Concepts

We now turn to the study of subtyping, perhaps the most characteristic feature of the static type systems of recently designed programming languages and a key feature needed to support the object-oriented programming style.

A Motivating Example

Suppose we are writing a program involving two record types defined as follows:

    Person  = {name:String, age:Nat}
    Student = {name:String, age:Nat, gpa:Nat}

In the simply typed lamdba-calculus with records, the term

    (λr:Person. (r.age)+1) {name="Pat",age=21,gpa=1}

is not typable: it involves an application of a function that wants a one-field record to an argument that actually provides two fields, while the T_App rule demands that the domain type of the function being applied must match the type of the argument precisely. But this is silly: we're passing the function a better argument than it needs! The only thing the body of the function can possibly do with its record argument r is project the field age from it: nothing else is allowed by the type, and the presence or absence of an extra gpa field makes no difference at all. So, intuitively, it seems that this function should be applicable to any record value that has at least an age field. Looking at the same thing from another point of view, a record with more fields is "at least as good in any context" as one with just a subset of these fields, in the sense that any value belonging to the longer record type can be used safely in any context expecting the shorter record type. If the context expects something with the shorter type but we actually give it something with the longer type, nothing bad will happen (formally, the program will not get stuck). The general principle at work here is called subtyping. We say that "S is a subtype of T", informally written S <: T, if a value of type S can safely be used in any context where a value of type T is expected. The idea of subtyping applies not only to records, but to all of the type constructors in the language — functions, pairs, etc.

Subtyping and Object-Oriented Languages

Subtyping plays a fundamental role in many programming languages — in particular, it is closely related to the notion of subclassing in object-oriented languages. An object in Java, C#, etc. can be thought of as a record, some of whose fields are functions ("methods") and some of whose fields are data values ("fields" or "instance variables"). Invoking a method m of an object o on some arguments a1..an consists of projecting out the m field of o and applying it to a1..an. The type of an object can be given as either a class or an interface. Both of these provide a description of which methods and which data fields the object offers. Classes and interfaces are related by the subclass and subinterface relations. An object belonging to a subclass (or subinterface) is required to provide all the methods and fields of one belonging to a superclass (or superinterface), plus possibly some more. The fact that an object from a subclass (or sub-interface) can be used in place of one from a superclass (or super-interface) provides a degree of flexibility that is is extremely handy for organizing complex libraries. For example, a GUI toolkit like Java's Swing framework might define an abstract interface Component that collects together the common fields and methods of all objects having a graphical representation that can be displayed on the screen and that can interact with the user. Examples of such object would include the buttons, checkboxes, and scrollbars of a typical GUI. A method that relies only on this common interface can now be applied to any of these objects. Of course, real object-oriented languages include many other features besides these. For example, fields can be updated. Fields and methods can be declared private. Classes also give code that is used when constructing objects and implementing their methods, and the code in subclasses cooperate with code in superclasses via inheritance. Classes can have static methods and fields, initializers, etc., etc. To keep things simple here, we won't deal with any of these issues — in fact, we won't even talk any more about objects or classes. (There is a lot of discussion in Types and Programming Languages, if you are interested.) Instead, we'll study the core concepts behind the subclass / subinterface relation in the simplified setting of the STLC.

Of course, real OO languages have lots of other features...

• mutable fields
• private
• method inheritance
• static fields
• etc., etc.

We'll ignore all these and focus on core mechanisms.

The Subsumption Rule

Our goal for this chapter is to add subtyping to the simply typed lambda-calculus (with some of the basic extensions from MoreStlc). This involves two steps:

• Defining a binary subtype relation between types.
• Enriching the typing relation to take subtyping into account.

The second step is actually very simple. We add just a single rule to the typing relation: the so-called rule of subsumption:

Γ ⊢ t : S     S <: T	(T_Sub)
Γ ⊢ t : T

This rule says, intuitively, that it is OK to "forget" some of what we know about a term. For example, we may know that t is a record with two fields (e.g., S = {x:A→A, y:B→B}), but choose to forget about one of the fields (T = {y:B→B}) so that we can pass t to a function that requires just a single-field record.

