TheHoTTGame/Trinitarianism/Quest1.md

Dependent Types

In a 'place to do maths' we would like to be able to express and 'prove' the statement

There exists a natural that is even.

This requires the notion of a predicate. In general a predicate on a type A : Type is a term of type A → Type. For example,

isEven : ℕ → Type
isEven n = ? 

• Do C-c C-l to load the file.

• Navigate to the hole.

• Input n in the hole and do C-c C-c. You should now see

  
  isEven : ℕ → Type
  isEven zero = {!!}
  isEven (suc n) = {!!}
  
  

  Explanation : 'to define a function on ℕ, it suffices to define the function on the _cases, zero and suc n, since these are the only constructors given in the definition of ℕ'. This has the following interpretations,

  • propositionally, this is the principle of mathematical induction.
  • categorically, this is the universal property of a natural numbers object.

• Navigate to the first hole and check the goal. You should see

  Goal: Type
  ———————————
  

  Fill the hole with ⊤, since we want zero to be even.

• Navigate to the second hole.

• Input n and do C-c C-c again. You should now see

  isEven : ℕ → Type
  isEven zero = ⊤
  isEven (suc zero) = {!!}
  isEven (suc (suc n)) = {!!} 
  

  Explanation : we have just used induction again.

• Navigate to the first hole and check the goal. Agda should be asking for a term of type Type, so fill the hole with ⊥, since we don't want suc zero to be even.

• Navigate to the next hole and check the goal. You should see in the 'agda information' window,

  Goal: Type
  ——————————————
  n : ℕ
  

  Explanation : We are in the 'inductive step', so we have access to the previous natural number.

• Fill the hole with isEven n, since we want suc (suc n) to be even precisely when n is even.

  The reason we have access to the term isEven n is again because we are in the 'inductive step'.

• There should now be nothing in the 'agda info' window. Everything is working!

There are three interpretations of isEven : ℕ → Type.

• Already mentioned, isEven is a predicate on ℕ.
• isEven is a dependent construction. Specifically, it is either ⊤ or ⊥ depending on n : ℕ.
• isEven is a bundle over ℕ, i.e. an object in the over-category Type↓ℕ. Pictorially, it looks like (insert picture).