ReferencesTyping Mutable References

(* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)

Require Export Smallstep.

So far, we have considered a variety of pure language features, including functional abstraction, basic types such as numbers and booleans, and structured types such as records and variants. These features form the backbone of most programming languages — including purely functional languages such as Haskell, "mostly functional" languages such as ML, imperative languages such as C, and object-oriented languages such as Java.
Most practical programming languages also include various impure features that cannot be described in the simple semantic framework we have used so far. In particular, besides just yielding results, evaluation of terms in these languages may assign to mutable variables (reference cells, arrays, mutable record fields, etc.), perform input and output to files, displays, or network connections, make non-local transfers of control via exceptions, jumps, or continuations, engage in inter-process synchronization and communication, and so on. In the literature on programming languages, such "side effects" of computation are more generally referred to as computational effects.
In this chapter, we'll see how one sort of computational effect — mutable references — can be added to the calculi we have studied. The main extension will be dealing explicitly with a store (or heap). This extension is straightforward to define; the most interesting part is the refinement we need to make to the statement of the type preservation theorem.


Pretty much every programming language provides some form of assignment operation that changes the contents of a previously allocated piece of storage. (Coq's internal language is a rare exception!)
In some languages — notably ML and its relatives — the mechanisms for name-binding and those for assignment are kept separate. We can have a variable x whose value is the number 5, or we can have a variable y whose value is a reference (or pointer) to a mutable cell whose current contents is 5. These are different things, and the difference is visible to the programmer. We can add x to another number, but not assign to it. We can use y directly to assign a new value to the cell that it points to (by writing y:=84), but we cannot use it directly as an argument to an operation like +. Instead, we must explicitly dereference it, writing !y to obtain its current contents.
In most other languages — in particular, in all members of the C family, including Java — every variable name refers to a mutable cell, and the operation of dereferencing a variable to obtain its current contents is implicit.
For purposes of formal study, it is useful to keep these mechanisms separate. The development in this chapter will closely follow ML's model. Applying the lessons learned here to C-like languages is a straightforward matter of collapsing some distinctions and rendering some operations such as dereferencing implicit instead of explicit.
In this chapter, we study adding mutable references to the simply-typed lambda calculus with natural numbers.


Module STLCRef.

The basic operations on references are allocation, dereferencing, and assignment.
  • To allocate a reference, we use the ref operator, providing an initial value for the new cell. For example, ref 5 creates a new cell containing the value 5, and evaluates to a reference to that cell.
  • To read the current value of this cell, we use the dereferencing operator !; for example, !(ref 5) evaluates to 5.
  • To change the value stored in a cell, we use the assignment operator. If r is a reference, r := 7 will store the value 7 in the cell referenced by r. However, r := 7 evaluates to the trivial value unit; it exists only to have the side effect of modifying the contents of a cell.


We start with the simply typed lambda calculus over the natural numbers. To the base natural number type and arrow types we need to add two more types to deal with references. First, we need the unit type, which we will use as the result type of an assignment operation. We then add reference types. If T is a type, then Ref T is the type of references which point to a cell holding values of type T.
      T ::= Nat
          | Unit
          | T  T
          | Ref T

Inductive ty : Type :=
  | TNat : ty
  | TUnit : ty
  | TArrow : ty ty ty
  | TRef : ty ty.


Besides variables, abstractions, applications, natural-number-related terms, and unit, we need four more sorts of terms in order to handle mutable references:
      t ::= ...              Terms
          | ref t              allocation
          | !t                 dereference
          | t := t             assignment
          | l                  location

Inductive tm : Type :=
  (* STLC with numbers: *)
  | 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
  (* New terms: *)
  | tunit : tm
  | tref : tm tm
  | tderef : tm tm
  | tassign : tm tm tm
  | tloc : nat tm.

  • ref t (formally, tref t) allocates a new reference cell with the value t and evaluates to the location of the newly allocated cell;
  • !t (formally, tderef t) evaluates to the contents of the cell referenced by t;
  • t1 := t2 (formally, tassign t1 t2) assigns t2 to the cell referenced by t1; and
  • l (formally, tloc l) is a reference to the cell at location l. We'll discuss locations later.
In informal examples, we'll also freely use the extensions of the STLC developed in the MoreStlc chapter; however, to keep the proofs small, we won't bother formalizing them again here. It would be easy to do so, since there are no very interesting interactions between those features and references.

Tactic Notation "t_cases" tactic(first) ident(c) :=
  [ Case_aux c "tvar" | Case_aux c "tapp"
  | Case_aux c "tabs" | Case_aux c "tzero"
  | Case_aux c "tsucc" | Case_aux c "tpred"
  | Case_aux c "tmult" | Case_aux c "tif0"
  | Case_aux c "tunit" | Case_aux c "tref"
  | Case_aux c "tderef" | Case_aux c "tassign"
  | Case_aux c "tloc" ].

Module ExampleVariables.

Definition x := Id 0.
Definition y := Id 1.
Definition r := Id 2.
Definition s := Id 3.

End ExampleVariables.

Typing (Preview)

Informally, the typing rules for allocation, dereferencing, and assignment will look like this:
Γ  t1 : T1 (T_Ref)  

Γ  ref t1 : Ref T1
Γ  t1 : Ref T11 (T_Deref)  

Γ  !t1 : T11
Γ  t1 : Ref T11
Γ  t2 : T11 (T_Assign)  

Γ  t1 := t2 : Unit
The rule for locations will require a bit more machinery, and this will motivate some changes to the other rules; we'll come back to this later.

Values and Substitution

Besides abstractions and numbers, we have two new types of values: the unit value, and locations.

Inductive value : tm Prop :=
  | v_abs : x T t,
      value (tabs x T t)
  | v_nat : n,
      value (tnat n)
  | v_unit :
      value tunit
  | v_loc : l,
      value (tloc l).

Hint Constructors value.

Extending substitution to handle the new syntax of terms is straightforward.

Fixpoint subst (x:id) (s:tm) (t:tm) : tm :=
  match t with
  | tvar x'
      if eq_id_dec x x' then s else t
  | tapp t1 t2
      tapp (subst x s t1) (subst x s t2)
  | tabs x' T t1
      if eq_id_dec x x' then t else tabs x' T (subst x s t1)
  | 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)
  | tunit
  | tref t1
      tref (subst x s t1)
  | tderef t1
      tderef (subst x s t1)
  | tassign t1 t2
      tassign (subst x s t1) (subst x s t2)
  | tloc _

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


Side Effects and Sequencing

The fact that the result of an assignment expression is the trivial value unit allows us to use a nice abbreviation for sequencing. For example, we can write
       r:=succ(!r); !r
as an abbreviation for
       (λx:Unit. !r) (r := succ(!r)).
This has the effect of evaluating two expressions in order and returning the value of the second. Restricting the type of the first expression to Unit helps the typechecker to catch some silly errors by permitting us to throw away the first value only if it is really guaranteed to be trivial.
Notice that, if the second expression is also an assignment, then the type of the whole sequence will be Unit, so we can validly place it to the left of another ; to build longer sequences of assignments:
       r:=succ(!r); r:=succ(!r); r:=succ(!r); r:=succ(!r); !r
Formally, we introduce sequencing as a "derived form" tseq that expands into an abstraction and an application.

