MoreStlcMore on the Simply Typed Lambda-Calculus

Set Warnings "-notation-overridden,-parsing".
Require Import Maps.
Require Import Types.
Require Import Smallstep.
Require Import Stlc.

Simple Extensions to STLC

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

Numbers

As we saw in exercise stlc_arith at the end of the StlcProp chapter, adding types, constants, and primitive operations for natural numbers is easy — basically just a matter of combining the Types and Stlc chapters. Adding more realistic numeric types like machine integers and floats is also straightforward, though of course the specifications of the numeric primitives become more fiddly.

Let Bindings

When writing a complex expression, it is useful to be able to give names to some of its subexpressions to avoid repetition and increase readability. Most languages provide one or more ways of doing this. In OCaml (and Coq), for example, we can write let x=t1 in t2 to mean "reduce the expression t1 to a value and bind the name x to this value while reducing t2."
Our let-binder follows OCaml in choosing a standard call-by-value evaluation order, where the let-bound term must be fully reduced before reduction 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 book, it's probably easier simply to look at the rules defining this new feature than to wade through a lot of English text conveying the same information. Here they are:
Syntax:
       t ::=                Terms
           | ...               (other terms same as before)
           | let x=t in t      let-binding
Reduction:
t1 ==> t1' (ST_Let1)  

let x=t1 in t2 ==> let x=t1' in t2
   (ST_LetValue)  

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

Gamma |- let x=t1 in t2 : T2

Pairs

Our functional programming examples in Coq have made frequent use of pairs of values. The type of such a pair is called a product type.
The formalization of pairs is almost too simple to be worth discussing. 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 is to take fst and snd — the first- and second-projection operators — as primitives. Just for fun, let's do our pairs 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 difference:
       \x : Nat*Nat.
          let sum = x.fst + x.snd in
          let diff = x.fst - x.snd in
          (sum,diff)
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.
Syntax:
       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 reduction, 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
   (ST_FstPair)  

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

t1.snd ==> t1'.snd
   (ST_SndPair)  

(v1,v2).snd ==> v2
Rules ST_FstPair and ST_SndPair say that, when a fully reduced 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 reduced. ST_Pair1 and ST_Pair2 reduce 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 reduced before the function body starts executing.
The typing rules for pairs and projections are straightforward.
Gamma |- t1 : T1       Gamma |- t2 : T2 (T_Pair)  

Gamma |- (t1,t2) : T1*T2
Gamma |- t1 : T11*T12 (T_Fst)  

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

Gamma |- t1.snd : T12
T_Pair says that (t1,t2) has type T1*T2 if t1 has type T1 and t2 has type T2. Conversely, T_Fst and T_Snd tell us that, if t1 has a product type T11*T12 (i.e., if it will reduce to a pair), then the types of the projections from this pair are T11 and T12.

Unit

Another handy base type, found especially in languages in the ML family, is the singleton type Unit. It has 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 values — indeed, unit is the only possible result of reducing an expression of type Unit.
Syntax:
       t ::=                Terms
           | unit              unit value
           | ...

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

       T ::=                Types
           | Unit              Unit type
           | ...
Typing:
   (T_Unit)  

Gamma |- unit : Unit
It may seem a little strange to bother defining a type that has just one element — after all, wouldn't every computation living in such a type be trivial?
This is a fair question, and indeed in the STLC the Unit type is not especially critical (though we'll see two uses for it below). Where Unit really comes in handy is in richer languages with side effects — e.g., assignment statements that mutate variables or pointers, exceptions and other sorts of nonlocal control structures, etc. In such languages, it is convenient to have a type for the (trivial) result of an expression that is evaluated only for its effect.

Sums

Many programs need to deal with values that can take two distinct forms. For example, we might identify employees in an accounting application using either their name or their id number. A search function might return either a matching value or an error code.
These are specific examples of a binary sum type (sometimes called a disjoint union), which describes a set of values drawn from one of two given types, e.g.:
       Nat + Bool
We create elements of these types by tagging elements of the component types. For example, if n is a Nat then inl n is an element of Nat+Bool; similarly, if b is a Bool then inr b is a Nat+Bool. The names of the tags inl and inr arise from thinking of them as functions
   inl : Nat -> Nat + Bool
   inr : Bool -> Nat + Bool
that "inject" elements of Nat or Bool into the left and right components of the sum type Nat+Bool. (But note that we don't actually treat them as functions in the way we formalize them: inl and inr are keywords, and inl t and inr t are primitive syntactic forms, not function applications.)
In general, the elements of a type T1 + T2 consist of the elements of T1 tagged with the token inl, plus the elements of T2 tagged with inr.
One important usage of sums is signaling errors:
    div : Nat -> Nat -> (Nat + Unit) =
    div =
      \x:Nat. \y:Nat.
        if iszero y then
          inr unit
        else
          inl ...
The type Nat + Unit above is in fact isomorphic to option nat in Coq — i.e., it's easy to write functions that translate back and forth.
To use elements of sum types, we introduce a case construct (a very simplified form of Coq's match) to destruct them. For example, the following procedure converts a Nat+Bool into a Nat:
    getNat =
      \x:Nat+Bool.
        case x of
          inl n => n
        | inr b => if b then 1 else 0
More formally...
Syntax:
       t ::=                Terms
           | inl T t           tagging (left)
           | inr T t           tagging (right)
           | case t of         case
               inl x => t
             | inr x => t
           | ...

       v ::=                Values
           | inl T v           tagged value (left)
           | inr T v           tagged value (right)
           | ...

       T ::=                Types
           | T + T             sum type
           | ...
Reduction:
t1 ==> t1' (ST_Inl)  

inl T t1 ==> inl T t1'
t1 ==> t1' (ST_Inr)  

inr T t1 ==> inr T t1'
t0 ==> t0' (ST_Case)  

case t0 of inl x1 ⇒ t1 | inr x2 ⇒ t2 ==>
case t0' of inl x1 ⇒ t1 | inr x2 ⇒ t2
   (ST_CaseInl)  

case (inl T v0) of inl x1 ⇒ t1 | inr x2 ⇒ t2
==>  [x1:=v0]t1
   (ST_CaseInr)  

case (inr T v0) of inl x1 ⇒ t1 | inr x2 ⇒ t2
==>  [x2:=v0]t2
Typing:
Gamma |- t1 :  T1 (T_Inl)  

Gamma |- inl T2 t1 : T1 + T2
Gamma |- t1 : T2 (T_Inr)  

Gamma |- inr T1 t1 : T1 + T2
Gamma |- t0 : T1+T2
Gamma , x1:T1 |- t1 : T
Gamma , x2:T2 |- t2 : T (T_Case)  

Gamma |- case t0 of inl x1 ⇒ t1 | inr x2 ⇒ t2 : T
We use the type annotation in inl and inr to make the typing relation simpler, similarly to what we did for functions.
Without this extra information, the typing rule T_Inl, for example, would have to say that, once we have shown that t1 is an element of type T1, we can derive that inl t1 is an element of T1 + T2 for any type T2. For example, we could derive both inl 5 : Nat + Nat and inl 5 : Nat + Bool (and infinitely many other types). This peculiarity (technically, a failure of uniqueness of types) would mean that we cannot build a typechecking algorithm simply by "reading the rules from bottom to top" as we could for all the other features seen so far.
There are various ways to deal with this difficulty. One simple one — which we've adopted here — forces the programmer to explicitly annotate the "other side" of a sum type when performing an injection. This is a bit heavy for programmers (so real languages adopt other solutions), but it is easy to understand and formalize.

