Terms and Types

There are three ways of looking at A : Type.

proof theoretically, 'A is a proposition'
type theoretically, 'A is a construction'
categorically, 'A is an object in category Type'

A first example of a type construction is the function type. Given types A : Type and B : Type, we have another type A → B : Type which can be seen as

the proposition 'A implies B'
the construction 'ways to convert A recipes to B recipes'
internal hom of the category Type

To give examples of this, let's make some types first! Here is how we define 'true'.

data ⊤ : Type where
  tt : ⊤

It reads '⊤ is an inductive type with a constructor tt', with three interpretations

⊤ is a proposition and there is a proof of it, called tt.
⊤ is a construction with a recipe called tt
⊤ is a terminal object: every object has a morphism into ⊤ given by · ↦ tt

In general, the expression a : A is read 'a is a term of type A', and has three interpretations,

a is a proof of the proposition A
a is a recipe for the construction A
a is a generalised element of the object A in the category Type.

The above tells you how we make a term of type ⊤. Let's see an example of using a term of type ⊤:

TrueToTrue : ⊤ → ⊤
TrueToTrue = {!!}

enter C-c C-l (this means Ctrl-c Ctrl-l) to load the document, and now you can fill the hole { }
navigate to the hole { } using C-c C-f (forward) or C-c C-b (backward)
enter C-c C-r and agda will try to help you (r stands for refine)
you should see λ x → { }. This is agda's notation for x ↦ { } and is called λ abstraction, λ for 'let'.
navigate to the new hole
enter C-c C-, to check the goal (C-c C-comma)
the Goal area ('agda information' window) should look like

Goal: ⊤
—————————————————————————
x : ⊤

you have a proof/recipe/generalized element x : ⊤ and you need to give a proof/recipe/generalized element of ⊤
you can give it a proof/recipe/generalized element of ⊤ and press C-c C-SPC to fill the hole (SPC means the space button)

There is more than one proof (see solutions) - are they 'the same'? What is 'the same'?

Here is an important solution:

TrueToTrue' : ⊤ → ⊤
TrueToTrue' x = {!!}

Naviagate to the hole and check the goal.
Note x is already taken out for you.
You can try type x in the hole and C-c C-c
c stands for 'cases'. Doing C-c C-c with x in the hole tells agda to 'do cases on x'. The only case is tt.

Built into the definition of ⊤ is agda's way of making a map out of ⊤ into another type A, which we have just used. It says 'to map out of ⊤ it suffices to do the case when x is tt', or

the only proof of ⊤ is tt
the only recipe for ⊤ is tt
the only one generalized element tt in ⊤

Let's define another type.


data ⊥ : Type u where

It reads '⊥ is an inductive type with no constructors', with three interepretations

⊥ is a proposition with no proofs
⊥ is a construction with no recipes
There are no generalized elements of ⊥ (it is a strict initial object)

Let's try mapping out of ⊥.

explosion : ⊥ → ⊤
explosion x = {!!}

Navigate to the hole and do cases on x.

Agda knows that there are no cases so there is nothing to do! This has three interpretations:

false implies anything (principle of explosion)
One can convert recipes of ⊥ to recipes of any other construction since there are no recipes of ⊥.
⊥ is initial in the category Type

We can also encode "natural numbers" as a type.

data ℕ : Type where
  zero : ℕ
  suc : ℕ → ℕ

As a construction, this reads :

ℕ is a type of construction
zero is a recipe for ℕ
suc takes an existing recipe for ℕ and gives another recipe for ℕ.

We can also see ℕ categorically : ℕ is a natural numbers object in the category Type. This means it is equipped with morphisms zero : ⊤ → ℕ and suc : ℕ → ℕ such that given any ⊤ → A → A there exist a unique morphism ℕ → A such that the diagram commutes:

ℕ has no interpretation as a proposition since there are 'too many proofs' - mathematicians classically don't distinguish between proofs of a single proposition. (ZFC doesn't even mention logic internally, unlike Type Theory!)

To see how to use terms of type ℕ, i.e. induction on ℕ, go to Quest1!

Universes

You may have noticed the notational similarities between zero : ℕ and ℕ : Type. This may have lead you to the question, Type : ?. In type theory, we simply assert Type : Type₁. But then we are chasing our tail, asking Type₁ : Type₂. Type theorists make sure that every type (i.e. anything the right side of :) itself is a term (i.e. anything on the left of :), and every term has a type. So what we really need is

Type : Type₁, Type₁ : Type₂, Type₂ : Type₃, ⋯

These are called universes. The numberings of universes are called levels. We will start using universes in the next quest.

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Terms and Types

Universes

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