. [x:=s]t12 if x <> y
[x:=s](t1 t2) = ([x:=s]t1) ([x:=s]t2)
[x:=s]true = true
[x:=s]false = false
[x:=s](if t1 then t2 else t3) =
if [x:=s]t1 then [x:=s]t2 else [x:=s]t3
]]
... and formally: *)
Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
match t with
| tvar x' =>
if beq_id x x' then s else t
| tabs x' T t1 =>
tabs x' T (if beq_id x x' then t1 else (subst x s t1))
| tapp t1 t2 =>
tapp (subst x s t1) (subst x s t2)
| ttrue =>
ttrue
| tfalse =>
tfalse
| tif t1 t2 t3 =>
tif (subst x s t1) (subst x s t2) (subst x s t3)
end.
Notation "'[' x ':=' s ']' t" := (subst x s t) (at level 20).
(** _Technical note_: Substitution becomes trickier to define if we
consider the case where [s], the term being substituted for a
variable in some other term, may itself contain free variables.
Since we are only interested here in defining the [step] relation
on closed terms (i.e., terms like [\x:Bool. x y] that mention
variables are not bound by some enclosing lambda), we can avoid
this extra complexity here. *)
(* ################################### *)
(** *** Reduction *)
(** The small-step reduction relation for STLC now follows the same
pattern as the ones we have seen before. Intuitively, to reduce a
function application, we first reduce its left-hand side until it
becomes a literal function; then we reduce its right-hand
side (the argument) until it is also a value; and finally we
substitute the argument for the bound variable in the body of the
function. This last rule, written informally as
(\x:T.t12) v2 ==> [x:=v2]t12
is traditionally called "beta-reduction". *)
(**
value v2
---------------------------- (ST_AppAbs)
(\x:T.t12) v2 ==> [x:=v2]t12
t1 ==> t1'
---------------- (ST_App1)
t1 t2 ==> t1' t2
value v1
t2 ==> t2'
---------------- (ST_App2)
v1 t2 ==> v1 t2'
*)
(** ... plus the usual rules for booleans:
-------------------------------- (ST_IfTrue)
(if true then t1 else t2) ==> t1
--------------------------------- (ST_IfFalse)
(if false then t1 else t2) ==> t2
t1 ==> t1'
---------------------------------------------------- (ST_If)
(if t1 then t2 else t3) ==> (if t1' then t2 else t3)
*)
Reserved Notation "t1 '==>' t2" (at level 40).
Inductive step : tm -> tm -> Prop :=
| ST_AppAbs : forall x T t12 v2,
value v2 ->
(tapp (tabs x T t12) v2) ==> [x:=v2]t12
| ST_App1 : forall t1 t1' t2,
t1 ==> t1' ->
tapp t1 t2 ==> tapp t1' t2
| ST_App2 : forall v1 t2 t2',
value v1 ->
t2 ==> t2' ->
tapp v1 t2 ==> tapp v1 t2'
| ST_IfTrue : forall t1 t2,
(tif ttrue t1 t2) ==> t1
| ST_IfFalse : forall t1 t2,
(tif tfalse t1 t2) ==> t2
| ST_If : forall t1 t1' t2 t3,
t1 ==> t1' ->
(tif t1 t2 t3) ==> (tif t1' t2 t3)
where "t1 '==>' t2" := (step t1 t2).
Tactic Notation "step_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ST_AppAbs" | Case_aux c "ST_App1"
| Case_aux c "ST_App2" | Case_aux c "ST_IfTrue"
| Case_aux c "ST_IfFalse" | Case_aux c "ST_If" ].
Hint Constructors step.
Notation multistep := (multi step).
Notation "t1 '==>*' t2" := (multistep t1 t2) (at level 40).
(* ##################################### *)
(** *** Examples *)
(** Example:
((\x:Bool->Bool. x) (\x:Bool. x)) ==>* (\x:Bool. x)
i.e.
(idBB idB) ==>* idB
*)
Lemma step_example1 :
(tapp idBB idB) ==>* idB.
Proof.
eapply multi_step.
apply ST_AppAbs.
apply v_abs.
simpl.
apply multi_refl. Qed.
(** A more automatic proof *)
Lemma step_example1' :
(tapp idBB idB) ==>* idB.
Proof. normalize. Qed.
Lemma step_example2 :
(tapp idBB (tapp idBB idB)) ==>* idB.
Proof.
eapply multi_step.
apply ST_App2. auto.
apply ST_AppAbs. auto.
eapply multi_step.
apply ST_AppAbs. simpl. auto.
simpl. apply multi_refl. Qed.
