MoreStlc: More on the Simply Typed Lambda-Calculus

(* Version of 4/23/2010 *)

Require Export Stlc.
Require Import Relations.

Typechecking for STLC

The has_type relation defines what it means for a term to belong to a type (in some context). But it doesn't, by itself, tell us how to check whether or not a term is well typed.
Fortunately, the rules defining has_type are syntax directed -- they exactly follow the shape of the term. This makes it straightforward to translate the typing rules into clauses of a typechecking function that takes a term and a context and either returns the term's type or else signals that the term is not typable.

Module STLCChecker.
Import STLC.

Comparing Types

First, we need a function to compare two types for equality...

Fixpoint beq_ty (T1 T2:ty) : bool :=
  match T1,T2 with
  | ty_base i, ty_base i' =>
      beq_id i i'
  | ty_arrow T11 T12, ty_arrow T21 T22 =>
      andb (beq_ty T11 T21) (beq_ty T12 T22)
  | _,_ =>

... and we need to establish the usual two-way connection between the boolean result returned by beq_ty and the logical proposition that its inputs are equal.

Lemma beq_ty_refl : forall T1,
  beq_ty T1 T1 = true.
  intros T1. induction T1; simpl.
    symmetry. apply beq_id_refl.
    rewrite IHT1_1. rewrite IHT1_2. reflexivity. Qed.

Lemma beq_ty__eq : forall T1 T2,
  beq_ty T1 T2 = true -> T1 = T2.
Proof with auto.
  intros T1. induction T1; intros T2 Hbeq; destruct T2; inversion Hbeq.
  Case "T1=ty_base i".
    symmetry in H0. apply beq_id_eq in H0. subst...
  Case "T1=ty_arrow T1_1 T1_2".
    apply andb_true in H0. destruct H0 as [Hbeq1 Hbeq2].
    apply IHT1_1 in Hbeq1. apply IHT1_2 in Hbeq2. subst... Qed.

The Typechecker

Now here's the typechecker. It works by walking over the structure of the given term, returning either Some T or None. Each time we make a recursive call to find out the types of the subterms, we need to pattern-match on the results to make sure that they are not None. Also, in the tm_app case, we use pattern matching to extract the left- and right-hand sides of the function's arrow type (and fail if the type of the function is not ty_arrow T11 T12 for some T1 and T2).

Fixpoint type_check (Gamma:context) (t:tm) : option ty :=
  match t with
  | tm_var x => Gamma x
  | tm_abs x T11 t12 => match type_check (extend Gamma x T11) t12 with
                          | Some T12 => Some (ty_arrow T11 T12)
                          | _ => None
  | tm_app t1 t2 => match type_check Gamma t1, type_check Gamma t2 with
                      | Some (ty_arrow T11 T12),Some T2 =>
                        if beq_ty T11 T2 then Some T12 else None
                      | _,_ => None


To verify that this typechecking algorithm is the correct one, we show that it is sound and complete for the original has_type relation -- that is, type_check and has_type define the same partial function.

Theorem type_checking_sound : forall Gamma t T,
  type_check Gamma t = Some T -> has_type Gamma t T.
Proof with eauto.
  intros Gamma t. generalize dependent Gamma.
  (tm_cases (induction t) Case); intros Gamma T Htc; inversion Htc.
  Case "tm_var"...
  Case "tm_app".
    remember (type_check Gamma t1) as TO1.
    remember (type_check Gamma t2) as TO2.
    destruct TO1 as [T1|]; try solve by inversion;
    destruct T1 as [|T11 T12]; try solve by inversion.
    destruct TO2 as [T2|]; try solve by inversion.
    remember (beq_ty T11 T2) as b.
    destruct b; try solve by inversion.
    symmetry in Heqb. apply beq_ty__eq in Heqb.
    inversion H0; subst...
  Case "tm_abs".
    rename i into y. rename t into T1.
    remember (extend Gamma y T1) as G'.
    remember (type_check G' t0) as TO2.
    destruct TO2; try solve by inversion.
    inversion H0; subst...