Lists

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.
In principle, we could encode lists using pairs, sums and recursive types. But giving semantics to recursive types is non-trivial. Instead, we'll just discuss the special case of lists directly.
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 lcase to destruct lists, to avoid dealing with questions like "what is the head of the empty list?"
For example, here is a function that calculates the sum of the first two elements of a list of numbers:
      \x:List Nat.
      lcase x of nil -> 0
         | a::x' -> lcase x' of nil -> a
                       | b::x'' -> a+b
Syntax:
       t ::=                Terms
           | nil T
           | cons t t
           | lcase 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
           | ...
Reduction:
t1 ==> t1' (ST_Cons1)  

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

cons v1 t2 ==> cons v1 t2'
t1 ==> t1' (ST_Lcase1)  

(lcase t1 of nil → t2 | xh::xt → t3) ==>
(lcase t1' of nil → t2 | xh::xt → t3)
   (ST_LcaseNil)  

(lcase nil T of nil → t2 | xh::xt → t3)
==> t2
   (ST_LcaseCons)  

(lcase (cons vh vt) of nil → t2 | xh::xt → t3)
==> [xh:=vh,xt:=vt]t3
Typing:
   (T_Nil)  

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_Lcase)  

Gamma |- (lcase 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)))
Note that the right-hand side of this binder mentions the variable being bound — something that is not allowed by our formalization of let above.
Directly formalizing this "recursive definition" mechanism is possible, but it requires a bit of extra effort: in particular, we'd have to pass around an "environment" of recursive function definitions in the definition of the step relation.
Here is another way of presenting recursive functions that is equally powerful (though not quite as convenient for the programmer) and more straightforward to formalize: instead of writing recursive definitions, we define a fixed-point operator called fix that performs the "unfolding" of the recursive definition in the right-hand side as needed, during reduction.
For example, instead of
      fact = \x:Nat.
                if x=0 then 1 else x * (fact (pred x)))
we will write:
      fact =
          fix
            (\f:Nat->Nat.
               \x:Nat.
                  if x=0 then 1 else x * (f (pred x)))
We can derive the latter from the former as follows:
  • In the right-hand side of the definition of fact, replace recursive references to fact by a fresh variable f.
  • Add an abstraction binding f at the front, with an appropriate type annotation. (Since we are using f in place of fact, which had type NatNat, we should require f to have the same type.) The new abstraction has type (NatNat) (NatNat).
  • Apply fix to this abstraction. This application has type NatNat.
  • Use all of this as the right-hand side of an ordinary let-binding for fact.
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 but we don't care what it does on inputs greater than n), then f returns a slightly 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, which is a function that gives the desired behavior for all inputs n.
(The term "fixed point" is used here in exactly the same sense as in ordinary mathematics, where a fixed point of a function f is an input x such that f(x) = x. Here, a fixed point of a function F of type (NatNat)->(NatNat) is a function f of type NatNat such that F f behaves the same as f.)
Syntax:
       t ::=                Terms
           | fix t             fixed-point operator
           | ...
Reduction:
t1 ==> t1' (ST_Fix1)  

fix t1 ==> fix t1'
   (ST_FixAbs)  

fix (\xf:T1.t2) ==> [xf:=fix (\xf:T1.t2)] t2
Typing:
Gamma |- t1 : T1->T1 (T_Fix)  

Gamma |- fix t1 : T1
Let's see how ST_FixAbs works by reducing fact 3 = fix F 3, where
    F = (\f. \x. if x=0 then 1 else x * (f (pred x)))
(type annotations are omitted for brevity).
    fix F 3
==> ST_FixAbs + ST_App1
    (\x. if x=0 then 1 else x * (fix F (pred x))) 3
==> ST_AppAbs
    if 3=0 then 1 else 3 * (fix F (pred 3))
==> ST_If0_Nonzero
    3 * (fix F (pred 3))
==> ST_FixAbs + ST_Mult2
    3 * ((\x. if x=0 then 1 else x * (fix F (pred x))) (pred 3))
==> ST_PredNat + ST_Mult2 + ST_App2
    3 * ((\x. if x=0 then 1 else x * (fix F (pred x))) 2)
==> ST_AppAbs + ST_Mult2
    3 * (if 2=0 then 1 else 2 * (fix F (pred 2)))
==> ST_If0_Nonzero + ST_Mult2
    3 * (2 * (fix F (pred 2)))
==> ST_FixAbs + 2 x ST_Mult2
    3 * (2 * ((\x. if x=0 then 1 else x * (fix F (pred x))) (pred 2)))
==> ST_PredNat + 2 x ST_Mult2 + ST_App2
    3 * (2 * ((\x. if x=0 then 1 else x * (fix F (pred x))) 1))
==> ST_AppAbs + 2 x ST_Mult2
    3 * (2 * (if 1=0 then 1 else 1 * (fix F (pred 1))))
==> ST_If0_Nonzero + 2 x ST_Mult2
    3 * (2 * (1 * (fix F (pred 1))))
==> ST_FixAbs + 3 x ST_Mult2
    3 * (2 * (1 * ((\x. if x=0 then 1 else x * (fix F (pred x))) (pred 1))))
==> ST_PredNat + 3 x ST_Mult2 + ST_App2
    3 * (2 * (1 * ((\x. if x=0 then 1 else x * (fix F (pred x))) 0)))
==> ST_AppAbs + 3 x ST_Mult2
    3 * (2 * (1 * (if 0=0 then 1 else 0 * (fix F (pred 0)))))
==> ST_If0Zero + 3 x ST_Mult2
    3 * (2 * (1 * 1))
==> ST_MultNats + 2 x ST_Mult2
    3 * (2 * 1)
==> ST_MultNats + ST_Mult2
    3 * 2
==> ST_MultNats
    6
One important point to note is that, unlike Fixpoint definitions in Coq, there is nothing to prevent functions defined using fix from diverging.

Exercise: 1 star, optional (halve_fix)

Translate this informal recursive definition into one using fix:
      halve =
        \x:Nat.
           if x=0 then 0
           else if (pred x)=0 then 0
           else 1 + (halve (pred (pred x))))
(* FILL IN HERE *)

Exercise: 1 star, optional (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).
(* FILL IN HERE *)
The ability to form the fixed point of a function of type TT 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 the term
    fix (\x:T.x)
By T_Fix and T_Abs, this term has type T. By ST_FixAbs it reduces to itself, over and over again. Thus it is a diverging element of T.
More usefully, here's an example using fix to define a two-argument recursive function:
    equal =
      fix
        (\eq:Nat->Nat->Bool.
           \m:Nat. \n:Nat.
             if m=0 then iszero n
             else if n=0 then false
             else eq (pred m) (pred n))
And finally, here is an example where fix is used to define a pair of recursive functions (illustrating the fact that the type T1 in the rule T_Fix need not be a function type):
      evenodd =
        fix
          (\eo: (Nat->Bool * Nat->Bool).
             let e = \n:Nat. if n=0 then true  else eo.snd (pred n) in
             let o = \n:Nat. if n=0 then false else eo.fst (pred n) in
             (e,o))

      even = evenodd.fst
      odd  = evenodd.snd

Records

As a final example of a basic extension of the STLC, let's look briefly at how to define records and their types. Intuitively, records can be obtained from pairs by two straightforward generalizations: they are n-ary (rather than just binary) and their fields are accessed by label (rather than position).
Syntax:
       t ::=                          Terms
           | {i1=t1, ..., in=tn}         record
           | t.i                         projection
           | ...

       v ::=                          Values
           | {i1=v1, ..., in=vn}         record value
           | ...