(** Again, we can use the [normalize] tactic from above to simplify
the proof. *)
Lemma step_example2' :
(tapp idBB (tapp idBB idB)) ==>* idB.
Proof.
normalize.
Qed.
(** **** Exercise: 2 stars (step_example3) *)
(* EXPECTED *)
(** Try to do this one both with and without [normalize]. *)
Lemma step_example3 :
(tapp (tapp idBBBB idBB) idB)
==>* idB.
Proof.
(* FILL IN HERE *) Admitted.
(* FILL IN HERE *)
(** [] *)
(* ###################################################################### *)
(** ** Typing *)
(* ################################### *)
(** *** Contexts *)
(** _Question_: What is the type of the term "[x y]"?
_Answer_: It depends on the types of [x] and [y]!
I.e., in order to assign a type to a term, we need to know
what assumptions we should make about the types of its free
variables.
This leads us to a three-place "typing judgment", informally
written [Gamma |- t : T], where [Gamma] is a
"typing context" -- a mapping from variables to their types. *)
(** We hide the definition of partial maps in a module since it is
actually defined in [SfLib]. *)
Module PartialMap.
Definition partial_map (A:Type) := id -> option A.
Definition empty {A:Type} : partial_map A := (fun _ => None).
(** Informally, we'll write [Gamma, x:T] for "extend the partial
function [Gamma] to also map [x] to [T]." Formally, we use the
function [extend] to add a binding to a partial map. *)
Definition extend {A:Type} (Gamma : partial_map A) (x:id) (T : A) :=
fun x' => if beq_id x x' then Some T else Gamma x'.
Lemma extend_eq : forall A (ctxt: partial_map A) x T,
(extend ctxt x T) x = Some T.
Proof.
intros. unfold extend. rewrite <- beq_id_refl. auto.
Qed.
Lemma extend_neq : forall A (ctxt: partial_map A) x1 T x2,
beq_id x2 x1 = false ->
(extend ctxt x2 T) x1 = ctxt x1.
Proof.
intros. unfold extend. rewrite H. auto.
Qed.
End PartialMap.
Definition context := partial_map ty.
(* ################################### *)
(** *** Typing Relation *)
(**
Gamma x = T
-------------- (T_Var)
Gamma |- x : T
Gamma , x:T11 |- t12 : T12
---------------------------- (T_Abs)
Gamma |- \x:T11.t12 : T11->T12
Gamma |- t1 : T11->T12
Gamma |- t2 : T11
---------------------- (T_App)
Gamma |- t1 t2 : T12
-------------------- (T_True)
Gamma |- true : Bool
--------------------- (T_False)
Gamma |- false : Bool
Gamma |- t1 : Bool Gamma |- t2 : T Gamma |- t3 : T
-------------------------------------------------------- (T_If)
Gamma |- if t1 then t2 else t3 : T
*)
Inductive has_type : context -> tm -> ty -> Prop :=
| T_Var : forall Gamma x T,
Gamma x = Some T ->
has_type Gamma (tvar x) T
| T_Abs : forall Gamma x T11 T12 t12,
has_type (extend Gamma x T11) t12 T12 ->
has_type Gamma (tabs x T11 t12) (TArrow T11 T12)
| T_App : forall T11 T12 Gamma t1 t2,
has_type Gamma t1 (TArrow T11 T12) ->
has_type Gamma t2 T11 ->
has_type Gamma (tapp t1 t2) T12
| T_True : forall Gamma,
has_type Gamma ttrue TBool
| T_False : forall Gamma,
has_type Gamma tfalse TBool
| T_If : forall t1 t2 t3 T Gamma,
has_type Gamma t1 TBool ->
has_type Gamma t2 T ->
has_type Gamma t3 T ->
has_type Gamma (tif t1 t2 t3) T.
Tactic Notation "has_type_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "T_Var" | Case_aux c "T_Abs"
| Case_aux c "T_App" | Case_aux c "T_True"
| Case_aux c "T_False" | Case_aux c "T_If" ].
Hint Constructors has_type.
(* ################################### *)
(** *** Examples *)
Example typing_example_1 :
has_type empty (tabs x TBool (tvar x)) (TArrow TBool TBool).
Proof.
apply T_Abs. apply T_Var. reflexivity. Qed.
(** Note that since we added the [has_type] constructors to the hints
database, auto can actually solve this one immediately. *)
Example typing_example_1' :
has_type empty (tabs x TBool (tvar x)) (TArrow TBool TBool).
Proof. auto. Qed.
Hint Unfold beq_id beq_nat extend.
(** Another example:
empty |- \x:A. \y:A->A. y (y x))
: A -> (A->A) -> A.