Theorem type_checking_complete : forall Gamma t T,
  has_type Gamma t T -> type_check Gamma t = Some T.
Proof with auto.
  intros Gamma t T Hty.
  (typing_cases (induction Hty) Case); simpl.
  Case "T_Var"...
  Case "T_Abs". rewrite IHHty...
  Case "T_App".
    rewrite IHHty1. rewrite IHHty2.
    rewrite (beq_ty_refl T11)...

End STLCChecker.

Simple Extensions to STLC

The simply typed lambda-calculus has enough structure to make its theoretical properties interesting, but it is not yet much of a programming language. In this chapter, we begin to close the gap with more familiar languages by introducing a number of familiar features that have straightforward treatments at the level of typing.

Numbers and Booleans

Adding types, constants, and primitive operations for numbers and booleans is easy -- just a matter of combining the two halves of the Stlc.v chapter.

The Unit Type

Another useful base type, found especially in languages in the ML family, is the singleton type Unit. In contrast to the uninterpreted base types of the simply typed lambda-calculus, this type is interpreted, in the simplest possible way: we explicitly introduce a single element---the term constant unit (with a small u)---and a typing rule making unit an element of Unit. We also add unit to the set of possible result values of computations---indeed, unit is the only possible result of evaluating an expression of type Unit.
       t ::=                Terms:
           | ...               (other terms same as before)
           | unit              unit value

       v ::=                Values:
           | ...     
           | unit              unit

       T ::=                Types:
           | ...
           | Unit              Unit type


When writing a complex expression, it is often useful---both for avoiding repetition and for increasing readability---to give names to some of its subexpressions. Most languages provide one or more ways of doing this. In OCaml, for example, we can write let x=t1 in t2 to mean ``evaluate the expression t1 and bind the name x to the resulting value while evaluating t2.''
Our let-binder follows ML's in choosing a call-by-value evaluation order, where the let-bound term must be fully evaluated before evaluation of the let-body can begin. The typing rule T_Let tells us that the type of a let can be calculated by calculating the type of the let-bound term, extending the context with a binding with this type, and in this enriched context calculating the type of the body, which is then the type of the whole let expression.
At this point in the course, it's probably just as easy to simply look at the rules defining this new feature as to wade through a lot of english text conveying the same information. Here they are:
       t ::=                Terms:
           | ...               
           | let x=t in t      let-binding
t1 --> t1' (ST_Let1)  

let x=t1 in t2 --> let x=t1' in t2

let x=v1 in t2 --> [v1/x] t2
Gamma |- t1 : T1      Gamma, x:T1 |- t2 : T2 (T_Let)  

Gamma |- let x=t1 in t2 : T2


Our functional programming examples have made frequent use of pairs of values. The type of such pairs is called a product type.
The formalization of pairs is almost too simple to be worth discussing---by this point in the book, it should be about as easy to read a set of inference rules (or a formal Inductive definition) as to wade through a description in English conveying the same information. However, let's look briefly at the various parts of the definition to emphasize the common pattern.
In Coq, the primitive way of extracting the components of a pair is pattern matching. An alternative style is to take fst and snd -- the first- and second-projection operators -- as primitives. Just for fun, let's do our products this way. For example, here's how we'd write a function that takes a pair of numbers and returns the pair of their sum and product:
          let sum = x.fst + x.snd in
          let prod = x.fst * x.snd in
Adding pairs to the simply typed lambda-calculus, then, involves adding two new forms of term---pairing, written (t1,t2), and projection, written t.fst for the first projection from t and t.snd for the second projection---plus one new type constructor, T1*T2, called the product of T1 and T2.
       t ::=                Terms:
           | ...               
           | (t,t)             pair
           | t.fst             first projection
           | t.snd             second projection

       v ::=                Values:
           | ...
           | (v,v)             pair value

