Our job for this chapter: Formalize a small functional language and its type system. Language: The simply typed lambda-calculus (STLC).

• A small subset of Coq's built-in functional language...
• ...but we'll use different informal syntax (to avoid confusion, and for consistency with standard treatments)

Main new technical challenges:

• variable binding
• substitution

Overview

Begin with some set of base types (here, just Bool) Add:

• variables
• function abstractions
• application

Informal concrete syntax:
       t ::= x (variable)
           | \x:T,t (abstraction)
           | t t (application)
           | true (constant true)
           | false (constant false)
           | if t then t else t (conditional)

Some examples:

• \x:Bool, x
  The identity function for booleans.
• (\x:Bool, x) true
  The identity function for booleans, applied to the boolean true.
• \x:Bool, if x then false else true
  The boolean "not" function.
• \x:Bool, true
  The constant function that takes every (boolean) argument to true.

• \x:Bool, \y:Bool, x
  A two-argument function that takes two booleans and returns the first one. (As in Coq, a two-argument function is really a one-argument function whose body is also a one-argument function.)
• (\x:Bool, \y:Bool, x) false true
  A two-argument function that takes two booleans and returns the first one, applied to the booleans false and true.
  As in Coq, application associates to the left -- i.e., this expression is parsed as ((\x:Bool, \y:Bool, x) false) true.
• \f:Bool→Bool, f (f true)
  A higher-order function that takes a function f (from booleans to booleans) as an argument, applies f to true, and applies f again to the result.
• (\f:Bool→Bool, f (f true)) (\x:Bool, false)
  The same higher-order function, applied to the constantly false function.

Note that all functions are anonymous. We'll see how to add named function declarations as "syntactic sugar" in the next chapter.

The types of the STLC include the base type Bool for boolean values and arrow types for functions.
      T ::= Bool
          | T → T For example:

• \x:Bool, false has type Bool→Bool
• \x:Bool, x has type Bool→Bool
• (\x:Bool, x) true has type Bool
• \x:Bool, \y:Bool, x has type Bool→Bool→Bool (i.e., Bool → (Bool→Bool))
• (\x:Bool, \y:Bool, x) false has type Bool→Bool
• (\x:Bool, \y:Bool, x) false true has type Bool

Syntax

We next formalize the syntax of the STLC.

Types

Terms

Now we need some notation magic to set up the concrete syntax, as we did in the Imp chapter...

Note that an abstraction \x:T,t (formally, tm_abs x T t) is always annotated with the type T of its parameter, in contrast to Coq (and other functional languages like ML, Haskell, etc.), which use type inference to fill in missing annotations. We're not considering type inference here.

Some examples...

Operational Semantics

To define the small-step semantics of STLC terms...

• We begin by defining the set of values.
• Next, we define free variables and substitution. These are used in the reduction rule for application expressions.
• Finally, we give the small-step relation itself.

Values

To define the values of the STLC, we have a few cases to consider. First, for the boolean part of the language, the situation is clear: true and false are the only values. An if expression is never a value.

Second, an application is not a value: it represents a function being invoked on some argument, which clearly still has work left to do.

Third, for abstractions, we have a choice:

• We can say that \x:T, t is a value only when t is a value -- i.e., only if the function's body has been reduced (as much as it can be without knowing what argument it is going to be applied to).
• Or we can say that \x:T, t is always a value, no matter whether t is one or not -- in other words, we can say that reduction stops at abstractions.

Our usual way of evaluating expressions in Gallina makes the first choice -- for example,
         Compute (fun x:bool ⇒ 3 + 4) yields:
         fun x:bool ⇒ 7 Most real-world functional programming languages make the second choice -- reduction of a function's body only begins when the function is actually applied to an argument. We also make the second choice here.

STLC Programs

Finally, we must consider what constitutes a complete program. Intuitively, a "complete program" must not refer to any undefined variables. We'll see shortly how to define the free variables in a STLC term. A complete program is closed -- that is, it contains no free variables. (Conversely, a term with free variables is often called an open term.)

Having made the choice not to reduce under abstractions, we don't need to worry about whether variables are values, since we'll always be reducing programs "from the outside in," and that means the step relation will always be working with closed terms.

Substitution

Now we come to the heart of the STLC: the operation of substituting one term for a variable in another term. This operation is used below to define the operational semantics of function application, where we will need to substitute the argument term for the function parameter in the function's body. For example, we reduce
       (\x:Bool, if x then true else x) false to
       if false then true else false by substituting false for the parameter x in the body of the function. In general, we need to be able to substitute some given term s for occurrences of some variable x in another term t. In informal discussions, this is usually written [x:=s]t and pronounced "substitute s for x in t."