*)
Example typing_example_2 :
has_type empty
(tabs x TBool
(tabs y (TArrow TBool TBool)
(tapp (tvar y) (tapp (tvar y) (tvar x)))))
(TArrow TBool (TArrow (TArrow TBool TBool) TBool)).
Proof with auto using extend_eq.
apply T_Abs.
apply T_Abs.
eapply T_App. apply T_Var...
eapply T_App. apply T_Var...
apply T_Var...
Qed.
(** **** Exercise: 2 stars, optional (typing_example_2_full) *)
(* EXPECTED *)
(** Prove the same result without using [auto], [eauto], or
[eapply]. *)
Example typing_example_2_full :
has_type empty
(tabs x TBool
(tabs y (TArrow TBool TBool)
(tapp (tvar y) (tapp (tvar y) (tvar x)))))
(TArrow TBool (TArrow (TArrow TBool TBool) TBool)).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 2 stars (typing_example_3) *)
(* EXPECTED *)
(** Formally prove the following typing derivation holds: *)
(**
empty |- (\x:Bool->B. \y:Bool->Bool. \z:Bool.
y (x z))
: T.
*)
Example typing_example_3 :
exists T,
has_type empty
(tabs x (TArrow TBool TBool)
(tabs y (TArrow TBool TBool)
(tabs z TBool
(tapp (tvar y) (tapp (tvar x) (tvar z))))))
T.
Proof with auto.
(* FILL IN HERE *) Admitted.
(** [] *)
(** We can also show that terms are _not_ typable. For example, let's
formally check that there is no typing derivation assigning a type
to the term [\x:Bool. \y:Bool, x y] -- i.e.,
~ exists T,
empty |- (\x:Bool. \y:Bool, x y) : T.
*)
Example typing_nonexample_1 :
~ exists T,
has_type empty
(tabs x TBool
(tabs y TBool
(tapp (tvar x) (tvar y))))
T.
Proof.
intros Hc. inversion Hc.
(* The [clear] tactic is useful here for tidying away bits of
the context that we're not going to need again. *)
inversion H. subst. clear H.
inversion H5. subst. clear H5.
inversion H4. subst. clear H4.
inversion H2. subst. clear H2.
inversion H5. subst. clear H5.
(* rewrite extend_neq in H1. rewrite extend_eq in H1. *)
inversion H1. Qed.
(** **** Exercise: 3 stars (typing_nonexample_3) *)
(* EXPECTED *)
(** Another nonexample:
~ (exists S, exists T,
empty |- (\x:S. x x) : T).
*)
Example typing_nonexample_3 :
~ (exists S, exists T,
has_type empty
(tabs x S
(tapp (tvar x) (tvar x)))
T).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, optional (typing_statements) *)
(** Which of the following propositions are provable?
- [y:Bool |- \x:Bool.x : Bool->Bool]
- [exists T, empty |- (\y:Bool->Bool. \x:Bool. y x) : T]
- [exists T, empty |- (\y:Bool->Bool. \x:Bool. x y) : T]
- [exists S, x:S |- (\y:Bool->Bool. y) x : S]
- [exists S, exists T, x:S |- (x x x) : T]
[]
*)
(** **** Exercise: 1 star (more_typing_statements) *)
(* EXPECTED *)
(** Which of the following propositions are provable? For the
ones that are, give witnesses for the existentially bound
variables.
- [exists T, empty |- (\y:B->B->B. \x:B, y x) : T]
- [exists T, empty |- (\x:A->B, \y:B->C, \z:A, y (x z)):T]
- [exists S, exists U, exists T, x:S, y:U |- \z:A. x (y z) : T]
- [exists S, exists T, x:S |- \y:A. x (x y) : T]
- [exists S, exists U, exists T, x:S |- x (\z:U. z x) : T]
[]
*)
(* ###################################################################### *)
(** ** Properties *)
(* ###################################################################### *)
(** *** Progress *)
(** The _progress_ theorem tells us that closed, well-typed
terms are not stuck: either a well-typed term is a value, or
it can take an evaluation step. *)
Theorem progress : forall t T,
has_type empty t T ->
value t \/ exists t', t ==> t'.
(** _Proof_: by induction on the derivation of [|- t : T].
- The last rule of the derivation cannot be [T_Var], since a
variable is never well typed in an empty context.
- The [T_True], [T_False], and [T_Abs] cases are trivial, since in
each of these cases we know immediately that [t] is a value.
- If the last rule of the derivation was [T_App], then [t = t1
t2], and we know that [t1] and [t2] are also well typed in the
empty context; in particular, there exists a type [T2] such that
[|- t1 : T2 -> T] and [|- t2 : T2]. By the induction
hypothesis, either [t1] is a value or it can take an evaluation
step.
