# ReferencesTyping Mutable References

So far, we have considered a variety of
Most practical programming languages also include various
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
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
In most other languages — in particular, in all members of the C
family, including Java —
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.

*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.*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*.*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.## Definitions

*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.*every*variable name refers to a mutable cell, and the operation of dereferencing a variable to obtain its current contents is implicit.## Syntax

The basic operations on references are
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

*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.

### Types

*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

### Terms

t ::= ... Terms | ref t allocation | !t dereference | t := t assignment | l location

Inductive tm : Type :=

(* STLC with numbers: *)

| tm_var : id → tm

| tm_app : tm → tm → tm

| tm_abs : id → ty → tm → tm

| tm_nat : nat → tm

| tm_succ : tm → tm

| tm_pred : tm → tm

| tm_mult : tm → tm → tm

| tm_if0 : tm → tm → tm → tm

(* New terms: *)

| tm_unit : tm

| tm_ref : tm → tm

| tm_deref : tm → tm

| tm_assign : tm → tm → tm

| tm_loc : nat → tm.

Intuitively...
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.

- ref t (formally, tm_ref t) allocates a new reference cell
with the value t and evaluates to the location of the newly
allocated cell;
- !t (formally, tm_deref t) evaluates to the contents of the
cell referenced by t;
- t1 := t2 (formally, tm_assign t1 t2) assigns t2 to the
cell referenced by t1; and
- l (formally, tm_loc l) is a reference to the cell at location l. We'll discuss locations later.

Tactic Notation "tm_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "tm_var" | Case_aux c "tm_app"

| Case_aux c "tm_abs" | Case_aux c "tm_zero"

| Case_aux c "tm_succ" | Case_aux c "tm_pred"

| Case_aux c "tm_mult" | Case_aux c "tm_if0"

| Case_aux c "tm_unit" | Case_aux c "tm_ref"

| Case_aux c "tm_deref" | Case_aux c "tm_assign"

| Case_aux c "tm_loc" ].

Module ExampleVariables.

Definition x := Id 0.

Definition y := Id 1.

Definition r := Id 2.

Definition s := Id 3.

End ExampleVariables.

### Typing (Preview)

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 |

### Values and Substitution

Inductive value : tm → Prop :=

| v_abs : ∀ x T t,

value (tm_abs x T t)

| v_nat : ∀ n,

value (tm_nat n)

| v_unit :

value tm_unit

| v_loc : ∀ l,

value (tm_loc 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

| tm_var x' =>

if beq_id x x' then s else t

| tm_app t1 t2 =>

tm_app (subst x s t1) (subst x s t2)

| tm_abs x' T t1 =>

if beq_id x x' then t else tm_abs x' T (subst x s t1)

| tm_nat n =>

t

| tm_succ t1 =>

tm_succ (subst x s t1)

| tm_pred t1 =>

tm_pred (subst x s t1)

| tm_mult t1 t2 =>

tm_mult (subst x s t1) (subst x s t2)

| tm_if0 t1 t2 t3 =>

tm_if0 (subst x s t1) (subst x s t2) (subst x s t3)

| tm_unit =>

t

| tm_ref t1 =>

tm_ref (subst x s t1)

| tm_deref t1 =>

tm_deref (subst x s t1)

| tm_assign t1 t2 =>

tm_assign (subst x s t1) (subst x s t2)

| tm_loc _ =>

t

end.

# Pragmatics

## Side Effects and Sequencing

*sequencing*. For example, we can write

r:=succ(!r); !ras an abbreviation for

(\x: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.

r:=succ(!r); r:=succ(!r); r:=succ(!r); r:=succ(!r); !r

## References and Aliasing

*reference*that is bound to r and the

*cell*in the store that is pointed to by this reference.

*reference*, not the contents of the cell itself.

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

r := 5; r := !sassigns 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

let c = ref 0 in let incc = \_:Unit. (c := succ (!c); !c) in let decc = \_:Unit. (c := pred (!c); !c) in ...

*thunks*.