       T ::=                          Types
           | {i1:T1, ..., in:Tn}         record type
           | ...
The generalization from products should be pretty obvious. But it's worth noticing the ways in which what we've actually written is even more informal than the informal syntax we've used in previous sections and chapters: we've used "..." in several places to mean "any number of these," and we've omitted explicit mention of the usual side condition that the labels of a record should not contain any repetitions.
Reduction:
ti ==> ti' (ST_Rcd)  

{i1=v1, ..., im=vm, in=ti, ...}
==> {i1=v1, ..., im=vm, in=ti', ...}
t1 ==> t1' (ST_Proj1)  

t1.i ==> t1'.i
   (ST_ProjRcd)  

{..., i=vi, ...}.i ==> vi
Again, these rules are a bit informal. For example, the first rule is intended to be read "if ti is the leftmost field that is not a value and if ti steps to ti', then the whole record steps..." In the last rule, the intention is that there should only be one field called i, and that all the other fields must contain values.
The typing rules are also simple:
Gamma |- t1 : T1     ...     Gamma |- tn : Tn (T_Rcd)  

Gamma |- {i1=t1, ..., in=tn} : {i1:T1, ..., in:Tn}
Gamma |- t : {..., i:Ti, ...} (T_Proj)  

Gamma |- t.i : Ti
There are several ways to approach formalizing the above definitions.
  • We can directly formalize the syntactic forms and inference rules, staying as close as possible to the form we've given them above. This is conceptually straightforward, and it's probably what we'd want to do if we were building a real compiler (in particular, it will allow us to print error messages in the form that programmers will find easy to understand). But the formal versions of the rules will not be very pretty or easy to work with, because all the ...s above will have to be replaced with explicit quantifications or comprehensions. For this reason, records are not included in the extended exercise at the end of this chapter. (It is still useful to discuss them informally here because they will help motivate the addition of subtyping to the type system when we get to the Sub chapter.)
  • Alternatively, we could look for a smoother way of presenting records — for example, a binary presentation with one constructor for the empty record and another constructor for adding a single field to an existing record, instead of a single monolithic constructor that builds a whole record at once. This is the right way to go if we are primarily interested in studying the metatheory of the calculi with records, since it leads to clean and elegant definitions and proofs. Chapter Records shows how this can be done.
  • Finally, if we like, we can avoid formalizing records altogether, by stipulating that record notations are just informal shorthands for more complex expressions involving pairs and product types. We sketch this approach in the next section.

Encoding Records (Optional)

Let's see how records can be encoded using just pairs and unit.
First, observe that we can encode arbitrary-size tuples using nested pairs and the unit value. To avoid overloading the pair notation (t1,t2), we'll use curly braces without labels to write down tuples, so {} is the empty tuple, {5} is a singleton tuple, {5,6} is a 2-tuple (morally the same as a pair), {5,6,7} is a triple, etc.
      {}                 ---->  unit
      {t1, t2, ..., tn}  ---->  (t1, trest)
                                where {t2, ..., tn} ----> trest
Similarly, we can encode tuple types using nested product types:
      {}                 ---->  Unit
      {T1, T2, ..., Tn}  ---->  T1 * TRest
                                where {T2, ..., Tn} ----> TRest
The operation of projecting a field from a tuple can be encoded using a sequence of second projections followed by a first projection:
      t.0        ---->  t.fst
      t.(n+1)    ---->  (t.snd).n
Next, suppose that there is some total ordering on record labels, so that we can associate each label with a unique natural number. This number is called the position of the label. For example, we might assign positions like this:
      LABEL   POSITION
      a       0
      b       1
      c       2
      ...     ...
      bar     1395
      ...     ...
      foo     4460
      ...     ...
We use these positions to encode record values as tuples (i.e., as nested pairs) by sorting the fields according to their positions. For example:
      {a=5, b=6}      ---->   {5,6}
      {a=5, c=7}      ---->   {5,unit,7}
      {c=7, a=5}      ---->   {5,unit,7}
      {c=5, b=3}      ---->   {unit,3,5}
      {f=8,c=5,a=7}   ---->   {7,unit,5,unit,unit,8}
      {f=8,c=5}       ---->   {unit,unit,5,unit,unit,8}
Note that each field appears in the position associated with its label, that the size of the tuple is determined by the label with the highest position, and that we fill in unused positions with unit.
We do exactly the same thing with record types:
      {a:Nat, b:Nat}      ---->   {Nat,Nat}
      {c:Nat, a:Nat}      ---->   {Nat,Unit,Nat}
      {f:Nat,c:Nat}       ---->   {Unit,Unit,Nat,Unit,Unit,Nat}
Finally, record projection is encoded as a tuple projection from the appropriate position:
      t.l  ---->  t.(position of l)
It is not hard to check that all the typing rules for the original "direct" presentation of records are validated by this encoding. (The reduction rules are "almost validated" — not quite, because the encoding reorders fields.)
Of course, this encoding will not be very efficient if we happen to use a record with label foo! But things are not actually as bad as they might seem: for example, if we assume that our compiler can see the whole program at the same time, we can choose the numbering of labels so that we assign small positions to the most frequently used labels. Indeed, there are industrial compilers that essentially do this!

Variants (Optional)

Just as products can be generalized to records, sums can be generalized to n-ary labeled types called variants. Instead of T1+T2, we can write something like <l1:T1,l2:T2,...ln:Tn> where l1,l2,... are field labels which are used both to build instances and as case arm labels.
These n-ary variants give us almost enough mechanism to build arbitrary inductive data types like lists and trees from scratch — the only thing missing is a way to allow recursion in type definitions. We won't cover this here, but detailed treatments can be found in many textbooks — e.g., Types and Programming Languages [Pierce 2002].

Exercise: Formalizing the Extensions

Exercise: 5 stars (STLC_extensions)

In this exercise, you will formalize some of the extensions described in this chapter. We've provided the necessary additions 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.
To get you started, we've provided implementations for:
  • numbers
  • sums
  • lists
  • unit
You need to complete the implementations for:
  • pairs
  • let (which involves binding)
  • fix
A good strategy is to work on the extensions one at a time, in two passes, rather than trying to work through the file from start to finish in a single pass. For each definition or proof, begin by reading carefully through the parts that are provided for you, referring to the text in the Stlc chapter for high-level intuitions and the embedded comments for detailed mechanics.
Module STLCExtended.

Syntax


Inductive ty : Type :=
  | TArrow : tytyty
  | TNat : ty
  | TUnit : ty
  | TProd : tytyty
  | TSum : tytyty
  | TList : tyty.