- If [t1] is a value, we now consider [t2], which by the other
induction hypothesis must also either be a value or take an
evaluation step.
- Suppose [t2] is a value. Since [t1] is a value with an
arrow type, it must be a lambda abstraction; hence [t1
t2] can take a step by [ST_AppAbs].
- Otherwise, [t2] can take a step, and hence so can [t1
t2] by [ST_App2].
- If [t1] can take a step, then so can [t1 t2] by [ST_App1].
- If the last rule of the derivation was [T_If], then [t = if t1
then t2 else t3], where [t1] has type [Bool]. By the IH, [t1]
is either a value or takes a step.
- If [t1] is a value, then since it has type [Bool] it must be
either [true] or [false]. If it is [true], then [t] steps
to [t2]; otherwise it steps to [t3].
- Otherwise, [t1] takes a step, and therefore so does [t] (by
[ST_If]).
*)
Proof with eauto.
intros t T Ht.
remember (@empty ty) as Gamma.
has_type_cases (induction Ht) Case; subst Gamma...
Case "T_Var".
(* contradictory: variables cannot be typed in an
empty context *)
inversion H.
Case "T_App".
(* [t] = [t1 t2]. Proceed by cases on whether [t1] is a
value or steps... *)
right. destruct IHHt1...
SCase "t1 is a value".
destruct IHHt2...
SSCase "t2 is also a value".
(* Since [t1] is a value and has an arrow type, it
must be an abs. Sometimes this is proved separately
and called a "canonical forms" lemma. *)
inversion H; subst. exists ([x0:=t2]t)...
solve by inversion. solve by inversion.
SSCase "t2 steps".
inversion H0 as [t2' Hstp]. exists (tapp t1 t2')...
SCase "t1 steps".
inversion H as [t1' Hstp]. exists (tapp t1' t2)...
Case "T_If".
right. destruct IHHt1...
SCase "t1 is a value".
(* Since [t1] is a value of boolean type, it must
be true or false *)
inversion H; subst. solve by inversion.
SSCase "t1 = true". eauto.
SSCase "t1 = false". eauto.
SCase "t1 also steps".
inversion H as [t1' Hstp]. exists (tif t1' t2 t3)...
Qed.
(** **** Exercise: 3 stars, optional (progress_from_term_ind) *)
(** Show that progress can also be proved by induction on terms
instead of induction on typing derivations. *)
Theorem progress' : forall t T,
has_type empty t T ->
value t \/ exists t', t ==> t'.
Proof.
intros t.
t_cases (induction t) Case; intros T Ht; auto.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################################### *)
(** *** Free Occurrences *)
(** A variable [x] _appears free in_ a term _t_ if [t] contains some
occurrence of [x] that is not under an abstraction labeled [x]. For example:
- [y] appears free, but [x] does not, in [\x:T->U. x y]
- both [x] and [y] appear free in [(\x:T->U. x y) x]
- no variables appear free in [\x:T->U. \y:T. x y] *)
Inductive appears_free_in : id -> tm -> Prop :=
| afi_var : forall x,
appears_free_in x (tvar x)
| afi_app1 : forall x t1 t2,
appears_free_in x t1 -> appears_free_in x (tapp t1 t2)
| afi_app2 : forall x t1 t2,
appears_free_in x t2 -> appears_free_in x (tapp t1 t2)
| afi_abs : forall x y T11 t12,
y <> x ->
appears_free_in x t12 ->
appears_free_in x (tabs y T11 t12)
| afi_if1 : forall x t1 t2 t3,
appears_free_in x t1 ->
appears_free_in x (tif t1 t2 t3)
| afi_if2 : forall x t1 t2 t3,
appears_free_in x t2 ->
appears_free_in x (tif t1 t2 t3)
| afi_if3 : forall x t1 t2 t3,
appears_free_in x t3 ->
appears_free_in x (tif t1 t2 t3).
Tactic Notation "afi_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "afi_var"
| Case_aux c "afi_app1" | Case_aux c "afi_app2"
| Case_aux c "afi_abs"
| Case_aux c "afi_if1" | Case_aux c "afi_if2"
| Case_aux c "afi_if3" ].
Hint Constructors appears_free_in.
(** A term in which no variables appear free is said to be _closed_. *)
Definition closed (t:tm) :=
forall x, ~ appears_free_in x t.
(* ###################################################################### *)
(** *** Substitution *)
(** We first need a technical lemma connecting free variables and
typing contexts. If a variable [x] appears free in a term [t],
and if we know [t] is well typed in context [Gamma], then it must
be the case that [Gamma] assigns a type to [x]. *)
Lemma free_in_context : forall x t T Gamma,
appears_free_in x t ->
has_type Gamma t T ->
exists T', Gamma x = Some T'.