       T ::=                Types:
           | ...
           | T * T             product type
For evaluation, we need several new rules specifying how pairs and projection behave.
t1 --> t1' (ST_Pair1)  

(t1,t2) --> (t1',t2)
t2 --> t2' (ST_Pair2)  

(v1,t2) --> (v1,t2')
t1 --> t1' (ST_Fst1)  

t1.fst --> t1'.fst

(v1,v2).fst --> v1
t1 --> t1' (ST_Snd1)  

t1.snd --> t1'.snd

(v1,v2).snd --> v2
Rules ST_FstPair and ST_SndPair specify that, when a fully evaluated pair meets a first or second projection, the result is the appropriate component. The congruence rules ST_Fst1 and ST_Snd1 allow reduction to proceed under projections, when the term being projected from has not yet been fully evaluated. ST_Pair1 and ST_Pair2 evaluate the parts of pairs: first the left part, and then---when a value appears on the left---the right part. The ordering arising from the use of the metavariables v and t in these rules enforces a left-to-right evaluation strategy for pairs. (Note the implicit convention that metavariables like v and v1 can only denote values.) We've also added a clause to the definition of values, above, specifying that (v1,v2) is a value. The fact that the components of a pair value must themselves be values ensures that a pair passed as an argument to a function will be fully evaluated before the function body starts executing.
The typing rules for pairs and projections are straightforward. The rule, T_Pair, says that (t1,t2) has type T1*T2 if t1 has type T1 and t2 has type T2. Conversely, the rules T_Fst and T_Snd tell us that, if t1 has a product type T11*T12 (i.e., if it will evaluate to a pair), then the types of the projections from this pair are T11 and T12.
Gamma |- t1 : T1       Gamma |- t2 : T2 (T_Pair)  

Gamma |- (t1,t2) : T1*T2
Gamma |- t1 : T11*1T2 (T_Fst)  

Gamma |- t1.fst : T11
Gamma |- t1 : T11*T12 (T_Snd)  

Gamma |- t1.snd : T12


The typing features we have seen can be classified into base types like Bool, and type constructors like -> and * that build new types from old ones. Another useful type constructor is list. For every type T, the type list T describes finite-length lists whose elements are drawn from T.
Below we give the syntax, semantics, and typing rules for lists. Except for the fact that explicit type annotations are mandatory on nil and cannot appear on cons, these lists are essentially identical to those we built in Coq. We use case (a very simplified form of Coq's match) to destruct lists, to avoid dealing with questions like "what is the head of the empty list?" So, for example, here is a function that calculates the sum of the first two elements of a list of numbers:
       \x:list nat.
          case x of
            nil -> 0
          | a::x' -> case x' of
                       nil -> a
                     | b::x'' -> a+b
While we say that cons v1 v2 is a value, we really mean that v2 should also be a list -- we'll have to enforce this in the formal definition of value.
       t ::=                Terms:
           | ...
           | nil T
           | cons t t
           | case t of nil -> t | x::x -> t

       v ::=                Values:
           | ...
           | nil T             nil value
           | cons v v          cons value

       T ::=                Types:
           | ...
           | list T            list of Ts
t1 --> t1' (ST_Cons1)  

cons t1 t2 --> cons t1' t2
t2 --> t2' (ST_Cons2)  

cons v1 t2 --> cons v1 t2'
t1 --> t1' (ST_Case1)  

(case t1 of nil -> t2 | h::t -> t3)
--> (case t1' of nil -> t2 | h::t -> t3)

(case nil T of nil -> t2 | h::t -> t3) --> t2

(case (cons vh vt) of nil -> t2 | h::t -> t3)
--> [vh/h,vt/t]t3

Gamma |- nil T : list T
Gamma |- t1 : T      Gamma |- t2 : list T (T_Cons)  

Gamma |- cons t1 t2: list T
Gamma |- t1 : list T1        Gamma |- t2 : T
Gamma, h:T1, t:list T1 |- t3 : T (T_Case)  