Inductive tm : Type :=
  (* pure STLC *)
  | tvar : idtm
  | tapp : tmtmtm
  | tabs : idtytmtm
  (* numbers *)
  | tnat : nattm
  | tsucc : tmtm
  | tpred : tmtm
  | tmult : tmtmtm
  | tif0 : tmtmtmtm
  (* pairs *)
  | tpair : tmtmtm
  | tfst : tmtm
  | tsnd : tmtm
  (* units *)
  | tunit : tm
  (* let *)
  | tlet : idtmtmtm
          (* i.e., let x = t1 in t2 *)
  (* sums *)
  | tinl : tytmtm
  | tinr : tytmtm
  | tcase : tmidtmidtmtm
          (* i.e., case t0 of inl x1 t1 | inr x2 t2 *)
  (* lists *)
  | tnil : tytm
  | tcons : tmtmtm
  | tlcase : tmtmididtmtm
           (* i.e., lcase t1 of | nil t2 | x::y t3 *)
  (* fix *)
  | tfix : tmtm.
Note that, for brevity, we've omitted booleans and instead provided a single if0 form combining a zero test and a conditional. That is, instead of writing
       if x = 0 then ... else ...
we'll write this:
       if0 x then ... else ...

Substitution


Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
  match t with
  | tvar y
      if beq_id x y then s else t
  | tabs y T t1
      tabs y T (if beq_id x y then t1 else (subst x s t1))
  | tapp t1 t2
      tapp (subst x s t1) (subst x s t2)
  | tnat n
      tnat n
  | tsucc t1
      tsucc (subst x s t1)
  | tpred t1
      tpred (subst x s t1)
  | tmult t1 t2
      tmult (subst x s t1) (subst x s t2)
  | tif0 t1 t2 t3
      tif0 (subst x s t1) (subst x s t2) (subst x s t3)
  (* FILL IN HERE *)
  | tunittunit
  (* FILL IN HERE *)
  | tinl T t1
      tinl T (subst x s t1)
  | tinr T t1
      tinr T (subst x s t1)
  | tcase t0 y1 t1 y2 t2
      tcase (subst x s t0)
         y1 (if beq_id x y1 then t1 else (subst x s t1))
         y2 (if beq_id x y2 then t2 else (subst x s t2))
  | tnil T
      tnil T
  | tcons t1 t2
      tcons (subst x s t1) (subst x s t2)
  | tlcase t1 t2 y1 y2 t3
      tlcase (subst x s t1) (subst x s t2) y1 y2
        (if beq_id x y1 then
           t3
         else if beq_id x y2 then t3
              else (subst x s t3))
  (* FILL IN HERE *)
  | _t (* ... and delete this line *)
  end.

Notation "'[' x ':=' s ']' t" := (subst x s t) (at level 20).

Reduction

Next we define the values of our language.
Inductive value : tmProp :=
  | v_abs : ∀ x T11 t12,
      value (tabs x T11 t12)
  (* Numbers are values: *)
  | v_nat : ∀ n1,
      value (tnat n1)
  (* A pair is a value if both components are: *)
  | v_pair : ∀ v1 v2,
      value v1
      value v2
      value (tpair v1 v2)
  (* A unit is always a value *)
  | v_unit : value tunit
  (* A tagged value is a value:  *)
  | v_inl : ∀ v T,
      value v
      value (tinl T v)
  | v_inr : ∀ v T,
      value v
      value (tinr T v)
  (* A list is a value iff its head and tail are values: *)
  | v_lnil : ∀ T, value (tnil T)
  | v_lcons : ∀ v1 vl,
      value v1
      value vl
      value (tcons v1 vl)
  .

Hint Constructors value.

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

Inductive step : tmtmProp :=
  | ST_AppAbs : ∀ x T11 t12 v2,
         value v2
         (tapp (tabs x T11 t12) v2) ==> [x:=v2]t12
  | ST_App1 : ∀ t1 t1' t2,
         t1 ==> t1'
         (tapp t1 t2) ==> (tapp t1' t2)
  | ST_App2 : ∀ v1 t2 t2',
         value v1
         t2 ==> t2'
         (tapp v1 t2) ==> (tapp v1 t2')
  (* nats *)
  | ST_Succ1 : ∀ t1 t1',
       t1 ==> t1'
       (tsucc t1) ==> (tsucc t1')
  | ST_SuccNat : ∀ n1,
       (tsucc (tnat n1)) ==> (tnat (S n1))
  | ST_Pred : ∀ t1 t1',
       t1 ==> t1'
       (tpred t1) ==> (tpred t1')
  | ST_PredNat : ∀ n1,
       (tpred (tnat n1)) ==> (tnat (pred n1))
  | ST_Mult1 : ∀ t1 t1' t2,
       t1 ==> t1'
       (tmult t1 t2) ==> (tmult t1' t2)
  | ST_Mult2 : ∀ v1 t2 t2',
       value v1
       t2 ==> t2'
       (tmult v1 t2) ==> (tmult v1 t2')
  | ST_MultNats : ∀ n1 n2,
       (tmult (tnat n1) (tnat n2)) ==> (tnat (mult n1 n2))
  | ST_If01 : ∀ t1 t1' t2 t3,
       t1 ==> t1'
       (tif0 t1 t2 t3) ==> (tif0 t1' t2 t3)
  | ST_If0Zero : ∀ t2 t3,
       (tif0 (tnat 0) t2 t3) ==> t2
  | ST_If0Nonzero : ∀ n t2 t3,
       (tif0 (tnat (S n)) t2 t3) ==> t3
  (* pairs *)
  (* FILL IN HERE *)
  (* let *)
  (* FILL IN HERE *)
  (* sums *)
  | ST_Inl : ∀ t1 t1' T,
        t1 ==> t1'
        (tinl T t1) ==> (tinl T t1')
  | ST_Inr : ∀ t1 t1' T,
        t1 ==> t1'
        (tinr T t1) ==> (tinr T t1')
  | ST_Case : ∀ t0 t0' x1 t1 x2 t2,
        t0 ==> t0'
        (tcase t0 x1 t1 x2 t2) ==> (tcase t0' x1 t1 x2 t2)
  | ST_CaseInl : ∀ v0 x1 t1 x2 t2 T,
        value v0
        (tcase (tinl T v0) x1 t1 x2 t2) ==> [x1:=v0]t1
  | ST_CaseInr : ∀ v0 x1 t1 x2 t2 T,
        value v0
        (tcase (tinr T v0) x1 t1 x2 t2) ==> [x2:=v0]t2
  (* lists *)
  | ST_Cons1 : ∀ t1 t1' t2,
       t1 ==> t1'
       (tcons t1 t2) ==> (tcons t1' t2)
  | ST_Cons2 : ∀ v1 t2 t2',
       value v1
       t2 ==> t2'
       (tcons v1 t2) ==> (tcons v1 t2')
  | ST_Lcase1 : ∀ t1 t1' t2 x1 x2 t3,
       t1 ==> t1'
       (tlcase t1 t2 x1 x2 t3) ==> (tlcase t1' t2 x1 x2 t3)
  | ST_LcaseNil : ∀ T t2 x1 x2 t3,
       (tlcase (tnil T) t2 x1 x2 t3) ==> t2
  | ST_LcaseCons : ∀ v1 vl t2 x1 x2 t3,
       value v1
       value vl
       (tlcase (tcons v1 vl) t2 x1 x2 t3) ==> (subst x2 vl (subst x1 v1 t3))
  (* fix *)
  (* FILL IN HERE *)

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

Notation multistep := (multi step).
Notation "t1 '==>*' t2" := (multistep t1 t2) (at level 40).

