References.v

(** * References: Typing Mutable References *)

(** Up to this point, 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 and "mostly functional" languages such as ML, as
    well as imperative languages such as C and object-oriented
    languages such as Java, C[#], and Scala.

    However, most practical 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, computation 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 collectively
    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_) and _pointers_ that name store locations.  This extension
    is fairly straightforward to define; the most interesting part is
    the refinement we need to make to the statement of the type
    preservation theorem. *)

Require Import Coq.Arith.Arith.
Require Import Coq.omega.Omega.
Require Import Coq.Lists.List.
Import ListNotations.
Require Import Maps.
Require Import Smallstep.

(* ################################################################# *)
(** * Definitions *)

(** 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 Gallina 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] to assign a new value to the
    cell that it points to (by writing [y:=84]), but we cannot use [y]
    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. *)

(* ################################################################# *)
(** * Syntax *)

(** 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 reduces to
         a reference to that cell.

       - To read the current value of this cell, we use the
         dereferencing operator [!]; for example, [!(ref 5)] reduces
         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]. *)

(* ----------------------------------------------------------------- *)
(** *** Types *)

(** We start with the simply typed lambda calculus over the
    natural numbers. Besides 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 to
    cells 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.

(* ----------------------------------------------------------------- *)
(** *** Terms *)

(** 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.

(** Intuitively:
    - [ref t] (formally, [tref t]) allocates a new reference cell
      with the value [t] and reduces to the location of the newly
      allocated cell;

    - [!t] (formally, [tderef t]) reduces 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.) *)

(* ----------------------------------------------------------------- *)
(** *** Typing (Preview) *)

(** Informally, the typing rules for allocation, dereferencing, and
    assignment will look like this:

                           Gamma |- t1 : T1
                       ------------------------                         (T_Ref)
                       Gamma |- ref t1 : Ref T1

                        Gamma |- t1 : Ref T11
                        ---------------------                         (T_Deref)
                          Gamma |- !t1 : T11

                        Gamma |- t1 : Ref T11
                          Gamma |- t2 : T11
                       ------------------------                      (T_Assign)
                       Gamma |- 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  : forall x T t,
      value (tabs x T t)
  | v_nat : forall n,
      value (tnat n)
  | v_unit :
      value tunit
  | v_loc : forall 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 beq_id x x' then s else t
  | tapp t1 t2    =>
      tapp (subst x s t1) (subst x s t2)
  | tabs x' T t1  =>
      if beq_id x x' then t else tabs x' T (subst x s t1)
  | tnat n        =>
      t
  | 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         =>
      t
  | 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 _        =>
      t
  end.

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

(* ################################################################# *)
(** * Pragmatics *)

(* ================================================================= *)
(** ** Side Effects and Sequencing *)

(** The fact that we've chosen the result of an assignment
    expression to be the trivial value [unit] allows a nice
    abbreviation for _sequencing_.  For example, we can write

       r:=succ(!r); !r

    as an abbreviation for

       (\x:Unit. !r) (r:=succ(!r)).

    This has the effect of reducing 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 some variable [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 reducing

      let r = ref 5 in
      let s = r in
      s := 82;
      (!r)+1

    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 = \_:Unit. (c := succ (!c); !c) in
      let decc = \_:Unit. (c := pred (!c); !c) in
      ...
*)

(** Note that, since their argument types are [Unit], the
    arguments to the abstractions in the definitions of [incc] and
    [decc] are not providing any useful information to the bodies of
    these functions (using the wildcard [_] as the name of the bound
    variable is a reminder of this).  Instead, their purpose of these
    abstractions is to "slow down" the execution of the function
    bodies.  Since function abstractions are values, the two [let]s are
    executed simply by binding these functions to the names [incc] and
    [decc], rather than by actually incrementing or decrementing [c].
    Later, each caddll 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]. *)

(* ================================================================= *)
(** ** Objects *)

(** 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 =
          \_:Unit.
             let c = ref 0 in
             let incc = \_:Unit. (c := succ (!c); !c) in
             let decc = \_: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 [let]s have finished and the third
    one is about to begin. *)

(* FILL IN HERE *)
(** [] *)

(* ================================================================= *)
(** ** 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 an (inefficient) implementation of arrays
    of numbers, as follows.  Write [NatArray] for the type
    [Ref (Nat->Nat)].