(** _Proof_: We show, by induction on the proof that [x] appears free
in [t], that, for all contexts [Gamma], if [t] is well typed
under [Gamma], then [Gamma] assigns some type to [x].
- If the last rule used was [afi_var], then [t = x], and from
the assumption that [t] is well typed under [Gamma] we have
immediately that [Gamma] assigns a type to [x].
- If the last rule used was [afi_app1], then [t = t1 t2] and [x]
appears free in [t1]. Since [t] is well typed under [Gamma],
we can see from the typing rules that [t1] must also be, and
the IH then tells us that [Gamma] assigns [x] a type.
- Almost all the other cases are similar: [x] appears free in a
subterm of [t], and since [t] is well typed under [Gamma], we
know the subterm of [t] in which [x] appears is well typed
under [Gamma] as well, and the IH gives us exactly the
conclusion we want.
- The only remaining case is [afi_abs]. In this case [t =
\y:T11.t12], and [x] appears free in [t12]; we also know that
[x] is different from [y]. The difference from the previous
cases is that whereas [t] is well typed under [Gamma], its
body [t12] is well typed under [(Gamma, y:T11)], so the IH
allows us to conclude that [x] is assigned some type by the
extended context [(Gamma, y:T11)]. To conclude that [Gamma]
assigns a type to [x], we appeal to lemma [extend_neq], noting
that [x] and [y] are different variables. *)
Proof.
intros x t T Gamma H H0. generalize dependent Gamma.
generalize dependent T.
afi_cases (induction H) Case;
intros; try solve [inversion H0; eauto].
Case "afi_abs".
inversion H1; subst.
apply IHappears_free_in in H7.
apply not_eq_beq_id_false in H.
rewrite extend_neq in H7; assumption.
Qed.
(** Next, we'll need the fact that any term [t] which is well typed in
the empty context is closed -- that is, it has no free variables. *)
(** **** Exercise: 2 stars (typable_empty__closed) *)
(* EXPECTED *)
Corollary typable_empty__closed : forall t T,
has_type empty t T ->
closed t.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Sometimes, when we have a proof [Gamma |- t : T], we will need to
replace [Gamma] by a different context [Gamma']. When is it safe
to do this? Intuitively, it must at least be the case that
[Gamma'] assigns the same types as [Gamma] to all the variables
that appear free in [t]. In fact, this is the only condition that
is needed. *)
Lemma context_invariance : forall Gamma Gamma' t T,
has_type Gamma t T ->
(forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
has_type Gamma' t T.
(** _Proof_: By induction on the derivation of [Gamma |- t : T].
- If the last rule in the derivation was [T_Var], then [t = x]
and [Gamma x = T]. By assumption, [Gamma' x = T] as well, and
hence [Gamma' |- t : T] by [T_Var].
- If the last rule was [T_Abs], then [t = \y:T11. t12], with [T
= T11 -> T12] and [Gamma, y:T11 |- t12 : T12]. The induction
hypothesis is that for any context [Gamma''], if [Gamma,
y:T11] and [Gamma''] assign the same types to all the free
variables in [t12], then [t12] has type [T12] under [Gamma''].
Let [Gamma'] be a context which agrees with [Gamma] on the
free variables in [t]; we must show [Gamma' |- \y:T11. t12 :
T11 -> T12].
By [T_Abs], it suffices to show that [Gamma', y:T11 |- t12 :
T12]. By the IH (setting [Gamma'' = Gamma', y:T11]), it
suffices to show that [Gamma, y:T11] and [Gamma', y:T11] agree
on all the variables that appear free in [t12].
Any variable occurring free in [t12] must either be [y], or
some other variable. [Gamma, y:T11] and [Gamma', y:T11]
clearly agree on [y]. Otherwise, we note that any variable
other than [y] which occurs free in [t12] also occurs free in
[t = \y:T11. t12], and by assumption [Gamma] and [Gamma']
agree on all such variables, and hence so do [Gamma, y:T11]
and [Gamma', y:T11].
- If the last rule was [T_App], then [t = t1 t2], with [Gamma |-
t1 : T2 -> T] and [Gamma |- t2 : T2]. One induction
hypothesis states that for all contexts [Gamma'], if [Gamma']
agrees with [Gamma] on the free variables in [t1], then [t1]
has type [T2 -> T] under [Gamma']; there is a similar IH for
[t2]. We must show that [t1 t2] also has type [T] under
[Gamma'], given the assumption that [Gamma'] agrees with
[Gamma] on all the free variables in [t1 t2]. By [T_App], it
suffices to show that [t1] and [t2] each have the same type
under [Gamma'] as under [Gamma]. However, we note that all
free variables in [t1] are also free in [t1 t2], and similarly
for free variables in [t2]; hence the desired result follows
by the two IHs.