Hint Constructors step.

Typing


Definition context := partial_map ty.
Next we define the typing rules. These are nearly direct transcriptions of the inference rules shown above.
Reserved Notation "Gamma '|-' t '∈' T" (at level 40).

Inductive has_type : contexttmtyProp :=
  (* Typing rules for proper terms *)
  | T_Var : ∀ Gamma x T,
      Gamma x = Some T
      Gamma |- (tvar x) ∈ T
  | T_Abs : ∀ Gamma x T11 T12 t12,
      (update Gamma x T11) |- t12T12
      Gamma |- (tabs x T11 t12) ∈ (TArrow T11 T12)
  | T_App : ∀ T1 T2 Gamma t1 t2,
      Gamma |- t1 ∈ (TArrow T1 T2) →
      Gamma |- t2T1
      Gamma |- (tapp t1 t2) ∈ T2
  (* nats *)
  | T_Nat : ∀ Gamma n1,
      Gamma |- (tnat n1) ∈ TNat
  | T_Succ : ∀ Gamma t1,
      Gamma |- t1TNat
      Gamma |- (tsucc t1) ∈ TNat
  | T_Pred : ∀ Gamma t1,
      Gamma |- t1TNat
      Gamma |- (tpred t1) ∈ TNat
  | T_Mult : ∀ Gamma t1 t2,
      Gamma |- t1TNat
      Gamma |- t2TNat
      Gamma |- (tmult t1 t2) ∈ TNat
  | T_If0 : ∀ Gamma t1 t2 t3 T1,
      Gamma |- t1TNat
      Gamma |- t2T1
      Gamma |- t3T1
      Gamma |- (tif0 t1 t2 t3) ∈ T1
  (* pairs *)
  (* FILL IN HERE *)
  (* unit *)
  | T_Unit : ∀ Gamma,
      Gamma |- tunitTUnit
  (* let *)
  (* FILL IN HERE *)
  (* sums *)
  | T_Inl : ∀ Gamma t1 T1 T2,
      Gamma |- t1T1
      Gamma |- (tinl T2 t1) ∈ (TSum T1 T2)
  | T_Inr : ∀ Gamma t2 T1 T2,
      Gamma |- t2T2
      Gamma |- (tinr T1 t2) ∈ (TSum T1 T2)
  | T_Case : ∀ Gamma t0 x1 T1 t1 x2 T2 t2 T,
      Gamma |- t0 ∈ (TSum T1 T2) →
      (update Gamma x1 T1) |- t1T
      (update Gamma x2 T2) |- t2T
      Gamma |- (tcase t0 x1 t1 x2 t2) ∈ T
  (* lists *)
  | T_Nil : ∀ Gamma T,
      Gamma |- (tnil T) ∈ (TList T)
  | T_Cons : ∀ Gamma t1 t2 T1,
      Gamma |- t1T1
      Gamma |- t2 ∈ (TList T1) →
      Gamma |- (tcons t1 t2) ∈ (TList T1)
  | T_Lcase : ∀ Gamma t1 T1 t2 x1 x2 t3 T2,
      Gamma |- t1 ∈ (TList T1) →
      Gamma |- t2T2
      (update (update Gamma x2 (TList T1)) x1 T1) |- t3T2
      Gamma |- (tlcase t1 t2 x1 x2 t3) ∈ T2
  (* fix *)
  (* FILL IN HERE *)

where "Gamma '|-' t '∈' T" := (has_type Gamma t T).

Hint Constructors has_type.

Examples

This section presents formalized versions of the examples from above (plus several more). The ones at the beginning focus on specific features; you can use these to make sure your definition of a given feature is reasonable before moving on to extending the proofs later in the file with the cases relating to this feature. The later examples require all the features together, so you'll need to come back to these when you've got all the definitions filled in.
Module Examples.

Preliminaries

First, let's define a few variable names:
Notation x := (Id "x").
Notation y := (Id "y").
Notation a := (Id "a").
Notation f := (Id "f").
Notation g := (Id "g").
Notation l := (Id "l").
Notation k := (Id "k").
Notation i1 := (Id "i1").
Notation i2 := (Id "i2").
Notation processSum := (Id "processSum").
Notation n := (Id "n").
Notation eq := (Id "eq").
Notation m := (Id "m").
Notation evenodd := (Id "evenodd").
Notation even := (Id "even").
Notation odd := (Id "odd").
Notation eo := (Id "eo").
Next, a bit of Coq hackery to automate searching for typing derivations. You don't need to understand this bit in detail — just have a look over it so that you'll know what to look for if you ever find yourself needing to make custom extensions to auto.
The following Hint declarations say that, whenever auto arrives at a goal of the form (Gamma |- (tapp e1 e1) T), it should consider eapply T_App, leaving an existential variable for the middle type T1, and similar for lcase. 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 _ (tapp _ _) _) ⇒
  eapply T_App; auto.
Hint Extern 2 (has_type _ (tlcase _ _ _ _ _) _) ⇒
  eapply T_Lcase; auto.
Hint Extern 2 (_ = _) ⇒ compute; reflexivity.

Numbers


Module Numtest.

(* if0 (pred (succ (pred (2 * 0))) then 5 else 6 *)
Definition test :=
  tif0
    (tpred
      (tsucc
        (tpred
          (tmult
            (tnat 2)
            (tnat 0)))))
    (tnat 5)
    (tnat 6).
Remove the comment braces once you've implemented enough of the definitions that you think this should work.
(* 
Example typechecks :
  empty |- test ∈ TNat.
Proof.
  unfold test.
  (* This typing derivation is quite deep, so we need 
     to increase the max search depth of auto from the 
     default 5 to 10. *)

  auto 10.
Qed.

Example numtest_reduces :
  test ==>* tnat 5.
Proof.
  unfold test. normalize.
Qed.
*)


End Numtest.

Products


Module Prodtest.

(* ((5,6),7).fst.snd *)
Definition test :=
  tsnd
    (tfst
      (tpair
        (tpair
          (tnat 5)
          (tnat 6))
        (tnat 7))).

(* 
Example typechecks :
  empty |- test ∈ TNat.
Proof. unfold test. eauto 15. Qed.

Example reduces :
  test ==>* tnat 6.
Proof. unfold test. normalize. Qed.
*)


End Prodtest.

let


Module LetTest.

(* let x = pred 6 in succ x *)
Definition test :=
  tlet
    x
    (tpred (tnat 6))
    (tsucc (tvar x)).

(* 
Example typechecks :
  empty |- test ∈ TNat.
Proof. unfold test. eauto 15. Qed.

Example reduces :
  test ==>* tnat 6.
Proof. unfold test. normalize. Qed.
*)


End LetTest.

Sums


Module Sumtest1.

(* case (inl Nat 5) of
     inl x => x
   | inr y => y *)


Definition test :=
  tcase (tinl TNat (tnat 5))
    x (tvar x)
    y (tvar y).