Definition tseq t1 t2 :=
  tapp (tabs (Id 0) TUnit t2) t1.

References and Aliasing

It is important to bear in mind the difference between the reference that is bound to r and the cell in the store that is pointed to by this reference.
If we make a copy of r, for example by binding its value to another variable s, what gets copied is only the reference, not the contents of the cell itself.
For example, after evaluating
      let r = ref 5 in
      let s = r in
      s := 82;
the cell referenced by r will contain the value 82, while the result of the whole expression will be 83. The references r and s are said to be aliases for the same cell.
The possibility of aliasing can make programs with references quite tricky to reason about. For example, the expression
      r := 5; r := !s
assigns 5 to r and then immediately overwrites it with s's current value; this has exactly the same effect as the single assignment
      r := !s
unless we happen to do it in a context where r and s are aliases for the same cell!

Shared State

Of course, aliasing is also a large part of what makes references useful. In particular, it allows us to set up "implicit communication channels" — shared state — between different parts of a program. For example, suppose we define a reference cell and two functions that manipulate its contents:
    let c = ref 0 in
    let incc = λ_:Unit. (c := succ (!c); !c) in
    let decc = λ_:Unit. (c := pred (!c); !c) in
Note that, since their argument types are Unit, the abstractions in the definitions of incc and decc are not providing any useful information to the bodies of the functions (using the wildcard _ as the name of the bound variable is a reminder of this). Instead, their purpose is to "slow down" the execution of the function bodies: since function abstractions are values, the two lets are executed simply by binding these functions to the names incc and decc, rather than by actually incrementing or decrementing c. Later, each call to one of these functions results in its body being executed once and performing the appropriate mutation on c. Such functions are often called thunks.
In the context of these declarations, calling incc results in changes to c that can be observed by calling decc. For example, if we replace the ... with (incc unit; incc unit; decc unit), the result of the whole program will be 1.


We can go a step further and write a function that creates c, incc, and decc, packages incc and decc together into a record, and returns this record:
    newcounter = 
           let c = ref 0 in
           let incc = λ_:Unit. (c := succ (!c); !c) in
           let decc = λ_:Unit. (c := pred (!c); !c) in
           {i=incc, d=decc}
Now, each time we call newcounter, we get a new record of functions that share access to the same storage cell c. The caller of newcounter can't get at this storage cell directly, but can affect it indirectly by calling the two functions. In other words, we've created a simple form of object.
    let c1 = newcounter unit in
    let c2 = newcounter unit in
    // Note that we've allocated two separate storage cells now!
    let r1 = c1.i unit in
    let r2 = c2.i unit in
    r2  // yields 1, not 2!

Exercise: 1 star (store_draw)

Draw (on paper) the contents of the store at the point in execution where the first two lets have finished and the third one is about to begin.


References to Compound Types

A reference cell need not contain just a number: the primitives we've defined above allow us to create references to values of any type, including functions. For example, we can use references to functions to give a (not very efficient) implementation of arrays of numbers, as follows. Write NatArray for the type Ref (NatNat).
Recall the equal function from the MoreStlc chapter:
    equal = 
           λm:Nat. λn:Nat.
             if m=0 then iszero n 
             else if n=0 then false
             else eq (pred m) (pred n))
Now, to build a new array, we allocate a reference cell and fill it with a function that, when given an index, always returns 0.
    newarray = λ_:Unit. ref (λn:Nat.0)
To look up an element of an array, we simply apply the function to the desired index.
    lookup = λa:NatArray. λn:Nat. (!a) n
The interesting part of the encoding is the update function. It takes an array, an index, and a new value to be stored at that index, and does its job by creating (and storing in the reference) a new function that, when it is asked for the value at this very index, returns the new value that was given to update, and on all other indices passes the lookup to the function that was previously stored in the reference.
    update = λa:NatArray. λm:Nat. λv:Nat. 
                 let oldf = !a in
                 a := (λn:Nat. if equal m n then v else oldf n);
References to values containing other references can also be very useful, allowing us to define data structures such as mutable lists and trees.

Exercise: 2 stars (compact_update)

If we defined update more compactly like this
    update = λa:NatArray. λm:Nat. λv:Nat. 
                a := (λn:Nat. if equal m n then v else (!a) n)
would it behave the same?


Null References

There is one more difference between our references and C-style mutable variables: in C-like languages, variables holding pointers into the heap may sometimes have the value NULL. Dereferencing such a "null pointer" is an error, and results in an exception (Java) or in termination of the program (C).
Null pointers cause significant trouble in C-like languages: the fact that any pointer might be null means that any dereference operation in the program can potentially fail. However, even in ML-like languages, there are occasionally situations where we may or may not have a valid pointer in our hands. Fortunately, there is no need to extend the basic mechanisms of references to achieve this: the sum types introduced in the MoreStlc chapter already give us what we need.
First, we can use sums to build an analog of the option types introduced in the Lists chapter. Define Option T to be an abbreviation for Unit + T.
Then a "nullable reference to a T" is simply an element of the type Option (Ref T).

Garbage Collection

A last issue that we should mention before we move on with formalizing references is storage de-allocation. We have not provided any primitives for freeing reference cells when they are no longer needed. Instead, like many modern languages (including ML and Java) we rely on the run-time system to perform garbage collection, collecting and reusing cells that can no longer be reached by the program.
This is not just a question of taste in language design: it is extremely difficult to achieve type safety in the presence of an explicit deallocation operation. The reason for this is the familiar dangling reference problem: we allocate a cell holding a number, save a reference to it in some data structure, use it for a while, then deallocate it and allocate a new cell holding a boolean, possibly reusing the same storage. Now we can have two names for the same storage cell — one with type Ref Nat and the other with type Ref Bool.

Exercise: 1 star (type_safety_violation)

Show how this can lead to a violation of type safety.


Operational Semantics


The most subtle aspect of the treatment of references appears when we consider how to formalize their operational behavior. One way to see why is to ask, "What should be the values of type Ref T?" The crucial observation that we need to take into account is that evaluating a ref operator should do something — namely, allocate some storage — and the result of the operation should be a reference to this storage.
What, then, is a reference?
The run-time store in most programming language implementations is essentially just a big array of bytes. The run-time system keeps track of which parts of this array are currently in use; when we need to allocate a new reference cell, we allocate a large enough segment from the free region of the store (4 bytes for integer cells, 8 bytes for cells storing Floats, etc.), mark it as being used, and return the index (typically, a 32- or 64-bit integer) of the start of the newly allocated region. These indices are references.
For present purposes, there is no need to be quite so concrete. We can think of the store as an array of values, rather than an array of bytes, abstracting away from the different sizes of the run-time representations of different values. A reference, then, is simply an index into the store. (If we like, we can even abstract away from the fact that these indices are numbers, but for purposes of formalization in Coq it is a bit more convenient to use numbers.) We'll use the word location instead of reference or pointer from now on to emphasize this abstract quality.
Treating locations abstractly in this way will prevent us from modeling the pointer arithmetic found in low-level languages such as C. This limitation is intentional. While pointer arithmetic is occasionally very useful, especially for implementing low-level services such as garbage collectors, it cannot be tracked by most type systems: knowing that location n in the store contains a float doesn't tell us anything useful about the type of location n+4. In C, pointer arithmetic is a notorious source of type safety violations.