Gamma |- (case t1 of nil -> t2 | h::t -> t3) : T

General Recursion

Another facility found in most programming languages (including Coq) is the ability to define recursive functions. For example, we might like to be able to define the factorial function like this:
   fact = \x:nat. 
             if x=0 then 1 else x * (fact (pred x)))    
But this would be require quite a bit of work to formalize: we'd have to introduce a notion of "function definitions" and carry around an "environment" of such definitions in the definition of the step relation.
Here is another way that is straightforward to formalize: instead of writing recursive definitions where the right-hand side can contain the identifier being defined, we can define a fixed-point operator that performs the "unfolding" of the recursive definition in the right-hand side lazily during reduction.
   fact = 
               if x=0 then 1 else x * (f (pred x)))    
The intuition is that the higher-order function f passed to fix is a generator for the fact function: if f is applied to a function that approximates the desired behavior of fact up to some number n (that is, a function that returns correct results on inputs less than or equal to n), then it returns a better approximation to fact---a function that returns correct results for inputs up to n+1. Applying fix to this generator returns its fixed point---a function that gives the desired behavior for all inputs n.
       t ::=                Terms:
           | ...
           | fix t             fixed-point operator
t1 --> t1' (ST_Fix1)  

fix t1 --> fix t1'

fix (\x:T1.t2) --> [(fix(\x:T1.t2)) / x] t2
Gamma |- t1 : T1->T1 (T_Fix)  

Gamma |- fix t1 : T1

Exercise: 1 star (halve_fix)

Translate this recursive definition into one using fix:
   halve = 
        if x=0 then 0 
        else if (pred x)=0 then 0
        else 1 + (halve (pred (pred x))))

Exercise: 1 star (fact_steps)

Write down the sequence of steps that the term fact 1 goes through to reduce to a normal form (assuming the usual reduction rules for arithmetic operations).
The ability to form the fixed point of a function of type T->T for any T has some surprising consequences. In particular, it implies that every type is inhabited by some term. To see this, observe that, for every type T, we can define a function diverge_T as follows:
      diverge_T = \x:Unit. fix (\x:Unit.x);
Whenever diverge_T is applied to a unit argument, we get a non-terminating evaluation sequence in which ST-AppAbs is applied over and over, always yielding the same term. That is, for every type T, the term diverge_T unit is an undefined element of T.

Formalizing the Extensions

Exercise: 5 stars (STLC_extensions)

Formalizing the extensions is left to you. We've provided the necessary extensions to the syntax of terms and types, and we've included a few examples that you can test your definitions with to make sure they are working as expected. You'll fill in the rest of the definitions and extend all the proofs accordingly. (A good strategy is to work on the extensions one at a time, in multiple passes, rather than trying to work through the file from start to finish in a single pass.)

Module STLCExtended.

Syntax and Operational Semantics

Inductive ty : Type :=
  | ty_base : id -> ty
  | ty_arrow : ty -> ty -> ty
  | ty_unit : ty
  | ty_pair : ty -> ty -> ty
  | ty_list : ty -> ty
  | ty_nat : ty.

Tactic Notation "ty_cases" tactic(first) tactic(c) :=
  [ c "ty_base" | c "ty_arrow" | c "ty_unit" |
    c "ty_pair" | c "ty_list" | c "ty_nat" ].

Inductive tm : Type :=
    (* pure STLC *)
  | tm_var : id -> tm
  | tm_app : tm -> tm -> tm
  | tm_abs : id -> ty -> tm -> tm
    (* unit *)
  | tm_unit : tm
    (* pairs *)
  | tm_pair : tm -> tm -> tm
  | tm_fst : tm -> tm
  | tm_snd : tm -> tm
    (* lists *)
  | tm_nil : ty -> tm
  | tm_cons : tm -> tm -> tm
  | tm_case : tm -> tm -> id -> id -> tm -> tm
          (* i.e., case t1 of | nil -> t2 | x::y -> t3 *)
    (* numbers *)
  | tm_nat : nat -> tm
  | tm_succ : tm -> tm
  | tm_pred : tm -> tm
  | tm_mult : tm -> tm -> tm
  | tm_if0 : tm -> tm -> tm -> tm
    (* let *)
  | tm_let : id -> tm -> tm -> tm
          (* i.e., let x = t1 in t2 *)
    (* fix *)
  | tm_fix : tm -> tm.
Note that, for brevity, we've chosen a compressed version of the numeric operations, with a single if0 form combining a zero test and a conditional.