## Objects

*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

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.(* FILL IN HERE *)

☐
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 (Nat→Nat).
Recall the equal function from the MoreStlc chapter:

## References to Compound Types

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))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 = \_: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) nThe 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 = \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

If we defined update more compactly like thisupdate = \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 *)

☐
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 Ref (Option T).
A last issue that we should mention before we move on with
formalizing references is storage
This is

## Null References

## Garbage Collection

*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.*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

Show how this can lead to a violation of type safety.(* FILL IN HERE *)

☐
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
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
Treating locations abstractly in this way will prevent us from
modeling the
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

# Operational Semantics

## Locations

*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.*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.*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

*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 tm_ref 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.)
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!)

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

| nil => x :: nil

| h :: t => h :: snoc t x

end.

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).

Proof.

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.

Proof.

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.

Qed.

Lemma nth_eq_snoc : ∀ A (l:list A) x d,

nth (length l) (snoc l x) d = x.

Proof.

induction l; intros; [ auto | simpl; rewrite IHl; auto ].

Qed.

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.

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 [] = [].

Proof.

destruct n; auto.

Qed.

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...

Qed.

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.

Qed.

Lemma lookup_replace_neq : ∀ 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.

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'...

Qed.

## Reduction

value v2 | (ST_AppAbs) |

(\a:T.t12) v2 / st ⇒ [v2/a]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' |

*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.

t1 / st ⇒ t1' / st' | (ST_Deref) |

!t1 / st ⇒ !t1' / st' |

l < |st| | (ST_DerefLoc) |

!(loc l) / st ⇒ lookup l st / st |

t1 / st ⇒ t1' / st' | (ST_Assign1) |

t1 := t2 / st ⇒ t1' := t2 / st' |

t2 / st ⇒ t2' / st' | (ST_Assign2) |

v1 := t2 / st ⇒ v1 := t2' / st' |

l < |st| | (ST_Assign) |

loc l := v2 / st ⇒ unit / [v2/l]st |

t1 / st ⇒ t1' / st' | (ST_Ref) |

ref t1 / st ⇒ ref t1' / st' |

(ST_RefValue) | |

ref v1 / st ⇒ loc |st| / st,v1 |

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 →

tm_app (tm_abs x T t12) v2 / st ⇒ subst x v2 t12 / st

| ST_App1 : ∀ t1 t1' t2 st st',

t1 / st ⇒ t1' / st' →

tm_app t1 t2 / st ⇒ tm_app t1' t2 / st'

| ST_App2 : ∀ v1 t2 t2' st st',

value v1 →

t2 / st ⇒ t2' / st' →

tm_app v1 t2 / st ⇒ tm_app v1 t2'/ st'

| ST_SuccNat : ∀ n st,

tm_succ (tm_nat n) / st ⇒ tm_nat (S n) / st

| ST_Succ : ∀ t1 t1' st st',

t1 / st ⇒ t1' / st' →

tm_succ t1 / st ⇒ tm_succ t1' / st'

| ST_PredNat : ∀ n st,

tm_pred (tm_nat n) / st ⇒ tm_nat (pred n) / st

| ST_Pred : ∀ t1 t1' st st',

t1 / st ⇒ t1' / st' →

tm_pred t1 / st ⇒ tm_pred t1' / st'

| ST_MultNats : ∀ n1 n2 st,

tm_mult (tm_nat n1) (tm_nat n2) / st ⇒ tm_nat (mult n1 n2) / st

| ST_Mult1 : ∀ t1 t2 t1' st st',

t1 / st ⇒ t1' / st' →

tm_mult t1 t2 / st ⇒ tm_mult t1' t2 / st'

| ST_Mult2 : ∀ v1 t2 t2' st st',

value v1 →

t2 / st ⇒ t2' / st' →

tm_mult v1 t2 / st ⇒ tm_mult v1 t2' / st'

| ST_If0 : ∀ t1 t1' t2 t3 st st',

t1 / st ⇒ t1' / st' →

tm_if0 t1 t2 t3 / st ⇒ tm_if0 t1' t2 t3 / st'