The Subtype Relation

The first step — the definition of the relation S <: T — is where all the action is. Let's look at each of the clauses of its definition.

Structural Rules

To start off, we impose two "structural rules" that are independent of any particular type constructor: a rule of transitivity, which says intuitively that, if S is better than U and U is better than T, then S is better than T...

S <: U    U <: T	(S_Trans)
S <: T

... and a rule of reflexivity, since any type T is always just as good as itself:

	(S_Refl)
T <: T

Products

Now we consider the individual type constructors, one by one, beginning with product types. We consider one pair to be "better than" another if each of its components is.

S1 <: T1    S2 <: T2	(S_Prod)
S1*S2 <: T1*T2

Arrows

Suppose we have two functions f and g with these types:

       f : C -> Student 
       g : (C -> Person) -> D

That is, f is a function that yields a record of type Student, and g is a (higher-order) function that expects its (function) argument to yield a record of type Person. Also suppose, even though we haven't yet discussed subtyping for records, that Student is a subtype of Person. Then the application g f is safe even though their types do not match up precisely, because the only thing g can do with f is to apply it to some argument (of type C); the result will actually be a Student, while g will be expecting a Person, but this is safe because the only thing g can then do is to project out the two fields that it knows about (name and age), and these will certainly be among the fields that are present. This example suggests that the subtyping rule for arrow types should say that two arrow types are in the subtype relation if their results are:

S2 <: T2	(S_Arrow_Co)
S1->S2 <: S1->T2

We can generalize this to allow the arguments of the two arrow types to be in the subtype relation as well:

T1 <: S1    S2 <: T2	(S_Arrow)
S1->S2 <: T1->T2

Notice that the argument types are subtypes "the other way round": in order to conclude that S₁→S₂ to be a subtype of T₁→T₂, it must be the case that T₁ is a subtype of S₁. The arrow constructor is said to be contravariant in its first argument and covariant in its second. Here is an example that illustrates this:

       f : Person -> C
       g : (Student -> C) -> D

The application g f is safe, because the only thing the body of g can do with f is to apply it to some argument of type Student. Since f requires records having (at least) the fields of a Person, this will always work. So Person → C is a subtype of Student → C since Student is a subtype of Person. The intuition is that, if we have a function f of type S₁→S₂, then we know that f accepts elements of type S₁; clearly, f will also accept elements of any subtype T₁ of S₁. The type of f also tells us that it returns elements of type S₂; we can also view these results belonging to any supertype T₂ of S₂. That is, any function f of type S₁→S₂ can also be viewed as having type T₁→T₂.

Exercise: 2 stars (arrow_sub_wrong)

Suppose we had incorrectly defined subtyping as covariant on both the right and the left of arrow types:

S1 <: T1    S2 <: T2	(S_Arrow_wrong)
S1->S2 <: T1->T2

Give a concrete example of functions f and g with types...

       f : Student -> Nat
       g : (Person -> Nat) -> Nat

... such that the application g f will get stuck during execution. ☐

Records

What about subtyping for record types? The basic intuition about subtyping for record types is that it is always safe to use a "bigger" record in place of a "smaller" one. That is, given a record type, adding extra fields will always result in a subtype. If some code is expecting a record with fields x and y, it is perfectly safe for it to receive a record with fields x, y, and z; the z field will simply be ignored. For example,

       {name:String, age:Nat, gpa:Nat} <: {name:String, age:Nat}
       {name:String, age:Nat} <: {name:String}
       {name:String} <: {}

This is known as "width subtyping" for records. We can also create a subtype of a record type by replacing the type of one of its fields with a subtype. If some code is expecting a record with a field x of type T, it will be happy with a record having a field x of type S as long as S is a subtype of T. For example,

       {x:Student} <: {x:Person}

This is known as "depth subtyping". Finally, although the fields of a record type are written in a particular order, the order does not really matter. For example,