Tactic Notation "tm_cases" tactic(first) tactic(c) :=
  [ c "tm_var" | c "tm_app" | c "tm_abs" |
    c "tm_unit" |
    c "tm_pair" | c "tm_fst" | c "tm_snd" |
    c "tm_nil" | c "tm_cons" | c "tm_case" |
    c "tm_nat" | c "tm_succ" | c "tm_pred" | c "tm_mult" | c "tm_if0" |
    c "tm_let" |
    c "tm_fix" ].

Some variables, for use in examples...

Notation a := (Id 0).
Notation f := (Id 1).
Notation g := (Id 2).
Notation l := (Id 3).
Notation A := (ty_base (Id 4)).
Notation B := (ty_base (Id 5)).
Notation k := (Id 6).
Notation i1 := (Id 7).
Notation i2 := (Id 8).


Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
  match t with
  | tm_var y => if beq_id x y then s else t
  | tm_abs y T t1 => tm_abs y T (if beq_id x y then t1 else (subst x s t1))
  | tm_app t1 t2 => tm_app (subst x s t1) (subst x s t2)
  (* FILL IN HERE *)
  | _ => t (* ... and delete this line *)


Next we define the values of our language.

Inductive value : tm -> Prop :=
  | v_abs : forall x T11 t12,
      value (tm_abs x T11 t12)
  (* FILL IN HERE *)

Hint Constructors value.

Reserved Notation "t1 '-->' t2" (at level 40).

Inductive step : tm -> tm -> Prop :=
  | ST_AppAbs : forall x T11 t12 v2,
         value v2 ->
         (tm_app (tm_abs x T11 t12) v2) --> (subst x v2 t12)
  | ST_App1 : forall t1 t1' t2,
         t1 --> t1' ->
         (tm_app t1 t2) --> (tm_app t1' t2)
  | ST_App2 : forall v1 t2 t2',
         value v1 ->
         t2 --> t2' ->
         (tm_app v1 t2) --> (tm_app v1 t2')
  (* FILL IN HERE *)

where "t1 '-->' t2" := (step t1 t2).

Tactic Notation "step_cases" tactic(first) tactic(c) :=
  [ c "ST_AppAbs" | c "ST_App1" | c "ST_App2" |
    (* FILL IN HERE *)

Notation stepmany := (refl_step_closure step).
Notation "t1 '-->*' t2" := (stepmany t1 t2) (at level 40).

Hint Constructors step.


(* Standard definitions for contexts *)
Definition context := id -> (option ty).
Definition empty : context := (fun _ => None).
Definition extend (Gamma : context) (x:id) (T : ty) :=
  fun x' => if beq_id x x' then Some T else Gamma x'.

Next we define the typing rules. These are nearly direct transcriptions of the inference rules shown above.

Inductive has_type : context -> tm -> ty -> Prop :=
  (* Typing rules for proper terms *)
  | T_Var : forall Gamma x T,
      Gamma x = Some T ->
      has_type Gamma (tm_var x) T
  | T_Abs : forall Gamma x T11 T12 t12,
      has_type (extend Gamma x T11) t12 T12 ->
      has_type Gamma (tm_abs x T11 t12) (ty_arrow T11 T12)
  | T_App : forall T1 T2 Gamma t1 t2,
      has_type Gamma t1 (ty_arrow T1 T2) ->
      has_type Gamma t2 T1 ->
      has_type Gamma (tm_app t1 t2) T2
  (* FILL IN HERE *)

Hint Constructors has_type.