Recall that, in the small-step operational semantics for IMP, the step relation needed to carry along an auxiliary state in addition to the program being executed. In the same way, once we have added reference cells to the STLC, our step relation must carry along a store to keep track of the contents of reference cells.
We could re-use the same functional representation we used for states in IMP, but for carrying out the proofs in this chapter it is actually more convenient to represent a store simply as a list of values. (The reason we couldn't use this representation before is that, in IMP, a program could modify any location at any time, so states had to be ready to map any variable to a value. However, in the STLC with references, the only way to create a reference cell is with tref t1, which puts the value of t1 in a new reference cell and evaluates to the location of the newly created reference cell. When evaluating such an expression, we can just add a new reference cell to the end of the list representing the store.)

Definition store := list tm.

We use store_lookup n st to retrieve the value of the reference cell at location n in the store st. Note that we must give a default value to nth in case we try looking up an index which is too large. (In fact, we will never actually do this, but proving it will of course require some work!)

Definition store_lookup (n:nat) (st:store) :=
  nth n st tunit.

To add a new reference cell to the store, we use snoc.

Fixpoint snoc {A:Type} (l:list A) (x:A) : list A :=
  match l with
  | nilx :: nil
  | h :: th :: snoc t x

We will need some boring lemmas about snoc. The proofs are routine inductions.

Lemma length_snoc : A (l:list A) x,
  length (snoc l x) = S (length l).
  induction l; intros; [ auto | simpl; rewrite IHl; auto ]. Qed.

(* The "solve by inversion" tactic is explained in Stlc.v. *)
Lemma nth_lt_snoc : A (l:list A) x d n,
  n < length l
  nth n l d = nth n (snoc l x) d.
  induction l as [|a l']; intros; try solve by inversion.
  Case "l = a :: l'".
    destruct n; auto.
    simpl. apply IHl'.
    simpl in H. apply lt_S_n in H. assumption.

Lemma nth_eq_snoc : A (l:list A) x d,
  nth (length l) (snoc l x) d = x.
  induction l; intros; [ auto | simpl; rewrite IHl; auto ].

To update the store, we use the replace function, which replaces the contents of a cell at a particular index.

Fixpoint replace {A:Type} (n:nat) (x:A) (l:list A) : list A :=
  match l with
  | nilnil
  | h :: t
    match n with
    | Ox :: t
    | S n'h :: replace n' x t

Of course, we also need some boring lemmas about replace, which are also fairly straightforward to prove.

Lemma replace_nil : A n (x:A),
  replace n x nil = nil.
  destruct n; auto.

Lemma length_replace : A n x (l:list A),
  length (replace n x l) = length l.
Proof with auto.
  intros A n x l. generalize dependent n.
  induction l; intros n.
    destruct n...
    destruct n...
      simpl. rewrite IHl...

Lemma lookup_replace_eq : l t st,
  l < length st
  store_lookup l (replace l t st) = t.
Proof with auto.
  intros l t st.
  unfold store_lookup.
  generalize dependent l.
  induction st as [|t' st']; intros l Hlen.
  Case "st = []".
   inversion Hlen.
  Case "st = t' :: st'".
    destruct l; simpl...
    apply IHst'. simpl in Hlen. omega.

Lemma lookup_replace_neq : l1 l2 t st,
  store_lookup l1 (replace l2 t st) = store_lookup l1 st.
Proof with auto.
  unfold store_lookup.
  induction l1 as [|l1']; intros l2 t st Hneq.
  Case "l1 = 0".
    destruct st.
    SCase "st = []". rewrite replace_nil...
    SCase "st = _ :: _". destruct l2... contradict Hneq...
  Case "l1 = S l1'".
    destruct st as [|t2 st2].
    SCase "st = []". destruct l2...
    SCase "st = t2 :: st2".
      destruct l2...
      simpl; apply IHl1'...


Next, we need to extend our operational semantics to take stores into account. Since the result of evaluating an expression will in general depend on the contents of the store in which it is evaluated, the evaluation rules should take not just a term but also a store as argument. Furthermore, since the evaluation of a term may cause side effects on the store that may affect the evaluation of other terms in the future, the evaluation rules need to return a new store. Thus, the shape of the single-step evaluation relation changes from t t' to t / st t' / st', where st and st' are the starting and ending states of the store.
To carry through this change, we first need to augment all of our existing evaluation rules with stores:
value v2 (ST_AppAbs)  

(λx:T.t12) v2 / st  [x:=v2]t12 / st
t1 / st  t1' / st' (ST_App1)  

t1 t2 / st  t1' t2 / st'
value v1     t2 / st  t2' / st' (ST_App2)  

v1 t2 / st  v1 t2' / st'
Note that the first rule here returns the store unchanged: function application, in itself, has no side effects. The other two rules simply propagate side effects from premise to conclusion.
Now, the result of evaluating a ref expression will be a fresh location; this is why we included locations in the syntax of terms and in the set of values.
It is crucial to note that making this extension to the syntax of terms does not mean that we intend programmers to write terms involving explicit, concrete locations: such terms will arise only as intermediate results of evaluation. This may initially seem odd, but really it follows naturally from our design decision to represent the result of every evaluation step by a modified term. If we had chosen a more "machine-like" model for evaluation, e.g. with an explicit stack to contain values of bound identifiers, then the idea of adding locations to the set of allowed values would probably seem more obvious.
In terms of this expanded syntax, we can state evaluation rules for the new constructs that manipulate locations and the store. First, to evaluate a dereferencing expression !t1, we must first reduce t1 until it becomes a value:
t1 / st  t1' / st' (ST_Deref)  

!t1 / st  !t1' / st'
Once t1 has finished reducing, we should have an expression of the form !l, where l is some location. (A term that attempts to dereference any other sort of value, such as a function or unit, is erroneous, as is a term that tries to derefence a location that is larger than the size |st| of the currently allocated store; the evaluation rules simply get stuck in this case. The type safety properties that we'll establish below assure us that well-typed terms will never misbehave in this way.)
l < |st| (ST_DerefLoc)  

!(loc l) / st  lookup l st / st
Next, to evaluate an assignment expression t1:=t2, we must first evaluate t1 until it becomes a value (a location), and then evaluate t2 until it becomes a value (of any sort):
t1 / st  t1' / st' (ST_Assign1)  

t1 := t2 / st  t1' := t2 / st'
t2 / st  t2' / st' (ST_Assign2)  

v1 := t2 / st  v1 := t2' / st'
Once we have finished with t1 and t2, we have an expression of the form l:=v2, which we execute by updating the store to make location l contain v2:
l < |st| (ST_Assign)  

loc l := v2 / st  unit / [l:=v2]st
The notation [l:=v2]st means "the store that maps l to v2 and maps all other locations to the same thing as st." Note that the term resulting from this evaluation step is just unit; the interesting result is the updated store.)
Finally, to evaluate an expression of the form ref t1, we first evaluate t1 until it becomes a value:
t1 / st  t1' / st' (ST_Ref)  

ref t1 / st  ref t1' / st'
Then, to evaluate the ref itself, we choose a fresh location at the end of the current store — i.e., location |st| — and yield a new store that extends st with the new value v1.

ref v1 / st  loc |st| / st,v1
The value resulting from this step is the newly allocated location itself. (Formally, st,v1 means snoc st v1.)
Note that these evaluation rules do not perform any kind of garbage collection: we simply allow the store to keep growing without bound as evaluation proceeds. This does not affect the correctness of the results of evaluation (after all, the definition of "garbage" is precisely parts of the store that are no longer reachable and so cannot play any further role in evaluation), but it means that a naive implementation of our evaluator might sometimes run out of memory where a more sophisticated evaluator would be able to continue by reusing locations whose contents have become garbage.