| ST_If0_Zero : ∀ t2 t3 st,

tm_if0 (tm_nat 0) t2 t3 / st ⇒ t2 / st

| ST_If0_Nonzero : ∀ n t2 t3 st,

tm_if0 (tm_nat (S n)) t2 t3 / st ⇒ t3 / st

| ST_RefValue : ∀ v1 st,

value v1 →

tm_ref v1 / st ⇒ tm_loc (length st) / snoc st v1

| ST_Ref : ∀ t1 t1' st st',

t1 / st ⇒ t1' / st' →

tm_ref t1 / st ⇒ tm_ref t1' / st'

| ST_DerefLoc : ∀ st l,

l < length st →

tm_deref (tm_loc l) / st ⇒ store_lookup l st / st

| ST_Deref : ∀ t1 t1' st st',

t1 / st ⇒ t1' / st' →

tm_deref t1 / st ⇒ tm_deref t1' / st'

| ST_Assign : ∀ v2 l st,

value v2 →

l < length st →

tm_assign (tm_loc l) v2 / st ⇒ tm_unit / replace l v2 st

| ST_Assign1 : ∀ t1 t1' t2 st st',

t1 / st ⇒ t1' / st' →

tm_assign t1 t2 / st ⇒ tm_assign t1' t2 / st'

| ST_Assign2 : ∀ v1 t2 t2' st st',

value v1 →

t2 / st ⇒ t2' / st' →

tm_assign v1 t2 / st ⇒ tm_assign v1 t2' / st'

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

Tactic Notation "step_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "ST_AppAbs" | Case_aux c "ST_App1"

| Case_aux c "ST_App2" | Case_aux c "ST_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 stepmany := (refl_step_closure step).

Notation "t1 '/' st '⇒*' t2 '/' st'" := (stepmany (t1,st) (t2,st'))

(at level 40, st at level 39, t2 at level 39).

# Typing

## Store typings

*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.

Γ ⊢ lookup l st : T1 | |

Γ ⊢ loc l : Ref T1 |

*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 |

*their*types each time they appear.

*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

Can you find a term whose evaluation will create this particular cyclic store?*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.

*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.

The store_ty_lookup function retrieves the type at a particular
index.

Suppose we are

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
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):

*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, Unit→Unit], then we may 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) |

*typing*rather than a concrete store. The rest of the typing rules are analogously augmented with store typings.## The Typing Relation

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 |

Inductive has_type : context → store_ty → tm → ty → Prop :=

| T_Var : ∀ Γ ST x T,

Γ x = Some T →

has_type Γ ST (tm_var x) T

| T_Abs : ∀ Γ ST x T11 T12 t12,

has_type (extend Γ x T11) ST t12 T12 →

has_type Γ ST (tm_abs x T11 t12) (ty_arrow T11 T12)

| T_App : ∀ T1 T2 Γ ST t1 t2,

has_type Γ ST t1 (ty_arrow T1 T2) →

has_type Γ ST t2 T1 →

has_type Γ ST (tm_app t1 t2) T2

| T_Nat : ∀ Γ ST n,

has_type Γ ST (tm_nat n) ty_Nat

| T_Succ : ∀ Γ ST t1,

has_type Γ ST t1 ty_Nat →

has_type Γ ST (tm_succ t1) ty_Nat

| T_Pred : ∀ Γ ST t1,

has_type Γ ST t1 ty_Nat →

has_type Γ ST (tm_pred t1) ty_Nat

| T_Mult : ∀ Γ ST t1 t2,

has_type Γ ST t1 ty_Nat →

has_type Γ ST t2 ty_Nat →

has_type Γ ST (tm_mult t1 t2) ty_Nat

| T_If0 : ∀ Γ ST t1 t2 t3 T,

has_type Γ ST t1 ty_Nat →

has_type Γ ST t2 T →

has_type Γ ST t3 T →

has_type Γ ST (tm_if0 t1 t2 t3) T

| T_Unit : ∀ Γ ST,

has_type Γ ST tm_unit ty_Unit

| T_Loc : ∀ Γ ST l,

l < length ST →

has_type Γ ST (tm_loc l) (ty_Ref (store_ty_lookup l ST))