Tactic Notation "has_type_cases" tactic(first) tactic(c) :=
  [ c "T_Var" | c "T_Abs" | c "T_App"
    (* FILL IN HERE *)


Exercise: 2 stars (examples)

Finish the proofs.
(* fact := fix
                   if a=0 then 1 else a * (f (pred a))) *)

Definition fact :=
    (tm_abs f (ty_arrow ty_nat ty_nat)
      (tm_abs a ty_nat
           (tm_var a)
           (tm_nat 1)
              (tm_var a)
              (tm_app (tm_var f) (tm_pred (tm_var a))))))).

(Warning: you may be able to typecheck fact but still have some rules wrong!)

(* These hints improve the automation for has_type

  It says that whenever auto arrives at a goal of the form
  (has_type G (tm_app e1 e1) T), it should consider eapply T_App,
  leaving an existential variable for the middle type T1, and similar
  for case. That variable will then be filled in during the search
  for type derivations for e1 and e2

  We also include a hint to "try harder" when solving equality goals;
  this is useful to automate uses of T_Var (which includes an equality
  as a precondition).

Hint Extern 2 (has_type _ (tm_app _ _) _) => eapply T_App; auto.
(* Uncomment this once you've defined the [T_Case] constructor for
   the typing relation: *)
Hint Extern 2 (has_type _ (tm_case _ _ _ _ _) _) => eapply T_Case; auto.
Hint Extern 2 (_ = _) => compute; reflexivity.

Example fact_typechecks :
  has_type empty fact (ty_arrow ty_nat ty_nat).
Proof with auto.
(* FILL IN HERE *) Admitted.

Example fact_example:
  (tm_app fact (tm_nat 4)) -->* (tm_nat 24).
  (* FILL IN HERE *) Admitted.

(* map :=
               case l of
               |  -> 
               | x::l -> (g x)::(f l)) *)

Definition map :=
  tm_abs g (ty_arrow ty_nat ty_nat)
      (tm_abs f (ty_arrow (ty_list ty_nat) (ty_list ty_nat))
        (tm_abs l (ty_list ty_nat)
          (tm_case (tm_var l)
            (tm_nil ty_nat)
            a l (tm_cons (tm_app (tm_var g) (tm_var a))
                         (tm_app (tm_var f) (tm_var l))))))).

Example map_typechecks :
  has_type empty map
    (ty_arrow (ty_arrow ty_nat ty_nat)
      (ty_arrow (ty_list ty_nat)
        (ty_list ty_nat))).
Proof with auto.
  (* FILL IN HERE *) Admitted.

Example map_example :
  tm_app (tm_app map (tm_abs a ty_nat (tm_succ (tm_var a))))
         (tm_cons (tm_nat 1) (tm_cons (tm_nat 2) (tm_nil ty_nat)))
  -->* (tm_cons (tm_nat 2) (tm_cons (tm_nat 3) (tm_nil ty_nat))).
Proof with auto.
  (* FILL IN HERE *) Admitted.

Properties of Typing

The proofs of progress and preservation for this system are essentially the same (though of course somewhat longer) as for the pure simply typed lambda-calculus.


Theorem progress : forall t T,
     has_type empty t T ->
     value t \/ exists t', t --> t'.
Proof with eauto.
  (* Theorem: Suppose empty |- t : T.  Then either
       1. t is a value, or
       2. t --> t' for some t'.
     Proof: By induction on the given typing derivation. *)

  intros t T Ht.
  remember empty as Gamma.
  generalize dependent HeqGamma.
  (has_type_cases (induction Ht) Case); intros HeqGamma; subst.
  Case "T_Var".
    (* The final rule in the given typing derivation cannot be T_Var,
       since it can never be the case that empty |- x : T (since the
       context is empty). *)

    inversion H.
  Case "T_Abs".
    (* If the T_Abs rule was the last used, then t = tm_abs x T11 t12,
       which is a value. *)