*)
Proof with eauto.
intros.
generalize dependent Gamma'.
has_type_cases (induction H) Case; intros; auto.
Case "T_Var".
apply T_Var. rewrite <- H0...
Case "T_Abs".
apply T_Abs.
apply IHhas_type. intros x1 Hafi.
(* the only tricky step... the [Gamma'] we use to
instantiate is [extend Gamma x T11] *)
unfold extend. remember (beq_id x0 x1) as e. destruct e...
Case "T_App".
apply T_App with T11...
Qed.
(** Now we come to the conceptual heart of the proof that reduction
preserves types -- namely, the observation that _substitution_
preserves types.
Formally, the so-called _Substitution Lemma_ says this: suppose we
have a term [t] with a free variable [x], and suppose we've been
able to assign a type [T] to [t] under the assumption that [x] has
some type [U]. Also, suppose that we have some other term [v] and
that we've shown that [v] has type [U]. Then, since [v] satisfies
the assumption we made about [x] when typing [t], we should be
able to substitute [v] for each of the occurrences of [x] in [t]
and obtain a new term that still has type [T]. *)
(** _Lemma_: If [Gamma,x:U |- t : T] and [|- v : U], then [Gamma |-
[x:=v]t : T]. *)
Lemma substitution_preserves_typing : forall Gamma x U t t' T,
has_type (extend Gamma x U) t T ->
has_type empty t' U ->
has_type Gamma ([x:=t']t) T.
(** One technical subtlety in the statement of the lemma is that we
assign [t'] the type [U] in the _empty_ context -- in other words,
we assume [t'] is closed. This assumption considerably simplifies
the [T_Abs] case of the proof (compared to assuming [Gamma |- t' :
U], which would be the other reasonable assumption at this point)
because the context invariance lemma then tells us that [t'] has
type [U] in any context at all -- we don't have to worry about
free variables in [t'] clashing with the variable being introduced
into the context by [T_Abs].
_Proof_: We prove, by induction on [t], that, for all [T] and
[Gamma], if [Gamma,x:U |- t : T] and [|- t' : U], then [Gamma |-
[x:=t']t : T].
- If [t] is a variable, there are two cases to consider, depending
on whether [t] is [x] or some other variable.
- If [t = x], then from the fact that [Gamma, x:U |- x : T] we
conclude that [U = T]. We must show that [[x:=t']x = t'] has
type [T] under [Gamma], given the assumption that [t'] has
type [U = T] under the empty context. This follows from
context invariance: if a closed term has type [T] in the
empty context, it has that type in any context.
- If [t] is some variable [y] that is not equal to [x], then
we need only note that [y] has the same type under [Gamma,
x:U] as under [Gamma].
- If [t] is an abstraction [\y:T11. t12], then the IH tells us,
for all [Gamma'] and [T'], that if [Gamma',x:U |- t12 : T']
and [|- t' : U], then [Gamma' |- [x:=t']t12 : T'].
The substitution in the conclusion behaves differently,
depending on whether [x] and [y] are the same variable name.
First, suppose [x = y]. Then, by the definition of
substitution, [[x:=t']t = t], so we just need to show [Gamma |-
t : T]. But we know [Gamma,x:U |- t : T], and since the
variable [y] does not appear free in [\y:T11. t12], the
context invariance lemma yields [Gamma |- t : T].
Second, suppose [x <> y]. We know [Gamma,x:U,y:T11 |- t12 :
T12] by inversion of the typing relation, and [Gamma,y:T11,x:U
|- t12 : T12] follows from this by the context invariance
lemma, so the IH applies, giving us [Gamma,y:T11 |- [x:=t']t12 :
T12]. By [T_Abs], [Gamma |- \y:T11. [x:=t']t12 : T11->T12], and
by the definition of substitution (noting that [x <> y]),
[Gamma |- \y:T11. [x:=t']t12 : T11->T12] as required.
- If [t] is an application [t1 t2], the result follows
straightforwardly from the definition of substitution and the
induction hypotheses.
- The remaining cases are similar to the application case.
Another technical note: This proof is a rare case where an
induction on terms, rather than typing derivations, yields a
simpler argument. The reason for this is that the assumption
[has_type (extend Gamma x U) t T] is not completely generic, in
the sense that one of the "slots" in the typing relation -- namely
the context -- is not just a variable, and this means that Coq's
native induction tactic does not give us the induction hypothesis
that we want. It is possible to work around this, but the needed
generalization is a little tricky. The term [t], on the other
hand, _is_ completely generic. *)
Proof with eauto.
intros Gamma x U t t' T Ht Ht'.
generalize dependent Gamma. generalize dependent T.
t_cases (induction t) Case; intros T Gamma H;
(* in each case, we'll want to get at the derivation of H *)
inversion H; subst; simpl...