    Recall the [equal] function from the [MoreStlc] chapter:

      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))

    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 = \_:Unit. ref (\n:Nat.0)

    To look up an element of an array, we simply apply
    the function to the desired index.

      lookup = \a:NatArray. \n: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], while on all other indices it passes the
    lookup to the function that was previously stored in the reference.

      update = \a:NatArray. \m:Nat. \v:Nat.
                   let oldf = !a in
                   a := (\n: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, recommended (compact_update)  *)
(** If we defined [update] more compactly like this

      update = \a:NatArray. \m:Nat. \v:Nat.
                  a := (\n:Nat. if equal m n then v else (!a) n)

would it behave the same? *)

(* FILL IN HERE *)
(** [] *)

(* ================================================================= *)
(** ** Null References *)

(** There is one final significant 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 either in a clean exception (Java and C[#]) or in
    arbitrary and possibly insecure behavior (C and relatives like
    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.

    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
    represent such situations: 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_, automatically identifying 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.  One 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. *)

(* FILL IN HERE *)
(** [] *)

(* ################################################################# *)
(** * Operational Semantics *)

(* ================================================================= *)
(** ** Locations *)

(** 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 reduci 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 [Float]s, etc.), record somewhere
    that it is 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 convenient to use
    numbers.)  We use the word _location_ instead of _reference_ or
    _pointer_ 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. *)

(* ================================================================= *)
(** ** Stores *)

(** 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 didn'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 reduces to the location of the newly
    created reference cell. When reducing 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
    that we don't will require a bit of work.) *)

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

(** 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
  | nil    => nil
  | h :: t =>
    match n with
    | O    => x :: t
    | S n' => h :: replace n' x t
    end
  end.

(** As might be expected, we will also need some technical
    lemmas about [replace]; they are straightforward to prove. *)

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

Lemma length_replace : forall 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...
Qed.

Lemma lookup_replace_eq : forall 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.
  - (* st = [] *)
   inversion Hlen.
  - (* st = t' :: st' *)
    destruct l; simpl...
    apply IHst'. simpl in Hlen. omega.
Qed.

Lemma lookup_replace_neq : forall l1 l2 t st,
  l1 <> l2 ->
  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.
  - (* l1 = 0 *)
    destruct st.
    + (* st = [] *) rewrite replace_nil...
    + (* st = _ :: _ *) destruct l2... contradict Hneq...
  - (* l1 = S l1' *)
    destruct st as [|t2 st2].
    + (* st = [] *) destruct l2...
    + (* st = t2 :: st2 *)
      destruct l2...
      simpl; apply IHl1'...
Qed.

(* ================================================================= *)
(** ** Reduction *)

(** Next, we need to extend the operational semantics to take
    stores into account.  Since the result of reducing an expression
    will in general depend on the contents of the store in which it is
    reduced, the evaluation rules should take not just a term but
    also a store as argument.  Furthermore, since the reduction of a
    term can cause side effects on the store, and these may affect the
    reduction of other terms in the future, the reduction rules need
    to return a new store.  Thus, the shape of the single-step
    reduction relation needs to change 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 reduction rules with stores:

                               value v2
                -------------------------------------- (ST_AppAbs) 
                (\x: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, since
    function application, in itself, has no side effects.  The other
    two rules simply propagate side effects from premise to
    conclusion.

    Now, the result of reducing 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 during reduction.  
    This may seem odd, but it follows naturally from our design decision
    to represent the result of every reduction step by a modified _term_. 
    If we had chosen a more "machine-like" model, e.g., with an explicit 
    stack to contain values of bound identifiers, then the idea of adding 
    locations to the set of allowed values might seem more obvious.