(* 
Example typechecks :
  empty |- test ∈ TNat.
Proof. unfold test. eauto 15. Qed.

Example reduces :
  test ==>* (tnat 5).
Proof. unfold test. normalize. Qed.
*)


End Sumtest1.

Module Sumtest2.

(* let processSum =
     \x:Nat+Nat.
        case x of
          inl n => n
          inr n => if0 n then 1 else 0 in
   (processSum (inl Nat 5), processSum (inr Nat 5))    *)


Definition test :=
  tlet
    processSum
    (tabs x (TSum TNat TNat)
      (tcase (tvar x)
         n (tvar n)
         n (tif0 (tvar n) (tnat 1) (tnat 0))))
    (tpair
      (tapp (tvar processSum) (tinl TNat (tnat 5)))
      (tapp (tvar processSum) (tinr TNat (tnat 5)))).

(* 
Example typechecks :
  empty |- test ∈ (TProd TNat TNat).
Proof. unfold test. eauto 15. Qed.

Example reduces :
  test ==>* (tpair (tnat 5) (tnat 0)).
Proof. unfold test. normalize. Qed.
*)


End Sumtest2.

Lists


Module ListTest.

(* let l = cons 5 (cons 6 (nil Nat)) in
   lcase l of
     nil => 0
   | x::y => x*x *)


Definition test :=
  tlet l
    (tcons (tnat 5) (tcons (tnat 6) (tnil TNat)))
    (tlcase (tvar l)
       (tnat 0)
       x y (tmult (tvar x) (tvar x))).

(* 
Example typechecks :
  empty |- test ∈ TNat.
Proof. unfold test. eauto 20. Qed.

Example reduces :
  test ==>* (tnat 25).
Proof. unfold test. normalize. Qed.
*)


End ListTest.

fix


Module FixTest1.

(* fact := fix
             (\f:nat->nat.
                \a:nat.
                   if a=0 then 1 else a * (f (pred a))) *)

Definition fact :=
  tfix
    (tabs f (TArrow TNat TNat)
      (tabs a TNat
        (tif0
           (tvar a)
           (tnat 1)
           (tmult
              (tvar a)
              (tapp (tvar f) (tpred (tvar a))))))).
(Warning: you may be able to typecheck fact but still have some rules wrong!)
(* 
Example fact_typechecks :
  empty |- fact ∈ (TArrow TNat TNat).
Proof. unfold fact. auto 10.
Qed.
*)


(* 
Example fact_example:
  (tapp fact (tnat 4)) ==>* (tnat 24).
Proof. unfold fact. normalize. Qed.
*)


End FixTest1.

Module FixTest2.

(* map :=
     \g:nat->nat.
       fix
         (\f:nat->nat.
            \l:nat.
               case l of
               |  -> 
               | x::l -> (g x)::(f l)) *)

Definition map :=
  tabs g (TArrow TNat TNat)
    (tfix
      (tabs f (TArrow (TList TNat) (TList TNat))
        (tabs l (TList TNat)
          (tlcase (tvar l)
            (tnil TNat)
            a l (tcons (tapp (tvar g) (tvar a))
                         (tapp (tvar f) (tvar l))))))).

(* 
(* Make sure you've uncommented the last Hint Extern above... *)
Example map_typechecks :
  empty |- map ∈
    (TArrow (TArrow TNat TNat)
      (TArrow (TList TNat)
        (TList TNat))).
Proof. unfold map. auto 10. Qed.

Example map_example :
  tapp (tapp map (tabs a TNat (tsucc (tvar a))))
         (tcons (tnat 1) (tcons (tnat 2) (tnil TNat)))
  ==>* (tcons (tnat 2) (tcons (tnat 3) (tnil TNat))).
Proof. unfold map. normalize. Qed.
*)


End FixTest2.

Module FixTest3.

(* equal =
      fix
        (\eq:Nat->Nat->Bool.
           \m:Nat. \n:Nat.
             if0 m then (if0 n then 1 else 0)
             else if0 n then 0
             else eq (pred m) (pred n))   *)


Definition equal :=
  tfix
    (tabs eq (TArrow TNat (TArrow TNat TNat))
      (tabs m TNat
        (tabs n TNat
          (tif0 (tvar m)
            (tif0 (tvar n) (tnat 1) (tnat 0))
            (tif0 (tvar n)
              (tnat 0)
              (tapp (tapp (tvar eq)
                              (tpred (tvar m)))
                      (tpred (tvar n)))))))).

(* 
Example equal_typechecks :
  empty |- equal ∈ (TArrow TNat (TArrow TNat TNat)).
Proof. unfold equal. auto 10.
Qed.
*)


(* 
Example equal_example1:
  (tapp (tapp equal (tnat 4)) (tnat 4)) ==>* (tnat 1).
Proof. unfold equal. normalize. Qed.
*)


(* 
Example equal_example2:
  (tapp (tapp equal (tnat 4)) (tnat 5)) ==>* (tnat 0).
Proof. unfold equal. normalize. Qed.
*)


End FixTest3.

Module FixTest4.

(* let evenodd =
         fix
           (\eo: (Nat->Nat * Nat->Nat).
              let e = \n:Nat. if0 n then 1 else eo.snd (pred n) in
              let o = \n:Nat. if0 n then 0 else eo.fst (pred n) in
              (e,o)) in
    let even = evenodd.fst in
    let odd  = evenodd.snd in
    (even 3, even 4)
*)


Definition eotest :=
  tlet evenodd
    (tfix
      (tabs eo (TProd (TArrow TNat TNat) (TArrow TNat TNat))
        (tpair
          (tabs n TNat
            (tif0 (tvar n)
              (tnat 1)
              (tapp (tsnd (tvar eo)) (tpred (tvar n)))))
          (tabs n TNat
            (tif0 (tvar n)
              (tnat 0)
              (tapp (tfst (tvar eo)) (tpred (tvar n))))))))
  (tlet even (tfst (tvar evenodd))
  (tlet odd (tsnd (tvar evenodd))
  (tpair
    (tapp (tvar even) (tnat 3))
    (tapp (tvar even) (tnat 4))))).

(* 
Example eotest_typechecks :
  empty |- eotest ∈ (TProd TNat TNat).
Proof. unfold eotest. eauto 30.
Qed.
*)


(* 
Example eotest_example1:
  eotest ==>* (tpair (tnat 0) (tnat 1)).
Proof. unfold eotest. normalize. Qed.
*)


End FixTest4.

End Examples.

Properties of Typing

The proofs of progress and preservation for this enriched system are essentially the same (though of course longer) as for the pure STLC.