Reserved Notation "t1 '/' st1 '' t2 '/' st2"
  (at level 40, st1 at level 39, t2 at level 39).

Inductive step : tm × store tm × store Prop :=
  | ST_AppAbs : x T t12 v2 st,
         value v2
         tapp (tabs x T t12) v2 / st [x:=v2]t12 / st
  | ST_App1 : t1 t1' t2 st st',
         t1 / st t1' / st'
         tapp t1 t2 / st tapp t1' t2 / st'
  | ST_App2 : v1 t2 t2' st st',
         value v1
         t2 / st t2' / st'
         tapp v1 t2 / st tapp v1 t2'/ st'
  | ST_SuccNat : n st,
         tsucc (tnat n) / st tnat (S n) / st
  | ST_Succ : t1 t1' st st',
         t1 / st t1' / st'
         tsucc t1 / st tsucc t1' / st'
  | ST_PredNat : n st,
         tpred (tnat n) / st tnat (pred n) / st
  | ST_Pred : t1 t1' st st',
         t1 / st t1' / st'
         tpred t1 / st tpred t1' / st'
  | ST_MultNats : n1 n2 st,
         tmult (tnat n1) (tnat n2) / st tnat (mult n1 n2) / st
  | ST_Mult1 : t1 t2 t1' st st',
         t1 / st t1' / st'
         tmult t1 t2 / st tmult t1' t2 / st'
  | ST_Mult2 : v1 t2 t2' st st',
         value v1
         t2 / st t2' / st'
         tmult v1 t2 / st tmult v1 t2' / st'
  | ST_If0 : t1 t1' t2 t3 st st',
         t1 / st t1' / st'
         tif0 t1 t2 t3 / st tif0 t1' t2 t3 / st'
  | ST_If0_Zero : t2 t3 st,
         tif0 (tnat 0) t2 t3 / st t2 / st
  | ST_If0_Nonzero : n t2 t3 st,
         tif0 (tnat (S n)) t2 t3 / st t3 / st
  | ST_RefValue : v1 st,
         value v1
         tref v1 / st tloc (length st) / snoc st v1
  | ST_Ref : t1 t1' st st',
         t1 / st t1' / st'
         tref t1 / st tref t1' / st'
  | ST_DerefLoc : st l,
         l < length st
         tderef (tloc l) / st store_lookup l st / st
  | ST_Deref : t1 t1' st st',
         t1 / st t1' / st'
         tderef t1 / st tderef t1' / st'
  | ST_Assign : v2 l st,
         value v2
         l < length st
         tassign (tloc l) v2 / st tunit / replace l v2 st
  | ST_Assign1 : t1 t1' t2 st st',
         t1 / st t1' / st'
         tassign t1 t2 / st tassign t1' t2 / st'
  | ST_Assign2 : v1 t2 t2' st st',
         value v1
         t2 / st t2' / st'
         tassign v1 t2 / st tassign v1 t2' / st'

where "t1 '/' st1 '' t2 '/' st2" := (step (t1,st1) (t2,st2)).

Tactic Notation "step_cases" tactic(first) ident(c) :=
  [ Case_aux c "ST_AppAbs" | Case_aux c "ST_App1"
  | Case_aux c "ST_App2" | Case_aux c "ST_SuccNat"
  | Case_aux c "ST_Succ" | Case_aux c "ST_PredNat"
  | Case_aux c "ST_Pred" | Case_aux c "ST_MultNats"
  | Case_aux c "ST_Mult1" | Case_aux c "ST_Mult2"
  | Case_aux c "ST_If0" | Case_aux c "ST_If0_Zero"
  | Case_aux c "ST_If0_Nonzero" | Case_aux c "ST_RefValue"
  | Case_aux c "ST_Ref" | Case_aux c "ST_DerefLoc"
  | Case_aux c "ST_Deref" | Case_aux c "ST_Assign"
  | Case_aux c "ST_Assign1" | Case_aux c "ST_Assign2" ].

Hint Constructors step.

Definition multistep := (multi step).
Notation "t1 '/' st '⇒*' t2 '/' st'" := (multistep (t1,st) (t2,st'))
  (at level 40, st at level 39, t2 at level 39).


Our contexts for free variables will be exactly the same as for the STLC, partial maps from identifiers to types.

Definition context := partial_map ty.

Store typings

Having extended our syntax and evaluation rules to accommodate references, our last job is to write down typing rules for the new constructs — and, of course, to check that they are sound. Naturally, the key question is, "What is the type of a location?"
First of all, notice that we do not need to answer this question for purposes of typechecking the terms that programmers actually write. Concrete location constants arise only in terms that are the intermediate results of evaluation; they are not in the language that programmers write. So we only need to determine the type of a location when we're in the middle of an evaluation sequence, e.g. trying to apply the progress or preservation lemmas. Thus, even though we normally think of typing as a static program property, it makes sense for the typing of locations to depend on the dynamic progress of the program too.
As a first try, note that when we evaluate a term containing concrete locations, the type of the result depends on the contents of the store that we start with. For example, if we evaluate the term !(loc 1) in the store [unit, unit], the result is unit; if we evaluate the same term in the store [unit, λx:Unit.x], the result is λx:Unit.x. With respect to the former store, the location 1 has type Unit, and with respect to the latter it has type UnitUnit. This observation leads us immediately to a first attempt at a typing rule for locations:
Γ  lookup  l st : T1  

Γ  loc l : Ref T1
That is, to find the type of a location l, we look up the current contents of l in the store and calculate the type T1 of the contents. The type of the location is then Ref T1.
Having begun in this way, we need to go a little further to reach a consistent state. In effect, by making the type of a term depend on the store, we have changed the typing relation from a three-place relation (between contexts, terms, and types) to a four-place relation (between contexts, stores, terms, and types). Since the store is, intuitively, part of the context in which we calculate the type of a term, let's write this four-place relation with the store to the left of the turnstile: Γ; st t : T. Our rule for typing references now has the form
Gamma; st  lookup l st : T1  

Gamma; st  loc l : Ref T1
and all the rest of the typing rules in the system are extended similarly with stores. The other rules do not need to do anything interesting with their stores — just pass them from premise to conclusion.
However, there are two problems with this rule. First, typechecking is rather inefficient, since calculating the type of a location l involves calculating the type of the current contents v of l. If l appears many times in a term t, we will re-calculate the type of v many times in the course of constructing a typing derivation for t. Worse, if v itself contains locations, then we will have to recalculate their types each time they appear.
Second, the proposed typing rule for locations may not allow us to derive anything at all, if the store contains a cycle. For example, there is no finite typing derivation for the location 0 with respect to this store:
   [λx:Nat. (!(loc 1)) x, λx:Nat. (!(loc 0)) x]

Exercise: 2 stars (cyclic_store)

Can you find a term whose evaluation will create this particular cyclic store?
Both of these problems arise from the fact that our proposed typing rule for locations requires us to recalculate the type of a location every time we mention it in a term. But this, intuitively, should not be necessary. After all, when a location is first created, we know the type of the initial value that we are storing into it. Suppose we are willing to enforce the invariant that the type of the value contained in a given location never changes; that is, although we may later store other values into this location, those other values will always have the same type as the initial one. In other words, we always have in mind a single, definite type for every location in the store, which is fixed when the location is allocated. Then these intended types can be collected together as a store typing —-a finite function mapping locations to types.
As usual, this conservative typing restriction on allowed updates means that we will rule out as ill-typed some programs that could evaluate perfectly well without getting stuck.
Just like we did for stores, we will represent a store type simply as a list of types: the type at index i records the type of the value stored in cell i.