Case "tvar".
rename i into y. remember (beq_id x y) as e. destruct e.
SCase "x=y".
apply beq_id_eq in Heqe. subst.
rewrite extend_eq in H2.
inversion H2; subst. clear H2.
eapply context_invariance... intros x Hcontra.
destruct (free_in_context _ _ T empty Hcontra) as [T' HT']...
inversion HT'.
SCase "x<>y".
apply T_Var. rewrite extend_neq in H2...
Case "tabs".
rename i into y. apply T_Abs.
remember (beq_id x y) as e. destruct e.
SCase "x=y".
eapply context_invariance...
apply beq_id_eq in Heqe. subst.
intros x Hafi. unfold extend.
destruct (beq_id y x)...
SCase "x<>y".
apply IHt. eapply context_invariance...
intros z Hafi. unfold extend.
remember (beq_id y z) as e0. destruct e0...
apply beq_id_eq in Heqe0. subst.
rewrite <- Heqe...
Qed.
(** The substitution lemma can be viewed as a kind of "commutation"
property. Intuitively, it says that substitution and typing can
be done in either order: we can either assign types to the terms
[t] and [v] separately (under suitable contexts) and then combine
them using substitution, or we can substitute first and then
assign a type to [ [x:=v] t ] -- the result is the same either
way. *)
(* ###################################################################### *)
(** *** Preservation *)
(** We now have the tools we need to prove _preservation_: if a closed
term [t] has type [T], and takes an evaluation step to [t'], then [t']
is also a closed term with type [T]. In other words, the small-step
evaluation relation preserves types.
*)
Theorem preservation : forall t t' T,
has_type empty t T ->
t ==> t' ->
has_type empty t' T.
(** _Proof_: by induction on the derivation of [|- t : T].
- We can immediately rule out [T_Var], [T_Abs], [T_True], and
[T_False] as the final rules in the derivation, since in each of
these cases [t] cannot take a step.
- If the last rule in the derivation was [T_App], then [t = t1
t2]. There are three cases to consider, one for each rule that
could have been used to show that [t1 t2] takes a step to [t'].
- If [t1 t2] takes a step by [ST_App1], with [t1] stepping to
[t1'], then by the IH [t1'] has the same type as [t1], and
hence [t1' t2] has the same type as [t1 t2].
- The [ST_App2] case is similar.
- If [t1 t2] takes a step by [ST_AppAbs], then [t1 =
\x:T11.t12] and [t1 t2] steps to [[x:=t2]t12]; the
desired result now follows from the fact that substitution
preserves types.
- If the last rule in the derivation was [T_If], then [t = if t1
then t2 else t3], and there are again three cases depending on
how [t] steps.
- If [t] steps to [t2] or [t3], the result is immediate, since
[t2] and [t3] have the same type as [t].
- Otherwise, [t] steps by [ST_If], and the desired conclusion
follows directly from the induction hypothesis.
*)
Proof with eauto.
remember (@empty ty) as Gamma.
intros t t' T HT. generalize dependent t'.
has_type_cases (induction HT) Case;
intros t' HE; subst Gamma; subst;
try solve [inversion HE; subst; auto].
Case "T_App".
inversion HE; subst...
(* Most of the cases are immediate by induction,
and [eauto] takes care of them *)
SCase "ST_AppAbs".
apply substitution_preserves_typing with T11...
inversion HT1...
Qed.
(** **** Exercise: 2 stars, recommended (subject_expansion_stlc) *)
(* EXPECTED *)
(** An exercise in Types.v asked about the subject
expansion property for the simple language of arithmetic and
boolean expressions. Does this property hold for STLC? That
is, is it always the case that, if [t ==> t'] and [has_type
t' T], then [has_type t T]? If so, prove it. If not, give a
counter-example.
(* FILL IN HERE *)
[]
*)
(* ###################################################################### *)
(** *** Type Soundness *)
(** **** Exercise: 2 stars, optional (type_soundness) *)
(* EXPECTED *)
(** Put progress and preservation together and show that a well-typed
term can _never_ reach a stuck state. *)
Definition stuck (t:tm) : Prop :=
(normal_form step) t /\ ~ value t.
Corollary soundness : forall t t' T,
has_type empty t T ->
t ==>* t' ->
~(stuck t').