    In terms of this expanded syntax, we can state reduction rules
    for the new constructs that manipulate locations and the store.
    First, to reduce 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 dereference a
    location that is larger than the size [|st|] of the currently
    allocated store; the reduction rules simply get stuck in this
    case.  The type-safety properties established 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 reduce an assignment expression [t1:=t2], we must first
    reduce [t1] until it becomes a value (a location), and then
    reduce [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 reduction step is just [unit];
    the interesting result is the updated store.

    Finally, to reduct an expression of the form [ref t1], we first
    reduce [t1] until it becomes a value:

                        t1 / st ==> t1' / st'
                    ----------------------------- (ST_Ref) 
                    ref t1 / st ==> ref t1' / st'

    Then, to reduce 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].

                   -------------------------------- (ST_RefValue) 
                   ref v1 / st ==> loc |st| / st,v1

    The value resulting from this step is the newly allocated location
    itself.  (Formally, [st,v1] means [st ++ v1::nil] -- i.e., to add
    a new reference cell to the store, we append it to the end.)  

    Note that these reduction rules do not perform any kind of
    garbage collection: we simply allow the store to keep growing
    without bound as reduction proceeds.  This does not affect the
    correctness of the results of reduction (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
    reduction), but it means that a naive implementation of our
    evaluator might run out of memory where a more sophisticated 
    evaluator would be able to continue by reusing locations whose 
    contents have become garbage.

    Here are the rules again, formally: *)

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

Import ListNotations.

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

(** One slightly ugly point should be noted here: In the [ST_RefValue]
    rule, we extend the state by writing [st ++ v1::nil] rather than
    the more natural [st ++ [v1]].  The reason for this is that the
    notation we've defined for substitution uses square brackets,
    which clash with the standard library's notation for lists. *)

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).

(* ################################################################# *)
(** * Typing *)

(** The contexts assigning types to free variables are 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 reduction rules to accommodate
    references, our last job is to write down typing rules for the new
    constructs (and, of course, to check that these rules are sound!).
    Naturally, the key question is, "What is the type of a location?"

    First of all, notice that this question doesn't arise when
    typechecking terms that programmers actually
    write.  Concrete location constants arise only in terms that are
    the intermediate results of reduction; 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 a reduction
    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 reduce 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 reduce the
    term [!(loc 1)] in the store [[unit, unit]], the result is [unit];
    if we reduce the same term in the store [[unit, \x:Unit.x]], the
    result is [\x:Unit.x].  With respect to the former store, the
    location [1] has type [Unit], and with respect to the latter it
    has type [Unit->Unit]. This observation leads us immediately to a
    first attempt at a typing rule for locations:

                             Gamma |- lookup  l st : T1
                            ----------------------------
                             Gamma |- 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: [Gamma; 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, this rule will not quite do.  For one thing, 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.  Worse yet, 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:

   [\x:Nat. (!(loc 1)) x, \x:Nat. (!(loc 0)) x]
*)

(** **** Exercise: 2 stars (cyclic_store)  *)
(** Can you find a term whose reduction will create this particular
    cyclic store? *)
(** [] *)

(** 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 with the other type systems we've seen, this conservative typing 
    restriction on allowed updates means that we will rule out as 
    ill-typed some programs that could reduce perfectly well without
    getting stuck. 

    Just as 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
    values that we expect to be 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 reduced.  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,
    Unit->Unit]], then we can immediately infer that [!(loc 1)] has
    type [Unit->Unit].  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, 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 formalize 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 '\in' T" (at level 40).