Definition store_ty := list ty.

The store_Tlookup function retrieves the type at a particular index.

Definition store_Tlookup (n:nat) (ST:store_ty) :=
  nth n ST TUnit.

Suppose we are given a store typing ST describing the store st in which some term t will be evaluated. Then we can use ST to calculate the type of the result of t without ever looking directly at st. For example, if ST is [Unit, UnitUnit], then we may immediately infer that !(loc 1) has type UnitUnit. More generally, the typing rule for locations can be reformulated in terms of store typings like this:
l < |ST|  

Gamma; ST  loc l : Ref (lookup l ST)
That is, as long as l is a valid location (it is less than the length of ST), we can compute the type of l just by looking it up in ST. Typing is again a four-place relation, but it is parameterized on a store typing rather than a concrete store. The rest of the typing rules are analogously augmented with store typings.

The Typing Relation

We can now give the typing relation for the STLC with references. Here, again, are the rules we're adding to the base STLC (with numbers and Unit):
l < |ST| (T_Loc)  

Gamma; ST  loc l : Ref (lookup l ST)
Gamma; ST  t1 : T1 (T_Ref)  

Gamma; ST  ref t1 : Ref T1
Gamma; ST  t1 : Ref T11 (T_Deref)  

Gamma; ST  !t1 : T11
Gamma; ST  t1 : Ref T11
Gamma; ST  t2 : T11 (T_Assign)  

Gamma; ST  t1 := t2 : Unit

Reserved Notation "Gamma ';' ST '' t '∈' T" (at level 40).

Inductive has_type : context store_ty tm ty Prop :=
  | T_Var : Γ ST x T,
      Γ x = Some T
      Γ; ST (tvar x) ∈ T
  | T_Abs : Γ ST x T11 T12 t12,
      (extend Γ x T11); ST t12T12
      Γ; ST (tabs x T11 t12) ∈ (TArrow T11 T12)
  | T_App : T1 T2 Γ ST t1 t2,
      Γ; ST t1 ∈ (TArrow T1 T2)
      Γ; ST t2T1
      Γ; ST (tapp t1 t2) ∈ T2
  | T_Nat : Γ ST n,
      Γ; ST (tnat n) ∈ TNat
  | T_Succ : Γ ST t1,
      Γ; ST t1TNat
      Γ; ST (tsucc t1) ∈ TNat
  | T_Pred : Γ ST t1,
      Γ; ST t1TNat
      Γ; ST (tpred t1) ∈ TNat
  | T_Mult : Γ ST t1 t2,
      Γ; ST t1TNat
      Γ; ST t2TNat
      Γ; ST (tmult t1 t2) ∈ TNat
  | T_If0 : Γ ST t1 t2 t3 T,
      Γ; ST t1TNat
      Γ; ST t2T
      Γ; ST t3T
      Γ; ST (tif0 t1 t2 t3) ∈ T
  | T_Unit : Γ ST,
      Γ; ST tunitTUnit
  | T_Loc : Γ ST l,
      l < length ST
      Γ; ST (tloc l) ∈ (TRef (store_Tlookup l ST))
  | T_Ref : Γ ST t1 T1,
      Γ; ST t1T1
      Γ; ST (tref t1) ∈ (TRef T1)
  | T_Deref : Γ ST t1 T11,
      Γ; ST t1 ∈ (TRef T11)
      Γ; ST (tderef t1) ∈ T11
  | T_Assign : Γ ST t1 t2 T11,
      Γ; ST t1 ∈ (TRef T11)
      Γ; ST t2T11
      Γ; ST (tassign t1 t2) ∈ TUnit

where "Gamma ';' ST '' t '∈' T" := (has_type Γ ST t T).

Hint Constructors has_type.

Tactic Notation "has_type_cases" tactic(first) ident(c) :=
  [ Case_aux c "T_Var" | Case_aux c "T_Abs" | Case_aux c "T_App"
  | Case_aux c "T_Nat" | Case_aux c "T_Succ" | Case_aux c "T_Pred"
  | Case_aux c "T_Mult" | Case_aux c "T_If0"
  | Case_aux c "T_Unit" | Case_aux c "T_Loc"
  | Case_aux c "T_Ref" | Case_aux c "T_Deref"
  | Case_aux c "T_Assign" ].

Of course, these typing rules will accurately predict the results of evaluation only if the concrete store used during evaluation actually conforms to the store typing that we assume for purposes of typechecking. This proviso exactly parallels the situation with free variables in the STLC: the substitution lemma promises us that, if Γ t : T, then we can replace the free variables in t with values of the types listed in Γ to obtain a closed term of type T, which, by the type preservation theorem will evaluate to a final result of type T if it yields any result at all. (We will see later how to formalize an analogous intuition for stores and store typings.)
However, for purposes of typechecking the terms that programmers actually write, we do not need to do anything tricky to guess what store typing we should use. Recall that concrete location constants arise only in terms that are the intermediate results of evaluation; they are not in the language that programmers write. Thus, we can simply typecheck the programmer's terms with respect to the empty store typing. As evaluation proceeds and new locations are created, we will always be able to see how to extend the store typing by looking at the type of the initial values being placed in newly allocated cells; this intuition is formalized in the statement of the type preservation theorem below.


Our final task is to check that standard type safety properties continue to hold for the STLC with references. The progress theorem ("well-typed terms are not stuck") can be stated and proved almost as for the STLC; we just need to add a few straightforward cases to the proof, dealing with the new constructs. The preservation theorem is a bit more interesting, so let's look at it first.

Well-Typed Stores

Since we have extended both the evaluation relation (with initial and final stores) and the typing relation (with a store typing), we need to change the statement of preservation to include these parameters. Clearly, though, we cannot just add stores and store typings without saying anything about how they are related:

Theorem preservation_wrong1 : ST T t st t' st',
  empty; ST tT
  t / st t' / st'
  empty; ST t'T.

If we typecheck with respect to some set of assumptions about the types of the values in the store and then evaluate with respect to a store that violates these assumptions, the result will be disaster. We say that a store st is well typed with respect a store typing ST if the term at each location l in st has the type at location l in ST. Since only closed terms ever get stored in locations (why?), it suffices to type them in the empty context. The following definition of store_well_typed formalizes this.

Definition store_well_typed (ST:store_ty) (st:store) :=
  length ST = length st
  (l, l < length st
     empty; ST (store_lookup l st) ∈ (store_Tlookup l ST)).

Informally, we will write ST st for store_well_typed ST st.
Intuitively, a store st is consistent with a store typing ST if every value in the store has the type predicted by the store typing. (The only subtle point is the fact that, when typing the values in the store, we supply the very same store typing to the typing relation! This allows us to type circular stores.)

