`Λ_T`

,
which is the smallest set such that
* each type `t` has an infinite set of distinct variables, {x^t}_1,
{x^t}_2, {x^t}_3, ...
* If a term `M` has type σ --> τ, and a term `N` has type
σ, then the application `(M N)` has type τ.
* If a variable `a` has type σ, and term `M` has type τ,
then the abstract `λ a M`

has type σ --> τ.
The definitions of types and of typed terms should be highly familiar
to semanticists, except that instead of writing σ --> τ,
linguists write <σ, τ>. We will use the arrow notation,
since it is more iconic.
Some examples (assume that `x` has type `o`):
x o
\x.x o --> o
((\x.x) x) o
Excercise: write down terms that have the following types:
o --> o --> o
(o --> o) --> o --> o
(o --> o --> o) --> o
#Associativity of types versus terms#
As we have seen many times, in the lambda calculus, function
application is left associative, so that `f x y z == (((f x) y) z)`.
Types, *THEREFORE*, are right associative: if `x`, `y`, and `z`
have types `a`, `b`, and `c`, respectively, then `f` has type
`a --> b --> c --> d == (a --> (b --> (c --> d)))`, where `d` is the
type of the complete term.
It is a serious faux pas to associate to the left for types. You may
as well use your salad fork to stir your tea.
#The simply-typed lambda calculus is strongly normalizing#
If `M` is a term with type τ in Λ_T, then `M` has a
normal form. The proof is not particularly complex, but we will not
present it here; see Berendregt or Hankin.
Since Ω does not have a normal form, it follows that Ω
cannot have a type in Λ_T. We can easily see why:
`Ω = (\x.xx)(\x.xx)`

Assume Ω has type τ, and `\x.xx` has type σ. Then
because `\x.xx` takes an argument of type σ and returns
something of type τ, `\x.xx` must also have type σ -->
τ. By repeating this reasoning, `\x.xx` must also have type
(σ --> τ) --> τ; and so on. Since variables have
finite types, there is no way to choose a type for the variable `x`
that can satisfy all of the requirements imposed on it.
In general, there is no way for a function to have a type that can
take itself for an argument. It follows that there is no way to
define the identity function in such a way that it can take itself as
an argument. Instead, there must be many different identity
functions, one for each type. Some of those types can be functions,
and some of those functions can be (type-restricted) identity
functions; but a simply-types identity function can never apply to itself.
#Typing numerals#
The Church numerals are well behaved with respect to types.
To see this, consider the first three Church numerals (starting with zero):
\s z . z
\s z . s z
\s z . s (s z)
Given the internal structure of the term we are using to represent
zero, its type must have the form ρ --> σ --> σ for
some ρ and σ. This type is consistent with term for one,
but the structure of the definition of one is more restrictive:
because the first argument (`s`) must apply to the second argument
(`z`), the type of the first argument must describe a function from
expressions of type σ to some result type. So we can refine
ρ by replacing it with the more specific type σ --> τ.
At this point, the overall type is (σ --> τ) --> σ -->
σ. Note that this refined type remains compatible with the
definition of zero. Finally, by examinining the definition of two, we
see that expressions of type τ must be suitable to serve as
arguments to functions of type σ --> τ, since the result of
applying `s` to `z` serves as the argument of `s`. The most general
way for that to be true is if τ ≡ σ. So at this
point, we have the overall type of (σ --> σ) --> σ
--> σ.
## Predecessor and lists are not representable in simply typed lambda-calculus ##
As Oleg Kiselyov points out, [[predecessor and lists can't be
represented in the simply-typed lambda
calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].
The reason is that ...
Need to digest the following, which is quoted from Oleg's page:
The predecessor of a Church-encoded numeral, or, generally, the encoding of a list with the car and cdr operations are both impossible in the simply typed lambda-calculus. Henk Barendregt's ``The impact of the lambda-calculus in logic and computer science'' (The Bulletin of Symbolic Logic, v3, N2, June 1997) has the following phrase, on p. 186:
Even for a function as simple as the predecessor lambda definability remained an open problem for a while. From our present knowledge it is tempting to explain this as follows. Although the lambda calculus was conceived as an untyped theory, typeable terms are more intuitive. Now the functions addition and multiplication are defineable by typeable terms, while [101] and [108] have characterized the lambda-defineable functions in the (simply) typed lambda calculus and the predecessor is not among them [the story of the removal of Kleene's four wisdom teeth is skipped...]
Ref 108 is R.Statman: The typed lambda calculus is not elementary recursive. Theoretical Comp. Sci., vol 9 (1979), pp. 73-81.
Since list is a generalization of numeral -- with cons being a successor, append being the addition, tail (aka cdr) being the predecessor -- it follows then the list cannot be encoded in the simply typed lambda-calculus.
To encode both operations, we need either inductive (generally, recursive) types, or System F with its polymorphism. The first approach is the most common. Indeed, the familiar definition of a list
data List a = Nil | Cons a (List a)
gives an (iso-) recursive data type (in Haskell. In ML, it is an inductive data type).
Lists can also be represented in System F. As a matter of fact, we do not need the full System F (where the type reconstruction is not decidable). We merely need the extension of the Hindley-Milner system with higher-ranked types, which requires a modicum of type annotations and yet is able to infer the types of all other terms. This extension is supported in Haskell and OCaml. With such an extension, we can represent a list by its fold, as shown in the code below. It is less known that this representation is faithful: we can implement all list operations, including tail, drop, and even zip.