Inductive has_type : context -> store_ty -> tm -> ty -> Prop :=
  | T_Var : forall Gamma ST x T,
      Gamma x = Some T ->
      Gamma; ST |- (tvar x) \in T
  | T_Abs : forall Gamma ST x T11 T12 t12,
      (update Gamma x T11); ST |- t12 \in T12 ->
      Gamma; ST |- (tabs x T11 t12) \in (TArrow T11 T12)
  | T_App : forall T1 T2 Gamma ST t1 t2,
      Gamma; ST |- t1 \in (TArrow T1 T2) ->
      Gamma; ST |- t2 \in T1 ->
      Gamma; ST |- (tapp t1 t2) \in T2
  | T_Nat : forall Gamma ST n,
      Gamma; ST |- (tnat n) \in TNat
  | T_Succ : forall Gamma ST t1,
      Gamma; ST |- t1 \in TNat ->
      Gamma; ST |- (tsucc t1) \in TNat
  | T_Pred : forall Gamma ST t1,
      Gamma; ST |- t1 \in TNat ->
      Gamma; ST |- (tpred t1) \in TNat
  | T_Mult : forall Gamma ST t1 t2,
      Gamma; ST |- t1 \in TNat ->
      Gamma; ST |- t2 \in TNat ->
      Gamma; ST |- (tmult t1 t2) \in TNat
  | T_If0 : forall Gamma ST t1 t2 t3 T,
      Gamma; ST |- t1 \in TNat ->
      Gamma; ST |- t2 \in T ->
      Gamma; ST |- t3 \in T ->
      Gamma; ST |- (tif0 t1 t2 t3) \in T
  | T_Unit : forall Gamma ST,
      Gamma; ST |- tunit \in TUnit
  | T_Loc : forall Gamma ST l,
      l < length ST ->
      Gamma; ST |- (tloc l) \in (TRef (store_Tlookup l ST))
  | T_Ref : forall Gamma ST t1 T1,
      Gamma; ST |- t1 \in T1 ->
      Gamma; ST |- (tref t1) \in (TRef T1)
  | T_Deref : forall Gamma ST t1 T11,
      Gamma; ST |- t1 \in (TRef T11) ->
      Gamma; ST |- (tderef t1) \in T11
  | T_Assign : forall Gamma ST t1 t2 T11,
      Gamma; ST |- t1 \in (TRef T11) ->
      Gamma; ST |- t2 \in T11 ->
      Gamma; ST |- (tassign t1 t2) \in TUnit

where "Gamma ';' ST '|-' t '\in' T" := (has_type Gamma ST t T).

Hint Constructors has_type.

(** Of course, these typing rules will accurately predict the results
    of reduction only if the concrete store used during reduction
    actually conforms to the store typing that we assume for purposes
    of typechecking.  This proviso exactly parallels the situation
    with free variables in the basic STLC: the substitution lemma
    promises that, if [Gamma |- t : T], then we can replace the free
    variables in [t] with values of the types listed in [Gamma] to
    obtain a closed term of type [T], which, by the type preservation
    theorem will reduce to a final result of type [T] if it yields
    any result at all.  We will see below 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.  Concrete locations arise only in
    terms that are the intermediate results of reduction; 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 reduction 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.  *)

(* ################################################################# *)
(** * Properties *)

(** 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 to deal 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 reduction 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.  But clearly we cannot just add stores
    and store typings without saying anything about how they are
    related -- i.e., this is wrong: *)

Theorem preservation_wrong1 : forall ST T t st t' st',
  empty; ST |- t \in T ->
  t / st ==> t' / st' ->
  empty; ST |- t' \in T.
Abort.

(** If we typecheck with respect to some set of assumptions about the
    types of the values in the store and then reduce 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 /\
  (forall l, l < length st ->
     empty; ST |- (store_lookup l st) \in (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 like the one we saw above. *)

(** **** 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]? *)

(* FILL IN HERE *)
(** [] *)

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

Theorem preservation_wrong2 : forall ST T t st t' st',
  empty; ST |- t \in T ->
  t / st ==> t' / st' ->
  store_well_typed ST st ->
  empty; ST |- t' \in T.
Abort.

(** This statement is fine for all of the reduction 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 reduction,
    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  : forall ST',
      extends ST' nil
  | extends_cons : forall 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 : forall 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'.
  - (* nil *) inversion Hlen.
  - (* cons *) unfold store_Tlookup in *.
    destruct ST'.
    + (* ST' = nil *) inversion HST'.
    + (* ST' = a' :: ST'2 *)
      inversion HST'; subst.
      destruct l as [|l'].
      * (* l = 0 *) auto.
      * (* l = S l' *) simpl. apply IHST2...
        simpl in Hlen; omega.
Qed.

(** Next, if [ST'] extends [ST], the length of [ST'] is at least that
    of [ST]. *)

Lemma length_extends : forall 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.
Qed.