| T_Ref : ∀ Γ ST t1 T1,

has_type Γ ST t1 T1 →

has_type Γ ST (tm_ref t1) (ty_Ref T1)

| T_Deref : ∀ Γ ST t1 T11,

has_type Γ ST t1 (ty_Ref T11) →

has_type Γ ST (tm_deref t1) T11

| T_Assign : ∀ Γ ST t1 t2 T11,

has_type Γ ST t1 (ty_Ref T11) →

has_type Γ ST t2 T11 →

has_type Γ ST (tm_assign t1 t2) ty_Unit.

Hint Constructors has_type.

Tactic Notation "has_type_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "T_Var" | Case_aux c "T_Abs" | Case_aux c "T_App"

| Case_aux c "T_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
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.
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:

*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.# Properties

## Well-Typed Stores

Theorem preservation_wrong1 : ∀ ST T t st t' st',

has_type empty ST t T →

t / st ⇒ t' / st' →

has_type empty ST t' T.

Admitted.

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 →

has_type empty ST (store_lookup l st) (store_ty_lookup 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

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 : ∀ ST T t st t' st',

has_type empty ST t T →

t / st ⇒ t' / st' →

store_well_typed ST st →

has_type empty ST t' T.

Admitted.

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.
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'

## Extending Store Typings

*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_ty_lookup l ST' = store_ty_lookup 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_ty_lookup in *.

destruct ST' as [|a' ST'2].

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.

Qed.

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.

Qed.

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...

simpl...

Qed.

Lemma extends_refl : ∀ ST,

extends ST ST.

Proof.

induction ST; auto.

Qed.

## Preservation, Finally

Definition preservation_theorem := ∀ ST t t' T st st',

has_type empty ST t T →

store_well_typed ST st →

t / st ⇒ t' / st' →

∃ ST',

(extends ST' ST ∧

has_type empty ST' t' T ∧

store_well_typed ST' st').

Note that the preservation theorem merely asserts that there is
In order to prove this, we'll need a few lemmas, as usual.
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.

*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."## Substitution lemma

Inductive appears_free_in : id → tm → Prop :=

| afi_var : ∀ x,

appears_free_in x (tm_var x)

| afi_app1 : ∀ x t1 t2,

appears_free_in x t1 → appears_free_in x (tm_app t1 t2)

| afi_app2 : ∀ x t1 t2,

appears_free_in x t2 → appears_free_in x (tm_app t1 t2)

| afi_abs : ∀ x y T11 t12,

y <> x →

appears_free_in x t12 →

appears_free_in x (tm_abs y T11 t12)

| afi_succ : ∀ x t1,

appears_free_in x t1 →

appears_free_in x (tm_succ t1)

| afi_pred : ∀ x t1,

appears_free_in x t1 →

appears_free_in x (tm_pred t1)

| afi_mult1 : ∀ x t1 t2,

appears_free_in x t1 →

appears_free_in x (tm_mult t1 t2)

| afi_mult2 : ∀ x t1 t2,

appears_free_in x t2 →

appears_free_in x (tm_mult t1 t2)

| afi_if0_1 : ∀ x t1 t2 t3,

appears_free_in x t1 →

appears_free_in x (tm_if0 t1 t2 t3)

| afi_if0_2 : ∀ x t1 t2 t3,

appears_free_in x t2 →

appears_free_in x (tm_if0 t1 t2 t3)

| afi_if0_3 : ∀ x t1 t2 t3,

appears_free_in x t3 →

appears_free_in x (tm_if0 t1 t2 t3)

| afi_ref : ∀ x t1,

appears_free_in x t1 → appears_free_in x (tm_ref t1)

| afi_deref : ∀ x t1,

appears_free_in x t1 → appears_free_in x (tm_deref t1)

| afi_assign1 : ∀ x t1 t2,

appears_free_in x t1 → appears_free_in x (tm_assign t1 t2)

| afi_assign2 : ∀ x t1 t2,

appears_free_in x t2 → appears_free_in x (tm_assign t1 t2).