Proof.
intros t t' T Hhas_type Hmulti. unfold stuck.
intros [Hnf Hnot_val]. unfold normal_form in Hnf.
induction Hmulti.
(* FILL IN HERE *) Admitted.
(* ###################################################################### *)
(** *** Uniqueness of Types *)
(** **** Exercise: 3 stars (types_unique) *)
(* EXPECTED *)
(** Another pleasant property of the STLC is that types are
unique: a given term (in a given context) has at most one
type. *)
(** Formalize this statement and prove it. *)
(* FILL IN HERE *)
(** [] *)
(* ###################################################################### *)
(** ** Additional Exercises *)
(** **** Exercise: 1 star (progress_preservation_statement) *)
(** Without peeking, write down the progress and preservation
theorems for the simply typed lambda-calculus. *)
(** [] *)
(** **** Exercise: 2 stars (stlc_variation1) *)
(* EXPECTED *)
(** suppose we add a new term [zap] with the following reduction rule:
--------- (ST_Zap)
t ==> zap
and the following typing rule:
---------------- (T_Zap)
Gamma |- zap : T
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
[]
*)
(** **** Exercise: 2 stars (stlc_variation2) *)
(** Suppose instead that we add a new term [foo] with the following reduction rules:
----------------- (ST_Foo1)
(\x:A. x) ==> foo
------------ (ST_Foo2)
foo ==> true
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
[]
*)
(** **** Exercise: 2 stars (stlc_variation3) *)
(** Suppose instead that we remove the rule [ST_App1] from the [step]
relation. Which of the following properties of the STLC remain
true in the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
[]
*)
(** **** Exercise: 2 stars (stlc_variation4) *)
(** Suppose instead that we add the following new rule to the reduction relation:
---------------------------------- (ST_FunnyIfTrue)
(if true then t1 else t2) ==> true
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
*)
(** **** Exercise: 2 stars (stlc_variation5) *)
(** Suppose instead that we add the following new rule to the typing relation:
Gamma |- t1 : Bool->Bool->Bool
Gamma |- t2 : Bool
------------------------------ (T_FunnyApp)
Gamma |- t1 t2 : Bool
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
*)
(** **** Exercise: 2 stars (stlc_variation6) *)
(** Suppose instead that we add the following new rule to the typing relation:
Gamma |- t1 : Bool
Gamma |- t2 : Bool
--------------------- (T_FunnyApp')
Gamma |- t1 t2 : Bool
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
*)
(** **** Exercise: 2 stars (stlc_variation7) *)
(** Suppose we add the following new rule to the typing
relation of the STLC:
------------------- (T_FunnyAbs)
|- \x:Bool.t : Bool
Which of the following properties of the STLC remain true in
the presence of this rule? For each one, write either
"remains true" or else "becomes false." If a property becomes
false, give a counterexample.
- Determinism of [step]
- Progress
- Preservation
[]
*)
End STLC.
(* ###################################################################### *)
(* ###################################################################### *)
(** * Optional Exercise: STLC with Arithmetic *)
(** To see how the STLC might function as the core of a real
programming language, let's extend it with a concrete base
type of numbers and some constants and primitive
operators. *)
Module STLCArith.
(* ###################################################################### *)
(** ** Syntax and Operational Semantics *)
(** To types, we add a base type of natural numbers (and remove
booleans, for brevity) *)
Inductive ty : Type :=
| TArrow : ty -> ty -> ty
| TNat : ty.
(** To terms, we add natural number constants, along with
successor, predecessor, multiplication, and zero-testing... *)
Inductive tm : Type :=
| tvar : id -> tm
| tapp : tm -> tm -> tm
| tabs : id -> ty -> tm -> tm
| tnat : nat -> tm
| tsucc : tm -> tm
| tpred : tm -> tm
| tmult : tm -> tm -> tm
| tif0 : tm -> tm -> tm -> tm.
Tactic Notation "t_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "tvar" | Case_aux c "tapp"
| Case_aux c "tabs" | Case_aux c "tnat"
| Case_aux c "tsucc" | Case_aux c "tpred"
| Case_aux c "tmult" | Case_aux c "tif0" ].
(** **** Exercise: 4 stars, optional (stlc_arith) *)
(** Finish formalizing the definition and properties of the STLC extended
with arithmetic. Specifically:
- Copy the whole development of STLC that we went through above (from
the definition of values through the Progress theorem), and
paste it into the file at this point.
- Extend the definitions of the [subst] operation and the [step]
relation to include appropriate clauses for the arithmetic operators.
- Extend the proofs of all the properties of the original STLC to deal
with the new syntactic forms. Make sure Coq accepts the whole file. *)
(* FILL IN HERE *)
(** [] *)
End STLCArith.