(** Finally, [ST ++ T] extends [ST], and [extends] is reflexive. *)

Lemma extends_app : forall ST T,
  extends (ST ++ T) ST.
Proof with auto.
  induction ST; intros T...
  simpl...
Qed.

Lemma extends_refl : forall ST,
  extends ST ST.
Proof.
  induction ST; auto.
Qed.

(* ================================================================= *)
(** ** Preservation, Finally *)

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

Definition preservation_theorem := forall ST t t' T st st',
  empty; ST |- t \in T ->
  store_well_typed ST st ->
  t / st ==> t' / st' ->
  exists ST',
    (extends ST' ST /\
     empty; ST' |- t' \in 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 exactly [ST ++ T1::nil], where
    [T1] is the type of the value [v1] in the extended store [st ++
    v1::nil], 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 reduction 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 : forall x,
      appears_free_in x (tvar x)
  | afi_app1 : forall x t1 t2,
      appears_free_in x t1 -> appears_free_in x (tapp t1 t2)
  | afi_app2 : forall x t1 t2,
      appears_free_in x t2 -> appears_free_in x (tapp t1 t2)
  | afi_abs : forall x y T11 t12,
      y <> x  ->
      appears_free_in x t12 ->
      appears_free_in x (tabs y T11 t12)
  | afi_succ : forall x t1,
      appears_free_in x t1 ->
      appears_free_in x (tsucc t1)
  | afi_pred : forall x t1,
      appears_free_in x t1 ->
      appears_free_in x (tpred t1)
  | afi_mult1 : forall x t1 t2,
      appears_free_in x t1 ->
      appears_free_in x (tmult t1 t2)
  | afi_mult2 : forall x t1 t2,
      appears_free_in x t2 ->
      appears_free_in x (tmult t1 t2)
  | afi_if0_1 : forall x t1 t2 t3,
      appears_free_in x t1 ->
      appears_free_in x (tif0 t1 t2 t3)
  | afi_if0_2 : forall x t1 t2 t3,
      appears_free_in x t2 ->
      appears_free_in x (tif0 t1 t2 t3)
  | afi_if0_3 : forall x t1 t2 t3,
      appears_free_in x t3 ->
      appears_free_in x (tif0 t1 t2 t3)
  | afi_ref : forall x t1,
      appears_free_in x t1 -> appears_free_in x (tref t1)
  | afi_deref : forall x t1,
      appears_free_in x t1 -> appears_free_in x (tderef t1)
  | afi_assign1 : forall x t1 t2,
      appears_free_in x t1 -> appears_free_in x (tassign t1 t2)
  | afi_assign2 : forall x t1 t2,
      appears_free_in x t2 -> appears_free_in x (tassign t1 t2).

Hint Constructors appears_free_in.

Lemma free_in_context : forall x t T Gamma ST,
   appears_free_in x t ->
   Gamma; ST |- t \in T ->
   exists T', Gamma x = Some T'.
Proof with eauto.
  intros. generalize dependent Gamma. generalize dependent T.
  induction H;
        intros; (try solve [ inversion H0; subst; eauto ]).
  - (* afi_abs *)
    inversion H1; subst.
    apply IHappears_free_in in H8.
    rewrite update_neq in H8; assumption.
Qed.

Lemma context_invariance : forall Gamma Gamma' ST t T,
  Gamma; ST |- t \in T ->
  (forall x, appears_free_in x t -> Gamma x = Gamma' x) ->
  Gamma'; ST |- t \in T.
Proof with eauto.
  intros.
  generalize dependent Gamma'.
  induction H; intros...
  - (* T_Var *)
    apply T_Var. symmetry. rewrite <- H...
  - (* T_Abs *)
    apply T_Abs. apply IHhas_type; intros.
    unfold update, t_update.
    destruct (beq_idP x x0)...
  - (* T_App *)
    eapply T_App.
      apply IHhas_type1...
      apply IHhas_type2...
  - (* T_Mult *)
    eapply T_Mult.
      apply IHhas_type1...
      apply IHhas_type2...
  - (* T_If0 *)
    eapply T_If0.
      apply IHhas_type1...
      apply IHhas_type2...
      apply IHhas_type3...
  - (* T_Assign *)
    eapply T_Assign.
      apply IHhas_type1...
      apply IHhas_type2...
Qed.