  Case "T_App".
    (* If the last rule applied was T_App, then t = t1 t2, and we know 
       from the form of the rule that
         empty |- t1 : T1 -> T2
         empty |- t2 : T1
       By the induction hypothesis, each of t1 and t2 either is a value 
       or can take a step. *)

    destruct IHHt1; subst...
    SCase "t1 is a value".
      destruct IHHt2; subst...
      SSCase "t2 is a value".
      (* If both t1 and t2 are values, then we know that 
         t1 = tm_abs x T11 t12, since abstractions are the only values
         that can have an arrow type.  But 
         (tm_abs x T11 t12) t2 --> subst x t2 t12 by ST_AppAbs. *)

        inversion H; subst; try (solve by inversion).
        exists (subst x t2 t12)...
      SSCase "t2 steps".
        (* If t1 is a value and t2 --> t2', then t1 t2 --> t1 t2' 
           by ST_App2. *)

        destruct H0 as [t2' Hstp]. exists (tm_app t1 t2')...
    SCase "t1 steps".
      (* Finally, If t1 --> t1', then t1 t2 --> t1' t2 by ST_App1. *)
      destruct H as [t1' Hstp]. exists (tm_app t1' t2)...
  (* FILL IN HERE *)

Context Invariance

Inductive appears_free_in : id -> tm -> Prop :=
  | afi_var : forall x,
      appears_free_in x (tm_var x)
  | afi_app1 : forall x t1 t2,
      appears_free_in x t1 -> appears_free_in x (tm_app t1 t2)
  | afi_app2 : forall x t1 t2,
      appears_free_in x t2 -> appears_free_in x (tm_app t1 t2)
  | afi_abs : forall x y T11 t12,
        y <> x ->
        appears_free_in x t12 ->
        appears_free_in x (tm_abs y T11 t12)
  (* FILL IN HERE *)

Hint Constructors appears_free_in.

Lemma context_invariance : forall Gamma Gamma' t S,
     has_type Gamma t S ->
     (forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
     has_type Gamma' t S.
Proof with eauto.
  intros. generalize dependent Gamma'.
  (has_type_cases (induction H) Case);
    intros Gamma' Heqv...
  Case "T_Var".
    apply T_Var... rewrite <- Heqv...
  Case "T_Abs".
    apply T_Abs... apply IHhas_type. intros y Hafi.
    unfold extend. remember (beq_id x y) as e.
    destruct e...
  (* FILL IN HERE *)

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 with eauto.
  intros x t T Gamma Hafi Htyp.
  (has_type_cases (induction Htyp) Case); inversion Hafi; subst...
  Case "T_Abs".
    destruct IHHtyp as [T' Hctx]... exists T'.
    unfold extend in Hctx.
    apply not_eq_beq_id_false in H2. rewrite H2 in Hctx...
  (* FILL IN HERE *)


Lemma substitution_preserves_typing : forall Gamma x U v t S,
     has_type (extend Gamma x U) t S ->
     has_type empty v U ->
     has_type Gamma (subst x v t) S.
Proof with eauto.
  (* Theorem: If Gamma,x:U |- t : S and empty |- v : U, then 
     Gamma |- (subst x v t) S. *)

  intros Gamma x U v t S Htypt Htypv.
  generalize dependent Gamma. generalize dependent S.
  (* Proof: By induction on the term t.  Most cases follow directly
     from the IH, with the exception of tm_var and tm_abs.
     The former aren't automatic because we must reason about how the
     variables interact. *)

  (tm_cases (induction t) Case);
    intros S Gamma Htypt; simpl; inversion Htypt; subst...
  Case "tm_var".
    simpl. rename i into y.
    (* If t = y, we know that
         empty |- v : U and
         Gamma,x:U |- y : S
       and, by inversion, extend Gamma x U y = Some S.  We want to
       show that Gamma |- subst x v y : S.