Progress


Theorem progress : ∀ t T,
     empty |- tT
     value t ∨ ∃ 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.
  induction Ht; intros HeqGamma; subst.
  - (* 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.
  - (* T_Abs *)
    (* If the T_Abs rule was the last used, then 
       t = tabs x T11 t12, which is a value. *)

    left...
  - (* 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. *)

    right.
    destruct IHHt1; subst...
    + (* t1 is a value *)
      destruct IHHt2; subst...
      * (* t2 is a value *)
        (* If both t1 and t2 are values, then we know that
           t1 = tabs x T11 t12, since abstractions are the 
           only values that can have an arrow type.  But
           (tabs x T11 t12) t2 ==> [x:=t2]t12 by ST_AppAbs. *)

        inversion H; subst; try solve_by_invert.
        ∃ (subst x t2 t12)...
      * (* t2 steps *)
        (* If t1 is a value and t2 ==> t2'
           then t1 t2 ==> t1 t2' by ST_App2. *)

        inversion H0 as [t2' Hstp]. ∃ (tapp t1 t2')...
    + (* t1 steps *)
      (* Finally, If t1 ==> t1', then t1 t2 ==> t1' t2 
         by ST_App1. *)

      inversion H as [t1' Hstp]. ∃ (tapp t1' t2)...
  - (* T_Nat *)
    left...
  - (* T_Succ *)
    right.
    destruct IHHt...
    + (* t1 is a value *)
      inversion H; subst; try solve_by_invert.
      ∃ (tnat (S n1))...
    + (* t1 steps *)
      inversion H as [t1' Hstp].
      ∃ (tsucc t1')...
  - (* T_Pred *)
    right.
    destruct IHHt...
    + (* t1 is a value *)
      inversion H; subst; try solve_by_invert.
      ∃ (tnat (pred n1))...
    + (* t1 steps *)
      inversion H as [t1' Hstp].
      ∃ (tpred t1')...
  - (* T_Mult *)
    right.
    destruct IHHt1...
    + (* t1 is a value *)
      destruct IHHt2...
      * (* t2 is a value *)
        inversion H; subst; try solve_by_invert.
        inversion H0; subst; try solve_by_invert.
        ∃ (tnat (mult n1 n0))...
      * (* t2 steps *)
        inversion H0 as [t2' Hstp].
        ∃ (tmult t1 t2')...
    + (* t1 steps *)
      inversion H as [t1' Hstp].
      ∃ (tmult t1' t2)...
  - (* T_If0 *)
    right.
    destruct IHHt1...
    + (* t1 is a value *)
      inversion H; subst; try solve_by_invert.
      destruct n1 as [|n1'].
      * (* n1=0 *)
        ∃ t2...
      * (* n1<>0 *)
        ∃ t3...
    + (* t1 steps *)
      inversion H as [t1' H0].
      ∃ (tif0 t1' t2 t3)...
  (* FILL IN HERE *)
  - (* T_Unit *)
    left...
  (* let *)
  (* FILL IN HERE *)
  - (* T_Inl *)
    destruct IHHt...
    + (* t1 steps *)
      right. inversion H as [t1' Hstp]...
      (* exists (tinl _ t1')... *)
  - (* T_Inr *)
    destruct IHHt...
    + (* t1 steps *)
      right. inversion H as [t1' Hstp]...
      (* exists (tinr _ t1')... *)
  - (* T_Case *)
    right.
    destruct IHHt1...
    + (* t0 is a value *)
      inversion H; subst; try solve_by_invert.
      * (* t0 is inl *)
        ∃ ([x1:=v]t1)...
      * (* t0 is inr *)
        ∃ ([x2:=v]t2)...
    + (* t0 steps *)
      inversion H as [t0' Hstp].
      ∃ (tcase t0' x1 t1 x2 t2)...
  - (* T_Nil *)
    left...
  - (* T_Cons *)
    destruct IHHt1...
    + (* head is a value *)
      destruct IHHt2...
      * (* tail steps *)
        right. inversion H0 as [t2' Hstp].
        ∃ (tcons t1 t2')...
    + (* head steps *)
      right. inversion H as [t1' Hstp].
      ∃ (tcons t1' t2)...
  - (* T_Lcase *)
    right.
    destruct IHHt1...
    + (* t1 is a value *)
      inversion H; subst; try solve_by_invert.
      * (* t1=tnil *)
        ∃ t2...
      * (* t1=tcons v1 vl *)
        ∃ ([x2:=vl]([x1:=v1]t3))...
    + (* t1 steps *)
      inversion H as [t1' Hstp].
      ∃ (tlcase t1' t2 x1 x2 t3)...
  (* fix *)
  (* FILL IN HERE *)
Qed.

Context Invariance


Inductive appears_free_in : idtmProp :=
  | afi_var : ∀ x,
      appears_free_in x (tvar x)
  | afi_app1 : ∀ x t1 t2,
      appears_free_in x t1appears_free_in x (tapp t1 t2)
  | afi_app2 : ∀ x t1 t2,
      appears_free_in x t2appears_free_in x (tapp t1 t2)
  | afi_abs : ∀ x y T11 t12,
        yx
        appears_free_in x t12
        appears_free_in x (tabs y T11 t12)
  (* nats *)
  | afi_succ : ∀ x t,
     appears_free_in x t
     appears_free_in x (tsucc t)
  | afi_pred : ∀ x t,
     appears_free_in x t
     appears_free_in x (tpred t)
  | afi_mult1 : ∀ x t1 t2,
     appears_free_in x t1
     appears_free_in x (tmult t1 t2)
  | afi_mult2 : ∀ x t1 t2,
     appears_free_in x t2
     appears_free_in x (tmult t1 t2)
  | afi_if01 : ∀ x t1 t2 t3,
     appears_free_in x t1
     appears_free_in x (tif0 t1 t2 t3)
  | afi_if02 : ∀ x t1 t2 t3,
     appears_free_in x t2
     appears_free_in x (tif0 t1 t2 t3)
  | afi_if03 : ∀ x t1 t2 t3,
     appears_free_in x t3
     appears_free_in x (tif0 t1 t2 t3)
  (* pairs *)
  (* FILL IN HERE *)
  (* let *)
  (* FILL IN HERE *)
  (* sums *)
  | afi_inl : ∀ x t T,
      appears_free_in x t
      appears_free_in x (tinl T t)
  | afi_inr : ∀ x t T,
      appears_free_in x t
      appears_free_in x (tinr T t)
  | afi_case0 : ∀ x t0 x1 t1 x2 t2,
      appears_free_in x t0
      appears_free_in x (tcase t0 x1 t1 x2 t2)
  | afi_case1 : ∀ x t0 x1 t1 x2 t2,
      x1x
      appears_free_in x t1
      appears_free_in x (tcase t0 x1 t1 x2 t2)
  | afi_case2 : ∀ x t0 x1 t1 x2 t2,
      x2x
      appears_free_in x t2
      appears_free_in x (tcase t0 x1 t1 x2 t2)
  (* lists *)
  | afi_cons1 : ∀ x t1 t2,
     appears_free_in x t1
     appears_free_in x (tcons t1 t2)
  | afi_cons2 : ∀ x t1 t2,
     appears_free_in x t2
     appears_free_in x (tcons t1 t2)
  | afi_lcase1 : ∀ x t1 t2 y1 y2 t3,
     appears_free_in x t1
     appears_free_in x (tlcase t1 t2 y1 y2 t3)
  | afi_lcase2 : ∀ x t1 t2 y1 y2 t3,
     appears_free_in x t2
     appears_free_in x (tlcase t1 t2 y1 y2 t3)
  | afi_lcase3 : ∀ x t1 t2 y1 y2 t3,
     y1x
     y2x
     appears_free_in x t3
     appears_free_in x (tlcase t1 t2 y1 y2 t3)
  (* fix *)
  (* FILL IN HERE *)
.