Lemma substitution_preserves_typing : forall Gamma ST x s S t T,
  empty; ST |- s \in S ->
  (update Gamma x S); ST |- t \in T ->
  Gamma; ST |- ([x:=s]t) \in T.
Proof with eauto.
  intros Gamma ST x s S t T Hs Ht.
  generalize dependent Gamma. generalize dependent T.
  induction t; intros T Gamma H;
    inversion H; subst; simpl...
  - (* tvar *)
    rename i into y.
    destruct (beq_idP x y).
    + (* x = y *)
      subst.
      rewrite update_eq in H3.
      inversion H3; subst.
      eapply context_invariance...
      intros x Hcontra.
      destruct (free_in_context _ _ _ _ _ Hcontra Hs)
        as [T' HT'].
      inversion HT'.
    + (* x <> y *)
      apply T_Var.
      rewrite update_neq in H3...
  - (* tabs *) subst.
    rename i into y.
    destruct (beq_idP x y).
    + (* x = y *)
      subst.
      apply T_Abs. eapply context_invariance...
      intros. rewrite update_shadow. reflexivity.
    + (* x <> x0 *)
      apply T_Abs. apply IHt.
      eapply context_invariance...
      intros. unfold update, t_update.
      destruct (beq_idP y x0)...
      subst.
      rewrite false_beq_id...
Qed.

(* ================================================================= *)
(** ** 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 : forall ST st l t,
  l < length st ->
  store_well_typed ST st ->
  empty; ST |- t \in (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'.
  - (* l' = l *)
    apply beq_nat_true in Heqll'; subst.
    rewrite lookup_replace_eq...
  - (* l' <> l *)
    apply beq_nat_false in Heqll'.
    rewrite lookup_replace_neq...
    rewrite length_replace in Hl'.
    apply H0...
Qed.

(* ================================================================= *)
(** ** 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 : forall Gamma ST ST' t T,
  extends ST' ST ->
  Gamma; ST |- t \in T ->
  Gamma; ST' |- t \in T.
Proof with eauto.
  intros. induction H0; eauto.
  - (* T_Loc *)
    erewrite <- extends_lookup...
    apply T_Loc.
    eapply length_extends...
Qed.

(** 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_app : forall ST st t1 T1,
  store_well_typed ST st ->
  empty; ST |- t1 \in T1 ->
  store_well_typed (ST ++ T1::nil) (st ++ t1::nil).
Proof with auto.
  intros.
  unfold store_well_typed in *.
  inversion H as [Hlen Hmatch]; clear H.
  rewrite app_length, plus_comm. simpl.
  rewrite app_length, plus_comm. simpl.
  split...
  - (* types match. *)
    intros l Hl.
    unfold store_lookup, store_Tlookup.
    apply le_lt_eq_dec in Hl; inversion Hl as [Hlt | Heq].
    + (* l < length st *)
      apply lt_S_n in Hlt.
      rewrite !app_nth1...
      * apply store_weakening with ST. apply extends_app.
        apply Hmatch...
      * rewrite Hlen...
    + (* l = length st *)
      inversion Heq.
      rewrite app_nth2; try omega.
      rewrite <- Hlen.
      rewrite minus_diag. simpl.
      apply store_weakening with ST...
      { apply extends_app. }
        rewrite app_nth2; try omega.
      rewrite minus_diag. simpl. trivial.
Qed.

(* ================================================================= *)
(** ** Preservation! *)

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

(** Begin with one technical lemma: *)

Lemma nth_eq_last : forall A (l:list A) x d,
  nth (length l) (l ++ x::nil) d = x.
Proof.
  induction l; intros; [ auto | simpl; rewrite IHl; auto ].
Qed.