       There are two cases to consider: either x=y or x<>y. *)

    remember (beq_id x y) as e. destruct e.
    SCase "x=y".
    (* If x = y, then we know that U = S, and that subst x v y = v.
       So what we really must show is that if empty |- v : U then
       Gamma |- v : U.  We have already proven a more general version
       of this theorem, called context invariance. *)

      apply beq_id_eq in Heqe. subst.
      unfold extend in H1. rewrite <- beq_id_refl in H1.
      inversion H1; subst. clear H1.
      eapply context_invariance...
      intros x Hcontra.
      destruct (free_in_context _ _ S empty Hcontra) as [T' HT']...
      inversion HT'.
    SCase "x<>y".
    (* If x <> y, then Gamma y = Some S and the substitution has no
       effect.  We can show that Gamma |- y : S by T_Var. *)

      apply T_Var... unfold extend in H1. rewrite <- Heqe in H1...
  Case "tm_abs".
    rename i into y. rename t into T11.
    (* If t = tm_abs y T11 t0, then we know that
         Gamma,x:U |- tm_abs y T11 t0 : T11->T12
         Gamma,x:U,y:T11 |- t0 : T12
         empty |- v : U
       As our IH, we know that forall S Gamma, 
         Gamma,x:U |- t0 : S -> Gamma |- subst x v t0 S.
       We can calculate that 
         subst x v t = tm_abs y T11 (if beq_id x y
                                      then t0 
                                      else subst x v t0)
       And we must show that Gamma |- subst x v t : T11->T12.  We know
       we will do so using T_Abs, so it remains to be shown that:
         Gamma,y:T11 |- if beq_id x y then t0 else subst x v t0 : T12
       We consider two cases: x = y and x <> y.

    apply T_Abs...
    remember (beq_id x y) as e. destruct e.
    SCase "x=y".
    (* If x = y, then the substitution has no effect.  Context
       invariance shows that Gamma,y:U,y:T11 and Gamma,y:T11 are
       equivalent.  Since the former context shows that t0 : T12, so
       does the latter. *)

      eapply context_invariance...
      apply beq_id_eq in Heqe. subst.
      intros x Hafi. unfold extend.
      destruct (beq_id y x)...
    SCase "x<>y".
    (* If x <> y, then the IH and context invariance allow us to show that
         Gamma,x:U,y:T11 |- t0 : T12       =>
         Gamma,y:T11,x:U |- t0 : T12       =>
         Gamma,y:T11 |- subst x v t0 : T12 *)

      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...
  (* FILL IN HERE *)

Theorem preservation : forall t t' T,
     has_type empty t T ->
     t --> t' ->
     has_type empty t' T.
Proof with eauto.
  intros t t' T HT.
  (* Theorem: If empty |- t : T and t --> t', then empty |- t' : T. *)
  remember empty as Gamma. generalize dependent HeqGamma.
  generalize dependent t'.
  (* Proof: By induction on the given typing derivation.  Many cases are
     contradictory (T_VarT_Abs).  We show just the interesting ones. *)

  (has_type_cases (induction HT) Case);
    intros t' HeqGamma HE; subst; inversion HE; subst...
  Case "T_App".
    (* If the last rule used was T_App, then t = t1 t2, and three rules
       could have been used to show t --> t'ST_App1ST_App2, and 
       ST_AppAbs. In the first two cases, the result follows directly from 
       the IH. *)

    inversion HE; subst...
    SCase "ST_AppAbs".
      (* For the third case, suppose 
           t1 = tm_abs x T11 t12
           t2 = v2.  
         We must show that empty |- subst x v2 t12 : T2
         We know by assumption that
             empty |- tm_abs x T11 t12 : T1->T2
         and by inversion
             x:T1 |- t12 : T2
         We have already proven that substitution_preserves_typing and 
             empty |- v2 : T1
         by assumption, so we are done. *)

      apply substitution_preserves_typing with T1...
      inversion HT1...
  (* FILL IN HERE *)