Hint Constructors appears_free_in.

Lemma context_invariance : ∀ Gamma Gamma' t S,
     Gamma |- tS
     (∀ x, appears_free_in x tGamma x = Gamma' x) →
     Gamma' |- tS.
Proof with eauto.
  intros. generalize dependent Gamma'.
  induction H;
    intros Gamma' Heqv...
  - (* T_Var *)
    apply T_Var... rewrite <- Heqv...
  - (* T_Abs *)
    apply T_Abs... apply IHhas_type. intros y Hafi.
    unfold update, t_update.
    destruct (beq_idP x y)...
  - (* T_Mult *)
    apply T_Mult...
  - (* T_If0 *)
    apply T_If0...
  (* pair *)
  (* FILL IN HERE *)
  (* let *)
  (* FILL IN HERE *)
  - (* T_Case *)
    eapply T_Case...
    + apply IHhas_type2. intros y Hafi.
      unfold update, t_update.
      destruct (beq_idP x1 y)...
    + apply IHhas_type3. intros y Hafi.
      unfold update, t_update.
      destruct (beq_idP x2 y)...
  - (* T_Cons *)
    apply T_Cons...
  - (* T_Lcase *)
    eapply T_Lcase... apply IHhas_type3. intros y Hafi.
    unfold update, t_update.
    destruct (beq_idP x1 y)...
    destruct (beq_idP x2 y)...
Qed.

Lemma free_in_context : ∀ x t T Gamma,
   appears_free_in x t
   Gamma |- tT
   ∃ 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_beq_id in Hctx...
  (* let *)
  (* FILL IN HERE *)
  (* T_Case *)
  - (* left *)
    destruct IHHtyp2 as [T' Hctx]... ∃ T'.
    unfold update, t_update in Hctx.
    rewrite false_beq_id in Hctx...
  - (* right *)
    destruct IHHtyp3 as [T' Hctx]... ∃ T'.
    unfold update, t_update in Hctx.
    rewrite false_beq_id in Hctx...
  - (* T_Lcase *)
    clear Htyp1 IHHtyp1 Htyp2 IHHtyp2.
    destruct IHHtyp3 as [T' Hctx]... ∃ T'.
    unfold update, t_update in Hctx.
    rewrite false_beq_id in Hctx...
    rewrite false_beq_id in Hctx...
Qed.

Substitution


Lemma substitution_preserves_typing : ∀ Gamma x U v t S,
     (update Gamma x U) |- tS
     empty |- vU
     Gamma |- ([x:=v]t) ∈ S.
Proof with eauto.
  (* Theorem: If Gamma,x:U |- t : S and empty |- v : U, then
     Gamma |- x:=vt : 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 tvar
     and tabs. These aren't automatic because we must
     reason about how the variables interact. *)

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

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

    unfold update, t_update in H1.
    destruct (beq_idP x y).
    + (* x=y *)
      (* If x = y, then we know that U = S, and that 
         [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. *)

      subst.
      inversion H1; subst. clear H1.
      eapply context_invariance...
      intros x Hcontra.
      destruct (free_in_context _ _ S empty Hcontra)
        as [T' HT']...
      inversion HT'.
    + (* 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...
  - (* tabs *)
    rename i into y. rename t into T11.
    (* If t = tabs y T11 t0, then we know that
         Gamma,x:U |- tabs y T11 t0 : T11T12
         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 |- [x:=v]t0 : S.

       We can calculate that
         x:=vt = tabs y T11 (if beq_id x y then t0 else x:=vt0)
       And we must show that Gamma |- [x:=v]t : T11T12.  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 [x:=v]t0 : T12
       We consider two cases: x = y and x y.
    *)

    apply T_Abs...
    destruct (beq_idP x y) as [Hxy|Hxy].
    + (* 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...
      subst.
      intros x Hafi. unfold update, t_update.
      destruct (beq_id y x)...
    + (* 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 |- [x:=v]t0 : T12 *)

      apply IHt. eapply context_invariance...
      intros z Hafi. unfold update, t_update.
      destruct (beq_idP y z) as [Hyz|Hyz]...
      subst.
      rewrite false_beq_id...
  (* let *)
  (* FILL IN HERE *)
  - (* tcase *)
    rename i into x1. rename i0 into x2.
    eapply T_Case...
    + (* left arm *)
      destruct (beq_idP x x1) as [Hxx1|Hxx1].
      * (* x = x1 *)
        eapply context_invariance...
        subst.
        intros z Hafi. unfold update, t_update.
        destruct (beq_id x1 z)...
      * (* x <> x1 *)
        apply IHt2. eapply context_invariance...
        intros z Hafi. unfold update, t_update.
        destruct (beq_idP x1 z) as [Hx1z|Hx1z]...
        subst. rewrite false_beq_id...
    + (* right arm *)
      destruct (beq_idP x x2) as [Hxx2|Hxx2].
      * (* x = x2 *)
        eapply context_invariance...
        subst.
        intros z Hafi. unfold update, t_update.
        destruct (beq_id x2 z)...
      * (* x <> x2 *)
        apply IHt3. eapply context_invariance...
        intros z Hafi. unfold update, t_update.
        destruct (beq_idP x2 z)...
        subst. rewrite false_beq_id...
  - (* tlcase *)
    rename i into y1. rename i0 into y2.
    eapply T_Lcase...
    destruct (beq_idP x y1).
    + (* x=y1 *)
      simpl.
      eapply context_invariance...
      subst.
      intros z Hafi. unfold update, t_update.
      destruct (beq_idP y1 z)...
    + (* x<>y1 *)
      destruct (beq_idP x y2).
      * (* x=y2 *)
        eapply context_invariance...
        subst.
        intros z Hafi. unfold update, t_update.
        destruct (beq_idP y2 z)...
      * (* x<>y2 *)
        apply IHt3. eapply context_invariance...
        intros z Hafi. unfold update, t_update.
        destruct (beq_idP y1 z)...
        subst. rewrite false_beq_id...
        destruct (beq_idP y2 z)...
        subst. rewrite false_beq_id...
Qed.

Preservation


Theorem preservation : ∀ t t' T,
     empty |- tT
     t ==> t'
     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. *)

  induction HT;
    intros t' HeqGamma HE; subst; inversion HE; subst...
  - (* 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...
    + (* ST_AppAbs *)
      (* For the third case, suppose
           t1 = tabs x T11 t12
         and
           t2 = v2.
         We must show that empty |- [x:=v2]t12 : T2.
         We know by assumption that
             empty |- tabs x T11 t12 : T1T2
         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...
  (* fst and snd *)
  (* FILL IN HERE *)
  (* let *)
  (* FILL IN HERE *)
  (* T_Case *)
  - (* ST_CaseInl *)
    inversion HT1; subst.
    eapply substitution_preserves_typing...
  - (* ST_CaseInr *)
    inversion HT1; subst.
    eapply substitution_preserves_typing...
  - (* T_Lcase *)
    + (* ST_LcaseCons *)
      inversion HT1; subst.
      apply substitution_preserves_typing with (TList T1)...
      apply substitution_preserves_typing with T1...
  (* fix *)
  (* FILL IN HERE *)
Qed.

End STLCExtended.