(** And here, at last, is the preservation theorem and proof: *)

Theorem preservation : forall ST t t' T st st',
  empty; ST |- t \in T ->
  store_well_typed ST st ->
  t / st ==> t' / st' ->
  exists ST',
    (extends ST' ST /\
     empty; ST' |- t' \in T /\
     store_well_typed ST' st').
Proof with eauto using store_weakening, extends_refl.
  remember (@empty ty) as Gamma.
  intros ST t t' T st st' Ht.
  generalize dependent t'.
  induction Ht; intros t' HST Hstep;
    subst; try solve_by_invert; inversion Hstep; subst;
    try (eauto using store_weakening, extends_refl).
  (* T_App *)
  - (* ST_AppAbs *) exists ST.
    inversion Ht1; subst.
    split; try split... eapply substitution_preserves_typing...
  - (* ST_App1 *)
    eapply IHHt1 in H0...
    inversion H0 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  - (* ST_App2 *)
    eapply IHHt2 in H5...
    inversion H5 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  - (* T_Succ *)
    + (* ST_Succ *)
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
      exists ST'...
  - (* T_Pred *)
    + (* ST_Pred *)
      eapply IHHt in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
      exists ST'...
  (* T_Mult *)
  - (* ST_Mult1 *)
    eapply IHHt1 in H0...
    inversion H0 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  - (* ST_Mult2 *)
    eapply IHHt2 in H5...
    inversion H5 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  - (* T_If0 *)
    + (* ST_If0_1 *)
      eapply IHHt1 in H0...
      inversion H0 as [ST' [Hext [Hty Hsty]]].
      exists ST'... split...
  (* T_Ref *)
  - (* ST_RefValue *)
    exists (ST ++ T1::nil).
    inversion HST; subst.
    split.
      apply extends_app.
    split.
      replace (TRef T1)
        with (TRef (store_Tlookup (length st) (ST ++ T1::nil))).
      apply T_Loc.
      rewrite <- H. rewrite app_length, plus_comm. simpl. omega.
      unfold store_Tlookup. rewrite <- H. rewrite nth_eq_last.
      reflexivity.
      apply store_well_typed_app; assumption.
  - (* ST_Ref *)
    eapply IHHt in H0...
    inversion H0 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  (* T_Deref *)
  - (* ST_DerefLoc *)
    exists ST. split; try split...
    inversion HST as [_ Hsty].
    replace T11 with (store_Tlookup l ST).
    apply Hsty...
    inversion Ht; subst...
  - (* ST_Deref *)
    eapply IHHt in H0...
    inversion H0 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  (* T_Assign *)
  - (* ST_Assign *)
    exists ST. split; try split...
    eapply assign_pres_store_typing...
    inversion Ht1; subst...
  - (* ST_Assign1 *)
    eapply IHHt1 in H0...
    inversion H0 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
  - (* ST_Assign2 *)
    eapply IHHt2 in H5...
    inversion H5 as [ST' [Hext [Hty Hsty]]].
    exists ST'...
Qed.

(** **** 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.

(* FILL IN HERE *)
[] *)

(* ================================================================= *)
(** ** Progress *)