Tactic Notation "afi_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "afi_var"

| Case_aux c "afi_app1" | Case_aux c "afi_app2" | Case_aux c "afi_abs"

| Case_aux c "afi_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 →

has_type Γ ST t T →

∃ 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.

apply not_eq_beq_id_false in H.

rewrite extend_neq in H8; assumption.

Qed.

Lemma context_invariance : ∀ Γ Gamma' ST t T,

has_type Γ ST t T →

(∀ x, appears_free_in x t → Γ x = Gamma' x) →

has_type Gamma' ST t T.

Proof with eauto.

intros.

generalize dependent Gamma'.

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.

remember (beq_id x x0) as e; destruct e...

apply H0. apply afi_abs. apply beq_id_false_not_eq... auto.

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...

Qed.

Lemma substitution_preserves_typing : ∀ Γ ST x s S t T,

has_type empty ST s S →

has_type (extend Γ x S) ST t T →

has_type Γ ST (subst x s t) T.

Proof with eauto.

intros Γ ST x s S t T Hs Ht.

generalize dependent Γ. generalize dependent T.

tm_cases (induction t) Case; intros T Γ H;

inversion H; subst; simpl...

Case "tm_var".

rename i into y.

remember (beq_id x y) as eq; destruct eq; subst.

SCase "x = y".

apply beq_id_eq in Heqeq; subst.

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 "tm_abs". subst.

rename i into y.

remember (beq_id x y) as eq; destruct eq.

SCase "x = y".

apply beq_id_eq in Heqeq; subst.

apply T_Abs. eapply context_invariance...

intros. apply extend_shadow.

SCase "x <> x0".

apply T_Abs. apply IHt.

eapply context_invariance...

intros. unfold extend.

remember (beq_id y x0) as e. destruct e...

apply beq_id_eq in Heqe; subst.

rewrite ← Heqeq...

Qed.

## Assignment Preserves Store Typing

Lemma assign_pres_store_typing : ∀ ST st l t,

l < length st →

store_well_typed ST st →

has_type empty ST t (store_ty_lookup 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'.

remember (beq_nat l' l) as ll'; destruct ll'.

Case "l' = l".

apply beq_nat_eq in Heqll'; subst.

rewrite lookup_replace_eq...

Case "l' <> l".

symmetry in Heqll'; apply beq_nat_false in Heqll'.

rewrite lookup_replace_neq...

rewrite length_replace in Hl'.

apply H0...

Qed.

## Weakening for Stores

Lemma store_weakening : ∀ Γ ST ST' t T,

extends ST' ST →

has_type Γ ST t T →

has_type Γ ST' t T.

Proof with eauto.

intros. has_type_cases (induction H0) Case; eauto.

Case "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_snoc : ∀ ST st t1 T1,

store_well_typed ST st →

has_type empty ST t1 T1 →

store_well_typed (snoc ST T1) (snoc st t1).

Proof with auto.

intros.

unfold store_well_typed in *.

inversion H as [Hlen Hmatch]; clear H.

rewrite !length_snoc.

split...

Case "types match.".

intros l Hl.

unfold store_lookup, store_ty_lookup.

apply le_lt_eq_dec in Hl; destruct 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.

Qed.

## Preservation!

Theorem preservation : ∀ ST t t' T st st',

has_type empty ST t T →

store_well_typed ST st →

t / st ⇒ t' / st' →

∃ ST',

(extends ST' ST ∧

has_type 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]]].

∃ ST'...

SCase "ST_App2".

eapply IHHt2 in H5...

inversion H5 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

Case "T_Succ".

SCase "ST_Succ".

eapply IHHt in H0...

inversion H0 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

Case "T_Pred".

SCase "ST_Pred".

eapply IHHt in H0...

inversion H0 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

Case "T_Mult".

SCase "ST_Mult1".

eapply IHHt1 in H0...

inversion H0 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

SCase "ST_Mult2".

eapply IHHt2 in H5...

inversion H5 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

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.

split.

apply extends_snoc.

split.

replace (ty_Ref T1)

with (ty_Ref (store_ty_lookup (length st) (snoc ST T1))).

apply T_Loc.

rewrite ← H. rewrite length_snoc. omega.

unfold store_ty_lookup. 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]]].