Exercise: 2 stars (store_not_unique)

Can you find a store st, and two different store typings ST1 and ST2 such that both ST1 st and ST2 st?

We can now state something closer to the desired preservation property:

Theorem preservation_wrong2 : ST T t st t' st',
  empty; ST tT
  t / st t' / st'
  store_well_typed ST st
  empty; ST t'T.

This statement is fine for all of the evaluation rules except the allocation rule ST_RefValue. The problem is that this rule yields a store with a larger domain than the initial store, which falsifies the conclusion of the above statement: if st' includes a binding for a fresh location l, then l cannot be in the domain of ST, and it will not be the case that t' (which definitely mentions l) is typable under ST.

Extending Store Typings

Evidently, since the store can increase in size during evaluation, we need to allow the store typing to grow as well. This motivates the following definition. We say that the store type ST' extends ST if ST' is just ST with some new types added to the end.

Inductive extends : store_ty store_ty Prop :=
  | extends_nil : ST',
      extends ST' nil
  | extends_cons : x ST' ST,
      extends ST' ST
      extends (x::ST') (x::ST).

Hint Constructors extends.

We'll need a few technical lemmas about extended contexts.
First, looking up a type in an extended store typing yields the same result as in the original:

Lemma extends_lookup : l ST ST',
  l < length ST
  extends ST' ST
  store_Tlookup l ST' = store_Tlookup l ST.
Proof with auto.
  intros l ST ST' Hlen H.
  generalize dependent ST'. generalize dependent l.
  induction ST as [|a ST2]; intros l Hlen ST' HST'.
  Case "nil". inversion Hlen.
  Case "cons". unfold store_Tlookup in ×.
    destruct ST'.
    SCase "ST' = nil". inversion HST'.
    SCase "ST' = a' :: ST'2".
      inversion HST'; subst.
      destruct l as [|l'].
      SSCase "l = 0"...
      SSCase "l = S l'". simpl. apply IHST2...
        simpl in Hlen; omega.

Next, if ST' extends ST, the length of ST' is at least that of ST.

Lemma length_extends : l ST ST',
  l < length ST
  extends ST' ST
  l < length ST'.
Proof with eauto.
  intros. generalize dependent l. induction H0; intros l Hlen.
    inversion Hlen.
    simpl in ×.
    destruct l; try omega.
      apply lt_n_S. apply IHextends. omega.

Finally, snoc ST T extends ST, and extends is reflexive.

Lemma extends_snoc : ST T,
  extends (snoc ST T) ST.
Proof with auto.
  induction ST; intros T...

Lemma extends_refl : ST,
  extends ST ST.
  induction ST; auto.

Preservation, Finally

We can now give the final, correct statement of the type preservation property:

Definition preservation_theorem := ST t t' T st st',
  empty; ST tT
  store_well_typed ST st
  t / st t' / st'
    (extends ST' ST
     empty; ST' t'T
     store_well_typed ST' st').

Note that the preservation theorem merely asserts that there is some store typing ST' extending ST (i.e., agreeing with ST on the values of all the old locations) such that the new term t' is well typed with respect to ST'; it does not tell us exactly what ST' is. It is intuitively clear, of course, that ST' is either ST or else it is exactly snoc ST T1, where T1 is the type of the value v1 in the extended store snoc st v1, but stating this explicitly would complicate the statement of the theorem without actually making it any more useful: the weaker version above is already in the right form (because its conclusion implies its hypothesis) to "turn the crank" repeatedly and conclude that every sequence of evaluation steps preserves well-typedness. Combining this with the progress property, we obtain the usual guarantee that "well-typed programs never go wrong."
In order to prove this, we'll need a few lemmas, as usual.

Substitution lemma

First, we need an easy extension of the standard substitution lemma, along with the same machinery about context invariance that we used in the proof of the substitution lemma for the STLC.

Inductive appears_free_in : id tm Prop :=
  | afi_var : x,
      appears_free_in x (tvar x)
  | afi_app1 : x t1 t2,
      appears_free_in x t1 appears_free_in x (tapp t1 t2)
  | afi_app2 : x t1 t2,
      appears_free_in x t2 appears_free_in x (tapp t1 t2)
  | afi_abs : x y T11 t12,
      appears_free_in x t12
      appears_free_in x (tabs y T11 t12)
  | afi_succ : x t1,
      appears_free_in x t1
      appears_free_in x (tsucc t1)
  | afi_pred : x t1,
      appears_free_in x t1
      appears_free_in x (tpred t1)
  | 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_if0_1 : x t1 t2 t3,
      appears_free_in x t1
      appears_free_in x (tif0 t1 t2 t3)
  | afi_if0_2 : x t1 t2 t3,
      appears_free_in x t2
      appears_free_in x (tif0 t1 t2 t3)
  | afi_if0_3 : x t1 t2 t3,
      appears_free_in x t3
      appears_free_in x (tif0 t1 t2 t3)
  | afi_ref : x t1,
      appears_free_in x t1 appears_free_in x (tref t1)
  | afi_deref : x t1,
      appears_free_in x t1 appears_free_in x (tderef t1)
  | afi_assign1 : x t1 t2,
      appears_free_in x t1 appears_free_in x (tassign t1 t2)
  | afi_assign2 : x t1 t2,
      appears_free_in x t2 appears_free_in x (tassign t1 t2).

Tactic Notation "afi_cases" tactic(first) ident(c) :=
  [ Case_aux c "afi_var"
  | Case_aux c "afi_app1" | Case_aux c "afi_app2" | Case_aux c "afi_abs"
  | Case_aux c "afi_succ" | Case_aux c "afi_pred"
  | Case_aux c "afi_mult1" | Case_aux c "afi_mult2"
  | Case_aux c "afi_if0_1" | Case_aux c "afi_if0_2" | Case_aux c "afi_if0_3"
  | Case_aux c "afi_ref" | Case_aux c "afi_deref"
  | Case_aux c "afi_assign1" | Case_aux c "afi_assign2" ].

Hint Constructors appears_free_in.

Lemma free_in_context : x t T Γ ST,
   appears_free_in x t
   Γ; ST tT
   T', Γ x = Some T'.
Proof with eauto.
  intros. generalize dependent Γ. generalize dependent T.
  afi_cases (induction H) Case;
        intros; (try solve [ inversion H0; subst; eauto ]).
  Case "afi_abs".
    inversion H1; subst.
    apply IHappears_free_in in H8.
    rewrite extend_neq in H8; assumption.

Lemma context_invariance : Γ Γ' ST t T,
  Γ; ST tT
  (x, appears_free_in x t Γ x = Γ' x)
  Γ'; ST tT.
Proof with eauto.
  generalize dependent Γ'.
  has_type_cases (induction H) Case; intros...
  Case "T_Var".
    apply T_Var. symmetry. rewrite H...
  Case "T_Abs".
    apply T_Abs. apply IHhas_type; intros.
    unfold extend.
    destruct (eq_id_dec x x0)...
  Case "T_App".
    eapply T_App.
      apply IHhas_type1...
      apply IHhas_type2...
  Case "T_Mult".
    eapply T_Mult.
      apply IHhas_type1...
      apply IHhas_type2...
  Case "T_If0".
    eapply T_If0.
      apply IHhas_type1...
      apply IHhas_type2...
      apply IHhas_type3...
  Case "T_Assign".
    eapply T_Assign.
      apply IHhas_type1...
      apply IHhas_type2...