       {name:String,age:Nat} <: {age:Nat,name:String}

This is known as "permutation subtyping". We could formalize these requirements in a single subtyping rule for records as follows:

for each jk in j1..jn,
∃ip in i1..im, such that
jk=ip and Sp <: Tk	(S_Rcd)
{i1:S1...im:Sm} <: {j1:T1...jn:Tn}

That is, the record on the left should have all the field labels of the one on the right (and possibly more), while the types of the common fields should be in the subtype relation. However, this rule is rather heavy and hard to read. If we like, we can decompose it into three simpler rules, which can be combined using S_Trans to achieve all the same effects. First, adding fields to the end of a record type gives a subtype:

n > m	(S_RcdWidth)
{i1:T1...in:Tn} <: {i1:T1...im:Tm}

We can use S_RcdWidth to drop later fields of a multi-field record while keeping earlier fields, showing for example that {age:Nat,name:String} <: {name:String}. Second, we can apply subtyping inside the components of a compound record type:

S1 <: T1  ...  Sn <: Tn	(S_RcdDepth)
{i1:S1...in:Sn} <: {i1:T1...in:Tn}

For example, we can use S_RcdDepth and S_RcdWidth together to show that {y:Student, x:Nat} <: {y:Person}. Third, we need to be able to reorder fields. For example, we might expect that {name:String, gpa:Nat, age:Nat} <: Person. We haven't quite achieved this yet: using just S_RcdDepth and S_RcdWidth we can only drop fields from the end of a record type. So we need:

{i1:S1...in:Sn} is a permutation of {i1:T1...in:Tn}	(S_RcdPerm)
{i1:S1...in:Sn} <: {i1:T1...in:Tn}

It is worth noting that full-blown language designs may choose not to adopt all of these subtyping rules. For example, in Java:

• A subclass may not change the argument or result types of a method of its superclass (i.e., no depth subtyping or no arrow subtyping, depending how you look at it).
• Each class has just one superclass ("single inheritance" of classes).
• Each class member (field or method) can be assigned a single index, adding new indices "on the right" as more members are added in subclasses (i.e., no permutation for classes).
• A class may implement multiple interfaces — so-called "multiple inheritance" of interfaces (i.e., permutation is allowed for interfaces).

Top

Finally, it is natural to give the subtype relation a maximal element — a type that lies above every other type and is inhabited by all (well-typed) values. We do this by adding to the language one new type constant, called Top, together with a subtyping rule that places it above every other type in the subtype relation:

	(S_Top)
S <: Top

The Top type is an analog of the Object type in Java and C#.

Summary

In summary, we form the STLC with subtyping by starting with the pure STLC (over some set of base types) and...

• adding a base type Top,
• adding the rule of subsumption

  Γ ⊢ t : S     S <: T	(T_Sub)
  Γ ⊢ t : T

  to the typing relation, and
• defining a subtype relation as follows:

  S <: U    U <: T	(S_Trans)
  S <: T

  	(S_Refl)
  T <: T

  	(S_Top)
  S <: Top

  S1 <: T1    S2 <: T2	(S_Prod)
  S1*S2 <: T1*T2

  T1 <: S1    S2 <: T2	(S_Arrow)
  S1->S2 <: T1->T2

  n > m	(S_RcdWidth)
  {i1:T1...in:Tn} <: {i1:T1...im:Tm}

  S1 <: T1  ...  Sn <: Tn	(S_RcdDepth)
  {i1:S1...in:Sn} <: {i1:T1...in:Tn}

  {i1:S1...in:Sn} is a permutation of {i1:T1...in:Tn}	(S_RcdPerm)
  {i1:S1...in:Sn} <: {i1:T1...in:Tn}

Exercises

Exercise: 1 star, optional (subtype_instances_tf_1)

Suppose we have types S, T, U, and V with S <: T and U <: V. Which of the following subtyping assertions are then true? Write true or false after each one. (A, B, and C here are base types.)

• T→S <: T→S
• Top→U <: S→Top
• (C→C) → (A*B) <: (C→C) → (Top*B)
• T→T→U <: S→S→V
• (T→T)→U <: (S→S)→V
• ((T→S)→T)→U <: ((S→T)→S)→V
• S*V <: T*U

☐

Exercise: 2 stars (subtype_order)

The following types happen to form a linear order with respect to subtyping:

• Top
• Top → Student
• Student → Person
• Student → Top
• Person → Student

Write these types in order from the most specific to the most general. Where does the type Top→Top→Student fit into this order?