(** As we've said, 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 : forall ST t T st,
  empty; ST |- t \in T ->
  store_well_typed ST st ->
  (value t \/ exists t', exists st', t / st ==> t' / st').
Proof with eauto.
  intros ST t T st Ht HST. remember (@empty ty) as Gamma.
  induction Ht; subst; try solve_by_invert...
  - (* T_App *)
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve_by_invert.
      destruct IHHt2 as [Ht2p | Ht2p]...
      * (* t2 steps *)
        inversion Ht2p as [t2' [st' Hstep]].
        exists (tapp (tabs x T t) t2'). exists st'...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tapp t1' t2). exists st'...
  - (* T_Succ *)
    right. destruct IHHt as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve [ inversion Ht ].
      * (* t1 is a tnat *)
        exists (tnat (S n)). exists st...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tsucc t1'). exists st'...
  - (* T_Pred *)
    right. destruct IHHt as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve [inversion Ht ].
      * (* t1 is a tnat *)
        exists (tnat (pred n)). exists st...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tpred t1'). exists st'...
  - (* T_Mult *)
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve [inversion Ht1].
      destruct IHHt2 as [Ht2p | Ht2p]...
      * (* t2 is a value *)
        inversion Ht2p; subst; try solve [inversion Ht2].
        exists (tnat (mult n n0)). exists st...
      * (* t2 steps *)
        inversion Ht2p as [t2' [st' Hstep]].
        exists (tmult (tnat n) t2'). exists st'...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tmult t1' t2). exists st'...
  - (* T_If0 *)
    right. destruct IHHt1 as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve [inversion Ht1].
      destruct n.
      * (* n = 0 *) exists t2. exists st...
      * (* n = S n' *) exists t3. exists st...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tif0 t1' t2 t3). exists st'...
  - (* T_Ref *)
    right. destruct IHHt as [Ht1p | Ht1p]...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tref t1'). exists st'...
  - (* T_Deref *)
    right. destruct IHHt as [Ht1p | Ht1p]...
    + (* t1 is a value *)
      inversion Ht1p; subst; try solve_by_invert.
      eexists. eexists. apply ST_DerefLoc...
      inversion Ht; subst. inversion HST; subst.
      rewrite <- H...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tderef t1'). exists st'...
  - (* T_Assign *)
    right. destruct IHHt1 as [Ht1p|Ht1p]...
    + (* t1 is a value *)
      destruct IHHt2 as [Ht2p|Ht2p]...
      * (* t2 is a value *)
        inversion Ht1p; subst; try solve_by_invert.
        eexists. eexists. apply ST_Assign...
        inversion HST; subst. inversion Ht1; subst.
        rewrite H in H5...
      * (* t2 steps *)
        inversion Ht2p as [t2' [st' Hstep]].
        exists (tassign t1 t2'). exists st'...
    + (* t1 steps *)
      inversion Ht1p as [t1' [st' Hstep]].
      exists (tassign t1' t2). exists st'...
Qed.

(* ################################################################# *)
(** * References and Nontermination *)

(** An important fact about the STLC (proved in chapter [Norm]) is
    that it is 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!

   (\r:Ref (Unit -> Unit).  
        r := (\x:Unit.(!r) unit); (!r) unit) 
   (ref (\x: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 [\x: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 start the 
   divergent loop, we execute the function stored in the cell by 
   evaluating [(!r) unit]. 

   Here is the divergent term in Coq: *)

Module ExampleVariables.

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

End ExampleVariables.

Module RefsAndNontermination.
Import ExampleVariables.

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

Definition loop :=
  tapp
    (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 : exists T, empty; nil |- loop \in T.
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 update, t_update. simpl. reflexivity. auto.
  eapply T_Assign.
    eapply T_Var. unfold update, t_update. simpl. reflexivity.
  eapply T_Abs.
    eapply T_App...
      eapply T_Deref. eapply T_Var. reflexivity.
Qed.

(** 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  : forall (x y : X),
                R x y -> step_closure R x y
  | sc_step : forall (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 |- ?x => idtac x end.
Ltac reduce :=
    repeat (print_goal; eapply multi_step ;
            [ (eauto 10; fail) | (instantiate; compute)];
            try solve [apply multi_refl]).

(** Next, we use [reduce] to show that [loop] steps to 
    [!(loc 0) unit], starting from the empty store. *)

Lemma loop_steps_to_loop_fun :
  loop / nil ==>*
  tapp (tderef (tloc 0)) tunit / cons ([r:=tloc 0]loop_fun) nil.
Proof.
  unfold loop.
  reduce.
Qed.

(** Finally, we show that 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...
Qed.

(** **** 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 uncomment the example below to make
    sure it gives the correct result when applied to the argument
    [4].) *)

Definition factorial : tm 
  (* REPLACE THIS LINE WITH   := _your_definition_ . *) . Admitted.

Lemma factorial_type : empty; nil |- factorial \in (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.
Proof.
  eexists. unfold factorial. reduce.
Qed.
*)
(** [] *)

(* ################################################################# *)
(** * 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.

(** $Date: 2016-07-13 12:41:41 -0400 (Wed, 13 Jul 2016) $ *)