Lemma substitution_preserves_typing : Γ ST x s S t T,
  empty; ST sS
  (extend Γ x S); ST tT
  Γ; ST ([x:=s]t) ∈ T.
Proof with eauto.
  intros Γ ST x s S t T Hs Ht.
  generalize dependent Γ. generalize dependent T.
  t_cases (induction t) Case; intros T Γ H;
    inversion H; subst; simpl...
  Case "tvar".
    rename i into y.
    destruct (eq_id_dec x y).
    SCase "x = y".
      rewrite extend_eq in H3.
      inversion H3; subst.
      eapply context_invariance...
      intros x Hcontra.
      destruct (free_in_context _ _ _ _ _ Hcontra Hs) as [T' HT'].
      inversion HT'.
    SCase "x ≠ y".
      apply T_Var.
      rewrite extend_neq in H3...
  Case "tabs". subst.
    rename i into y.
    destruct (eq_id_dec x y).
    SCase "x = y".
      apply T_Abs. eapply context_invariance...
      intros. apply extend_shadow.
    SCase "x ≠ x0".
      apply T_Abs. apply IHt.
      eapply context_invariance...
      intros. unfold extend.
      destruct (eq_id_dec y x0)...
      rewrite neq_id...

Assignment Preserves Store Typing

Next, we must show that replacing the contents of a cell in the store with a new value of appropriate type does not change the overall type of the store. (This is needed for the ST_Assign rule.)

Lemma assign_pres_store_typing : ST st l t,
  l < length st
  store_well_typed ST st
  empty; ST t ∈ (store_Tlookup l ST)
  store_well_typed ST (replace l t st).
Proof with auto.
  intros ST st l t Hlen HST Ht.
  inversion HST; subst.
  split. rewrite length_replace...
  intros l' Hl'.
  destruct (beq_nat l' l) eqn: Heqll'.
  Case "l' = l".
    apply beq_nat_true in Heqll'; subst.
    rewrite lookup_replace_eq...
  Case "l' ≠ l".
    apply beq_nat_false in Heqll'.
    rewrite lookup_replace_neq...
    rewrite length_replace in Hl'.
    apply H0...

Weakening for Stores

Finally, we need a lemma on store typings, stating that, if a store typing is extended with a new location, the extended one still allows us to assign the same types to the same terms as the original.
(The lemma is called store_weakening because it resembles the "weakening" lemmas found in proof theory, which show that adding a new assumption to some logical theory does not decrease the set of provable theorems.)

Lemma store_weakening : Γ ST ST' t T,
  extends ST' ST
  Γ; ST tT
  Γ; ST' tT.
Proof with eauto.
  intros. has_type_cases (induction H0) Case; eauto.
  Case "T_Loc".
    erewrite extends_lookup...
    apply T_Loc.
    eapply length_extends...

We can use the store_weakening lemma to prove that if a store is well typed with respect to a store typing, then the store extended with a new term t will still be well typed with respect to the store typing extended with t's type.

Lemma store_well_typed_snoc : ST st t1 T1,
  store_well_typed ST st
  empty; ST t1T1
  store_well_typed (snoc ST T1) (snoc st t1).
Proof with auto.
  unfold store_well_typed in ×.
  inversion H as [Hlen Hmatch]; clear H.
  rewrite !length_snoc.
  Case "types match.".
    intros l Hl.
    unfold store_lookup, store_Tlookup.
    apply le_lt_eq_dec in Hl; inversion Hl as [Hlt | Heq].
    SCase "l < length st".
      apply lt_S_n in Hlt.
      rewrite !nth_lt_snoc...
      apply store_weakening with ST. apply extends_snoc.
      apply Hmatch...
      rewrite Hlen...
    SCase "l = length st".
      inversion Heq.
      rewrite nth_eq_snoc.
      rewrite Hlen. rewrite nth_eq_snoc...
      apply store_weakening with ST... apply extends_snoc.


Now that we've got everything set up right, the proof of preservation is actually quite straightforward.

Theorem preservation : ST t t' T st st',
  empty; ST tT
  store_well_typed ST st
  t / st t' / st'
    (extends ST' ST
     empty; ST' t'T
     store_well_typed ST' st').
Proof with eauto using store_weakening, extends_refl.
    remember (@empty ty) as Γ.
  intros ST t t' T st st' Ht.
  generalize dependent t'.
  has_type_cases (induction Ht) Case; intros t' HST Hstep;
    subst; try (solve by inversion); inversion Hstep; subst;
    try (eauto using store_weakening, extends_refl).
  Case "T_App".
    SCase "ST_AppAbs". ST.
      inversion Ht1; subst.
      split; try split... eapply substitution_preserves_typing...
    SCase "ST_App1".
      eapply IHHt1 in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
    SCase "ST_App2".
      eapply IHHt2 in H5...
      inversion H5 as [ST' [Hext [Hty Hsty]]].
  Case "T_Succ".
    SCase "ST_Succ".
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
  Case "T_Pred".
    SCase "ST_Pred".
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
  Case "T_Mult".
    SCase "ST_Mult1".
      eapply IHHt1 in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
    SCase "ST_Mult2".
      eapply IHHt2 in H5...
      inversion H5 as [ST' [Hext [Hty Hsty]]].
  Case "T_If0".
    SCase "ST_If0_1".
      eapply IHHt1 in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
      ST'... split...
  Case "T_Ref".
    SCase "ST_RefValue".
      (snoc ST T1).
      inversion HST; subst.
        apply extends_snoc.
        replace (TRef T1)
          with (TRef (store_Tlookup (length st) (snoc ST T1))).
        apply T_Loc.
        rewrite H. rewrite length_snoc. omega.
        unfold store_Tlookup. rewrite H. rewrite nth_eq_snoc...
        apply store_well_typed_snoc; assumption.
    SCase "ST_Ref".
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
  Case "T_Deref".
    SCase "ST_DerefLoc".
      ST. split; try split...
      inversion HST as [_ Hsty].
      replace T11 with (store_Tlookup l ST).
      apply Hsty...
      inversion Ht; subst...
    SCase "ST_Deref".
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
  Case "T_Assign".
    SCase "ST_Assign".
      ST. split; try split...
      eapply assign_pres_store_typing...
      inversion Ht1; subst...
    SCase "ST_Assign1".
      eapply IHHt1 in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
    SCase "ST_Assign2".
      eapply IHHt2 in H5...
      inversion H5 as [ST' [Hext [Hty Hsty]]].

Exercise: 3 stars (preservation_informal)

Write a careful informal proof of the preservation theorem, concentrating on the T_App, T_Deref, T_Assign, and T_Ref cases.


Fortunately, progress for this system is pretty easy to prove; the proof is very similar to the proof of progress for the STLC, with a few new cases for the new syntactic constructs.