Exercise: 1 star (subtype_instances_tf_2)

Which of the following statements are true? Write true or false after each one. ☐

Exercise: 1 star (subtype_concepts_tf)

Which of the following statements are true, and which are false?

• There exists a type that is a supertype of every other type.
• There exists a type that is a subtype of every other type.
• There exists a pair type that is a supertype of every other pair type.
• There exists a pair type that is a subtype of every other pair type.
• There exists an arrow type that is a supertype of every other arrow type.
• There exists an arrow type that is a subtype of every other arrow type.
• There is an infinite descending chain of distinct types in the subtype relation—-that is, an infinite sequence of types S0, S₁, etc., such that all the Si's are different and each S(i+1) is a subtype of Si.
• There is an infinite ascending chain of distinct types in the subtype relation—-that is, an infinite sequence of types S0, S₁, etc., such that all the Si's are different and each S(i+1) is a supertype of Si.

☐

Exercise: 2 stars (proper_subtypes)

Is the following statement true or false? Briefly explain your answer. ☐

Exercise: 2 stars (small_large_1)

• What is the smallest type T ("smallest" in the subtype relation) that makes the following assertion true? (Assume we have Unit among the base types and unit as a constant of this type.)
    empty ⊢ (λp:T*Top. p.fst) ((λz:A.z), unit) : A→A
• What is the largest type T that makes the same assertion true?

☐

Exercise: 2 stars (small_large_2)

• What is the smallest type T that makes the following assertion true?
    empty ⊢ (λp:(A→A * B→B). p) ((λz:A.z), (λz:B.z)) : T
• What is the largest type T that makes the same assertion true?

☐

Exercise: 2 stars, optional (small_large_3)

• What is the smallest type T that makes the following assertion true?
    a:A ⊢ (λp:(A*T). (p.snd) (p.fst)) (a , λz:A.z) : A
• What is the largest type T that makes the same assertion true?

☐

Exercise: 2 stars (small_large_4)

• What is the smallest type T that makes the following assertion true?
    ∃S,
      empty ⊢ (λp:(A*T). (p.snd) (p.fst)) : S
• What is the largest type T that makes the same assertion true?

☐

Exercise: 2 stars (smallest_1)

What is the smallest type T that makes the following assertion true? ☐

Exercise: 2 stars (smallest_2)

What is the smallest type T that makes the following assertion true? ☐

Exercise: 3 stars, optional (count_supertypes)

How many supertypes does the record type {x:A, y:C→C} have? That is, how many different types T are there such that {x:A, y:C→C} <: T? (We consider two types to be different if they are written differently, even if each is a subtype of the other. For example, {x:A,y:B} and {y:B,x:A} are different.) ☐

Exercise: 2 stars (pair_permutation)

The subtyping rule for product types

S1 <: T1    S2 <: T2	(S_Prod)
S1*S2 <: T1*T2

intuitively corresponds to the "depth" subtyping rule for records. Extending the analogy, we might consider adding a "permutation" rule

T1*T2 <: T2*T1

for products. Is this a good idea? Briefly explain why or why not. ☐

Formal Definitions

Most of the definitions — in particular, the syntax and operational semantics of the language — are identical to what we saw in the last chapter. We just need to extend the typing relation with the subsumption rule and add a new Inductive definition for the subtyping relation. Let's first do the identical bits.

Core Definitions

Syntax

For the sake of more interesting examples below, we'll allow an arbitrary set of additional base types like String, Float, etc. We won't bother adding any constants belonging to these types or any operators on them, but we could easily do so. In the rest of the chapter, we formalize just base types, booleans, arrow types, Unit, and Top, omitting record types and leaving product types as an exercise.

Substitution

The definition of substitution remains the same as for the ordinary STLC.

Reduction

Likewise the definitions of the value property and the step relation.