∃ ST'...

Case "T_Deref".

SCase "ST_DerefLoc".

∃ ST. split; try split...

destruct HST as [_ Hsty].

replace T11 with (store_ty_lookup l ST).

apply Hsty...

inversion Ht; subst...

SCase "ST_Deref".

eapply IHHt in H0...

inversion H0 as [ST' [Hext [Hty Hsty]]].

∃ ST'...

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]]].

∃ ST'...

SCase "ST_Assign2".

eapply IHHt2 in H5...

inversion H5 as [ST' [Hext [Hty Hsty]]].

∃ 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.☐

## Progress

Theorem progress : ∀ ST t T st,

has_type empty ST t T →

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]].

∃ (tm_app (tm_abs x T t) t2'). ∃ st'...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_app 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 tm_nat".

∃ (tm_nat (S n)). ∃ st...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_succ 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 tm_nat".

∃ (tm_nat (pred n)). ∃ st...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_pred 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].

∃ (tm_nat (mult n n0)). ∃ st...

SSCase "t2 steps".

inversion Ht2p as [t2' [st' Hstep]].

∃ (tm_mult (tm_nat n) t2'). ∃ st'...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_mult 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]].

∃ (tm_if0 t1' t2 t3). ∃ st'...

Case "T_Ref".

right. destruct IHHt as [Ht1p | Ht1p]...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_ref 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]].

∃ (tm_deref 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]].

∃ (tm_assign t1 t2'). ∃ st'...

SCase "t1 steps".

inversion Ht1p as [t1' [st' Hstep]].

∃ (tm_assign t1' t2). ∃ st'...

Qed.

We know that the simply typed lambda calculus is
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!
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 get
the ball rolling we finally execute this function with (!r)
unit.

*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!(\r:Ref (Unit -> Unit). r := (\x:Unit.(!r) unit); (!r) unit) (ref (\x:Unit.unit))

Definition loop_fun :=

tm_abs x ty_Unit (tm_app (tm_deref (tm_var r)) tm_unit).

Definition loop :=

tm_app

(tm_abs r (ty_Ref (ty_arrow ty_Unit ty_Unit))

(tm_seq (tm_assign (tm_var r) loop_fun)

(tm_app (tm_deref (tm_var r)) tm_unit)))

(tm_ref (tm_abs x ty_Unit tm_unit)).

This term is well-typed:

Lemma loop_typeable : ∃ T, has_type empty [] loop 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 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.

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 : ∀ (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 stepmany1 := (step_closure step).

Notation "t1 '/' st '⇒+' t2 '/' st'" := (stepmany1 (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 [(loc 0) / r]
loop_fun.
As a convenience, we introduce a slight variant of the normalize
tactic, called reduce, which tries solving the goal with
rsc_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 rsc_step ;

[ (eauto 10; fail) | (instantiate; compute)];

try solve [apply rsc_refl]).

Lemma loop_steps_to_loop_fun :

loop / [] ⇒*

tm_app (tm_deref (tm_loc 0)) tm_unit / [subst r (tm_loc 0) loop_fun].

Proof with eauto.

unfold loop.

reduce.

Qed.

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

Lemma loop_fun_step_self :

tm_app (tm_deref (tm_loc 0)) tm_unit / [subst r (tm_loc 0) loop_fun] ⇒+

tm_app (tm_deref (tm_loc 0)) tm_unit / [subst r (tm_loc 0) loop_fun].

Proof with eauto.

unfold loop_fun; simpl.

eapply sc_step. apply ST_App1...

eapply sc_one. compute. apply ST_AppAbs...

Qed.

#### Exercise: 4 stars

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 : has_type empty [] factorial (ty_arrow ty_Nat ty_Nat).

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,

tm_app factorial (tm_nat 4) / ==>* tm_nat 24 / st.

Proof.

eexists. unfold factorial. reduce.

Qed.

*)