Here are some examples:

• [x:=true] (if x then x else false) yields if true then true else false
• [x:=true] x yields true
• [x:=true] (if x then x else y) yields if true then true else y
• [x:=true] y yields y
• [x:=true] false yields false (vacuous substitution)
• [x:=true] (\y:Bool, if y then x else false) yields \y:Bool, if y then true else false
• [x:=true] (\y:Bool, x) yields \y:Bool, true
• [x:=true] (\y:Bool, y) yields \y:Bool, y
• [x:=true] (\x:Bool, x) yields \x:Bool, x

The last example is very important: substituting x with true in \x:Bool, x does not yield \x:Bool, true! The reason for this is that the x in the body of \x:Bool, x is bound by the abstraction: it is a new, local name that just happens to be spelled the same as some global name x.

Here is the definition, informally...
       [x:=s]x = s
       [x:=s]y = y if x ≠ y
       [x:=s](\x:T, t) = \x:T, t
       [x:=s](\y:T, t) = \y:T, [x:=s]t if x ≠ y
       [x:=s](t₁ t₂) = ([x:=s]t₁) ([x:=s]t₂)
       [x:=s]true = true
       [x:=s]false = false
       [x:=s](if t₁ then t₂ else t₃) =
              if [x:=s]t₁ then [x:=s]t₂ else [x:=s]t₃

... and formally:

Technical note: Substitution becomes trickier to define if we consider the case where s, the term being substituted for a variable in some other term, may itself contain free variables. Since we are only interested here in defining the step relation on closed terms (i.e., terms like \x:Bool, x that include binders for all of the variables they mention), we can sidestep this extra complexity, but it must be dealt with when formalizing richer languages.

For example, using the definition of substitution above to substitute the open term s = \x:Bool, r, where r is a free reference to some global resource, for the variable z in the term t = \r:Bool, z, where r is a bound variable, we would get \r:Bool, \x:Bool, r, where the free reference to r in s has been "captured" by the binder at the beginning of t. Why would this be bad? Because it violates the principle that the names of bound variables do not matter. For example, if we rename the bound variable in t, e.g., let t' = \w:Bool, z, then [x:=s]t' is \w:Bool, \x:Bool, r, which does not behave the same as [x:=s]t = \r:Bool, \x:Bool, r. That is, renaming a bound variable changes how t behaves under substitution.

Reduction

value v2	(ST_AppAbs)
(\x:T2,t1) v2 --> [x:=v2]t1

t1 --> t1'	(ST_App1)
t1 t2 --> t1' t2

value v1
t2 --> t2'	(ST_App2)
v1 t2 --> v1 t2'

(plus the usual rules for conditionals). This is call by value reduction: to reduce an application (t₁ t₂),

• first reduce t₁ to a value (a function)
• then reduce the argument t₂ to a value
• then reduce the application itself by substituting t₂ for the bound variable x in the body t₁.

The ST_AppAbs rule is often called beta-reduction.

Typing

Next we consider the typing relation of the STLC.

Contexts

Question: What is the type of the term "x y"? Answer: It depends on the types of x and y! I.e., in order to assign a type to a term, we need to know what assumptions we should make about the types of its free variables. This leads us to a three-place typing judgment, informally written Gamma ⊢ t \in T, where Gamma is a "typing context" -- a mapping from variables to their types. Following the usual notation for partial maps, we write (X ⊢> T, Gamma) for "update the partial function Gamma so that it maps x to T."

Typing Relation

Gamma x = T1	(T_Var)
Gamma ⊢ x ∈ T1

x ⊢> T2 ; Gamma ⊢ t1 ∈ T1	(T_Abs)
Gamma ⊢ \x:T2,t1 ∈ T2->T1

Gamma ⊢ t1 ∈ T2->T1
Gamma ⊢ t2 ∈ T2	(T_App)
Gamma ⊢ t1 t2 ∈ T1

	(T_True)
Gamma ⊢ true ∈ Bool

	(T_False)
Gamma ⊢ false ∈ Bool

Gamma ⊢ t1 ∈ Bool    Gamma ⊢ t2 ∈ T1    Gamma ⊢ t3 ∈ T1	(T_If)
Gamma ⊢ if t1 then t2 else t3 ∈ T1

We can read the three-place relation Gamma ⊢ t \in T as: "under the assumptions in Gamma, the term t has the type T."

Examples

Note that, since we added the has_type constructors to the hints database, auto can actually solve this one immediately.

More examples:
       empty ⊢ \x:A, \y:A→A, y (y x)
             \in A → (A→A) → A.

       empty ⊢ \x:Bool→B, \y:Bool→Bool, \z:Bool,
                   y (x z)
             \in T.

We can also show that some terms are not typable. For example, let's check that there is no typing derivation assigning a type to the term \x:Bool, \y:Bool, x y -- i.e.,
    ¬∃ T,
        empty ⊢ \x:Bool, \y:Bool, x y \in T.

Another nonexample:
    ¬(∃ S T,
          empty ⊢ \x:S, x x \in T).