Subtyping

Now we come to the most interesting part. We begin by defining the subtyping relation and developing some of its important technical properties. The definition of subtyping is just what we sketched in the motivating discussion.

Note that we don't need any special rules for base types: they are automatically subtypes of themselves (by S_Refl) and Top (by S_Top), and that's all we want.

Exercise: 2 stars, optional (subtyping_judgements)

(Do this exercise after you have added product types to the language, at least up to this point in the file). Using the encoding of records into pairs, define pair types representing the record types

☐

The following facts are mostly easy to prove in Coq. To get full benefit from the exercises, make sure you also understand how to prove them on paper!

Exercise: 1 star, optional (subtyping_example_1)

☐

Exercise: 1 star, optional (subtyping_example_2)

☐

Typing

The only change to the typing relation is the addition of the rule of subsumption, T_Sub.

Typing examples

Do the following exercises after you have added product types to the language. For each informal typing judgement, write it as a formal statement in Coq and prove it.

Exercise: 1 star, optional (typing_example_0)

☐

Exercise: 2 stars, optional (typing_example_1)

☐

Exercise: 2 stars, optional (typing_example_2)

☐

Properties

The fundamental properties of the system that we want to check are the same as always: progress and preservation. Unlike the extension of the STLC with references, we don't need to change the statements of these properties to take subtyping into account. However, their proofs do become a little bit more involved.

Inversion Lemmas for Subtyping

Before we look at the properties of the typing relation, we need to record a couple of critical structural properties of the subtype relation:

• Bool is the only subtype of Bool
• every subtype of an arrow type is an arrow type.

These are called inversion lemmas because they play the same role in later proofs as the built-in inversion tactic: given a hypothesis that there exists a derivation of some subtyping statement S <: T and some constraints on the shape of S and/or T, each one reasons about what this derivation must look like to tell us something further about the shapes of S and T and the existence of subtype relations between their parts.

Exercise: 2 stars, optional (sub_inversion_Bool)

Exercise: 3 stars, optional (sub_inversion_arrow)

☐

Canonical Forms

We'll see first that the proof of the progress theorem doesn't change too much — we just need one small refinement. When we're considering the case where the term in question is an application t₁ t₂ where both t₁ and t₂ are values, we need to know that t₁ has the form of a lambda-abstraction, so that we can apply the ST_AppAbs reduction rule. In the ordinary STLC, this is obvious: we know that t₁ has a function type T₁₁→T₁₂, and there is only one rule that can be used to give a function type to a value — rule T_Abs — and the form of the conclusion of this rule forces t₁ to be an abstraction. In the STLC with subtyping, this reasoning doesn't quite work because there's another rule that can be used to show that a value has a function type: subsumption. Fortunately, this possibility doesn't change things much: if the last rule used to show Γ ⊢ t₁ : T₁₁→T₁₂ is subsumption, then there is some sub-derivation whose subject is also t₁, and we can reason by induction until we finally bottom out at a use of T_Abs. This bit of reasoning is packaged up in the following lemma, which tells us the possible "canonical forms" (i.e. values) of function type.

Exercise: 3 stars, optional (canonical_forms_of_arrow_types)

☐ Similarly, the canonical forms of type Bool are the constants true and false.

Progress

The proof of progress proceeds like the one for the pure STLC, except that in several places we invoke canonical forms lemmas... Theorem (Progress): For any term t and type T, if empty ⊢ t : T then t is a value or t ⇒ t' for some term t'. Proof: Let t and T be given, with empty ⊢ t : T. Proceed by induction on the typing derivation. The cases for T_Abs, T_Unit, T_True and T_False are immediate because abstractions, unit, true, and false are already values. The T_Var case is vacuous because variables cannot be typed in the empty context. The remaining cases are more interesting:

• If the last step in the typing derivation uses rule T_App, then there are terms t₁ t₂ and types T₁ and T₂ such that t = t₁ t₂, T = T₂, empty ⊢ t₁ : T₁ → T₂, and empty ⊢ t₂ : T₁. Moreover, by the induction hypothesis, either t₁ is a value or it steps, and either t₂ is a value or it steps. There are three possibilities to consider:
  • Suppose t₁ ⇒ t₁' for some term t₁'. Then t₁ t₂ ⇒ t₁' t₂ by ST_App1.
  • Suppose t₁ is a value and t₂ ⇒ t₂' for some term t₂'. Then t₁ t₂ ⇒ t₁ t₂' by rule ST_App2 because t₁ is a value.
  • Finally, suppose t₁ and t₂ are both values. By Lemma canonical_forms_for_arrow_types, we know that t₁ has the form λx:S₁.s₂ for some x, S₁, and s₂. But then (λx:S₁.s₂) t₂ ⇒ [x:=t₂]s₂ by ST_AppAbs, since t₂ is a value.
• If the final step of the derivation uses rule T_If, then there are terms t₁, t₂, and t₃ such that t = if t₁ then t₂ else t₃, with empty ⊢ t₁ : Bool and with empty ⊢ t₂ : T and empty ⊢ t₃ : T. Moreover, by the induction hypothesis, either t₁ is a value or it steps.
  • If t₁ is a value, then by the canonical forms lemma for booleans, either t₁ = true or t₁ = false. In either case, t can step, using rule ST_IfTrue or ST_IfFalse.
  • If t₁ can step, then so can t, by rule ST_If.
• If the final step of the derivation is by T_Sub, then there is a type S such that S <: T and empty ⊢ t : S. The desired result is exactly the induction hypothesis for the typing subderivation.

Inversion Lemmas for Typing

The proof of the preservation theorem also becomes a little more complex with the addition of subtyping. The reason is that, as with the "inversion lemmas for subtyping" above, there are a number of facts about the typing relation that are "obvious from the definition" in the pure STLC (and hence can be obtained directly from the inversion tactic) but that require real proofs in the presence of subtyping because there are multiple ways to derive the same has_type statement. The following "inversion lemma" tells us that, if we have a derivation of some typing statement Γ ⊢ λx:S₁.t₂ : T whose subject is an abstraction, then there must be some subderivation giving a type to the body t₂. Lemma: If Γ ⊢ λx:S₁.t₂ : T, then there is a type S₂ such that Γ, x:S₁ ⊢ t₂ : S₂ and S₁ → S₂ <: T. (Notice that the lemma does not say, "then T itself is an arrow type" — this is tempting, but false!) Proof: Let Γ, x, S₁, t₂ and T be given as described. Proceed by induction on the derivation of Γ ⊢ λx:S₁.t₂ : T. Cases T_Var, T_App, are vacuous as those rules cannot be used to give a type to a syntactic abstraction.

• If the last step of the derivation is a use of T_Abs then there is a type T₁₂ such that T = S₁ → T₁₂ and Γ, x:S₁ ⊢ t₂ : T₁₂. Picking T₁₂ for S₂ gives us what we need: S₁ → T₁₂ <: S₁ → T₁₂ follows from S_Refl.
• If the last step of the derivation is a use of T_Sub then there is a type S such that S <: T and Γ ⊢ λx:S₁.t₂ : S. The IH for the typing subderivation tell us that there is some type S₂ with S₁ → S₂ <: S and Γ, x:S₁ ⊢ t₂ : S₂. Picking type S₂ gives us what we need, since S₁ → S₂ <: T then follows by S_Trans.

Similarly...

The inversion lemmas for typing and for subtyping between arrow types can be packaged up as a useful "combination lemma" telling us exactly what we'll actually require below.

Context Invariance

The context invariance lemma follows the same pattern as in the pure STLC.

Substitution

The substitution lemma is proved along the same lines as for the pure STLC. The only significant change is that there are several places where, instead of the built-in inversion tactic, we need to use the inversion lemmas that we proved above to extract structural information from assumptions about the well-typedness of subterms.

    {a:Nat, b:Nat} ----> {Nat,Nat}       i.e. (Nat,(Nat,Top))
    {c:Nat, a:Nat} ----> {Nat,Top,Nat}   i.e. (Nat,(Top,(Nat,Top)))