Theorem progress : ST t T st,
  empty; ST tT
  store_well_typed ST st
  (value t t', st', t / st t' / st').
Proof with eauto.
    intros ST t T st Ht HST. remember (@empty ty) as Γ.
  has_type_cases (induction Ht) Case; subst; try solve by inversion...
  Case "T_App".
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve by inversion.
      destruct IHHt2 as [Ht2p | Ht2p]...
      SSCase "t2 steps".
        inversion Ht2p as [t2' [st' Hstep]].
        (tapp (tabs x T t) t2'). st'...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tapp t1' t2). st'...
  Case "T_Succ".
    right. destruct IHHt as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve [ inversion Ht ].
      SSCase "t1 is a tnat".
        (tnat (S n)). st...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tsucc t1'). st'...
  Case "T_Pred".
    right. destruct IHHt as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve [inversion Ht ].
      SSCase "t1 is a tnat".
        (tnat (pred n)). st...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tpred t1'). st'...
  Case "T_Mult".
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve [inversion Ht1].
      destruct IHHt2 as [Ht2p | Ht2p]...
      SSCase "t2 is a value".
        inversion Ht2p; subst; try solve [inversion Ht2].
        (tnat (mult n n0)). st...
      SSCase "t2 steps".
        inversion Ht2p as [t2' [st' Hstep]].
        (tmult (tnat n) t2'). st'...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tmult t1' t2). st'...
  Case "T_If0".
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve [inversion Ht1].
      destruct n.
      SSCase "n = 0". t2. st...
      SSCase "n = S n'". t3. st...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tif0 t1' t2 t3). st'...
  Case "T_Ref".
    right. destruct IHHt as [Ht1p | Ht1p]...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tref t1'). st'...
  Case "T_Deref".
    right. destruct IHHt as [Ht1p | Ht1p]...
    SCase "t1 is a value".
      inversion Ht1p; subst; try solve by inversion.
      eexists. eexists. apply ST_DerefLoc...
      inversion Ht; subst. inversion HST; subst.
      rewrite H...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tderef t1'). st'...
  Case "T_Assign".
    right. destruct IHHt1 as [Ht1p|Ht1p]...
    SCase "t1 is a value".
      destruct IHHt2 as [Ht2p|Ht2p]...
      SSCase "t2 is a value".
        inversion Ht1p; subst; try solve by inversion.
        eexists. eexists. apply ST_Assign...
        inversion HST; subst. inversion Ht1; subst.
        rewrite H in H5...
      SSCase "t2 steps".
        inversion Ht2p as [t2' [st' Hstep]].
        (tassign t1 t2'). st'...
    SCase "t1 steps".
      inversion Ht1p as [t1' [st' Hstep]].
      (tassign t1' t2). st'...

References and Nontermination

Section RefsAndNontermination.
Import ExampleVariables.

We know that the simply typed lambda calculus is normalizing, that is, every well-typed term can be reduced to a value in a finite number of steps. What about STLC + references? Surprisingly, adding references causes us to lose the normalization property: there exist well-typed terms in the STLC + references which can continue to reduce forever, without ever reaching a normal form!
How can we construct such a term? The main idea is to make a function which calls itself. We first make a function which calls another function stored in a reference cell; the trick is that we then smuggle in a reference to itself!
   (λr:Ref (Unit -> Unit). 
        r := (λx:Unit.(!r) unit); (!r) unit) 
   (ref (λx:Unit.unit))
First, ref x:Unit.unit) creates a reference to a cell of type Unit Unit. We then pass this reference as the argument to a function which binds it to the name r, and assigns to it the function (λx:Unit.(!r) unit) — that is, the function which ignores its argument and calls the function stored in r on the argument unit; but of course, that function is itself! To get the ball rolling we finally execute this function with (!r) unit.

Definition loop_fun :=
  tabs x TUnit (tapp (tderef (tvar r)) tunit).

Definition loop :=
  (tabs r (TRef (TArrow TUnit TUnit))
    (tseq (tassign (tvar r) loop_fun)
            (tapp (tderef (tvar r)) tunit)))
  (tref (tabs x TUnit tunit)).

This term is well typed:

Lemma loop_typeable : T, empty; nil loopT.
Proof with eauto.
  eexists. unfold loop. unfold loop_fun.
  eapply T_App...
  eapply T_Abs...
  eapply T_App...
    eapply T_Abs. eapply T_App. eapply T_Deref. eapply T_Var.
    unfold extend. simpl. reflexivity. auto.
  eapply T_Assign.
    eapply T_Var. unfold extend. simpl. reflexivity.
  eapply T_Abs.
    eapply T_App...
      eapply T_Deref. eapply T_Var. reflexivity.

To show formally that the term diverges, we first define the step_closure of the single-step reduction relation, written ⇒+. This is just like the reflexive step closure of single-step reduction (which we're been writing ⇒*), except that it is not reflexive: t ⇒+ t' means that t can reach t' by one or more steps of reduction.

Inductive step_closure {X:Type} (R: relation X) : X X Prop :=
  | sc_one : (x y : X),
                R x y step_closure R x y
  | sc_step : (x y z : X),
                R x y
                step_closure R y z
                step_closure R x z.

Definition multistep1 := (step_closure step).
Notation "t1 '/' st '⇒+' t2 '/' st'" := (multistep1 (t1,st) (t2,st'))
  (at level 40, st at level 39, t2 at level 39).

Now, we can show that the expression loop reduces to the expression !(loc 0) unit and the size-one store [r:=(loc 0)] loop_fun.
As a convenience, we introduce a slight variant of the normalize tactic, called reduce, which tries solving the goal with multi_refl at each step, instead of waiting until the goal can't be reduced any more. Of course, the whole point is that loop doesn't normalize, so the old normalize tactic would just go into an infinite loop reducing it forever!

Ltac print_goal := match goal with ?xidtac x end.
Ltac reduce :=
    repeat (print_goal; eapply multi_step ;
            [ (eauto 10; fail) | (instantiate; compute)];
            try solve [apply multi_refl]).

Lemma loop_steps_to_loop_fun :
  loop / nil ⇒*
  tapp (tderef (tloc 0)) tunit / cons ([r:=tloc 0]loop_fun) nil.
Proof with eauto.
  unfold loop.

Finally, the latter expression reduces in two steps to itself!

Lemma loop_fun_step_self :
  tapp (tderef (tloc 0)) tunit / cons ([r:=tloc 0]loop_fun) nil ⇒+
  tapp (tderef (tloc 0)) tunit / cons ([r:=tloc 0]loop_fun) nil.
Proof with eauto.
  unfold loop_fun; simpl.
  eapply sc_step. apply ST_App1...
  eapply sc_one. compute. apply ST_AppAbs...

Exercise: 4 stars (factorial_ref)

Use the above ideas to implement a factorial function in STLC with references. (There is no need to prove formally that it really behaves like the factorial. Just use the example below to make sure it gives the correct result when applied to the argument 4.)

Definition factorial : tm :=
  (* FILL IN HERE *) admit.

Lemma factorial_type : empty; nil factorial ∈ (TArrow TNat TNat).
Proof with eauto.
  (* FILL IN HERE *) Admitted.

If your definition is correct, you should be able to just uncomment the example below; the proof should be fully automatic using the reduce tactic.

Lemma factorial_4 : exists st, 
  tapp factorial (tnat 4) / nil ==>* tnat 24 / st.
  eexists. unfold factorial. reduce.

Additional Exercises

Exercise: 5 stars, optional (garabage_collector)

Challenge problem: modify our formalization to include an account of garbage collection, and prove that it satisfies whatever nice properties you can think to prove about it.

End RefsAndNontermination.
End STLCRef.