Quest 2 - Sigma Types

We are still trying to express and “prove” the statement

The statement

There exists a natural that is even.

We will achieve this by the end of this quest.

Part 0 - Existence / Dependent Pair / Total Space of Bundles

Recall from Quest 1 - Dependent Types that we defined isEven. What’s left is to be able write down “existence”. In maths we might write

∃ x ∈ ℕ, isEven x

which in agda notation is

Σ  isEven

This is called a sigma type, which has three interpretations:

  • the proposition “there exists an even natural”

  • the construction “keep a recipe n of naturals together with a recipe of isEven n

  • the total space of the bundle isEven over , which is the space obtained by putting together all the fibers. Pictorially, it looks like

    SigmaTypeOfIsEven

    which can also be viewed as the subset of even naturals, since the fibers are either empty or singleton. (It is a subsingleton bundle).

Part 1 - Making terms in Sigma Types

Making a term of this type has three interpretations:

  • (giving a proof that there existence of an even natural amounts to giving) a natural n : and a proof hn : isEven n that n is even.

  • pairing a recipe n : with a recipe hn : isEven n.

  • (giving a point in the total space is giving) a point n : downstairs together with a point hn : isEven n in its fiber.

Now you can prove that there exists an even natural:

  • Formulate the statement you need. Make sure you have it of the form

    Name : Statement
    Name = {!!}
    
  • Load the file, go to the hole and refine the goal.

  • If you formulated the statement right it should split into {!!} , {!!} and you can check the types of terms the holes require.

  • Fill the holes. There are many proofs you can do!

In general when A : Type is a type and B : A Type is a predicate/dependent construction/bundle over A, we can write the sigma type Σ A B whose terms are pairs a , b where a : A and b : B a. In the special case when B is not dependent on a : A, i.e. it looks like λ a C for some C : Type then Σ A B is just

  • the proposition “A and C” since giving a proof of this is the same as giving a proof of A and a proof of C

  • a recipe a : A together with a recipe c : C

  • B is now a trivial bundle since the fibers B a are constant with respect to a : A. In other words it is just a product Σ A B A × C. For this reason, some refer to the sigma type as the dependent product, but we will avoid this terminology.

_×_ : Type  Type  Type
A × C = Σ A (λ a  C)

agda supports the notation _×_ (without spaces) which means from now on you can write A × C (with spaces).

Part 2 - Using Terms in Sigma Types

There are two ways of using a term in a sigma type. We can extract the first part using fst or the second part using snd. Given x : Σ A B there are three interpretations of fst and snd:

  • Viewing x as a proof of existence fst x provides the witness of existence and snd provides the proof that the witness fst x has the desired property

  • Viewing x as a recipe fst extracts the first component and snd extracts the second component

  • Viewing x as a point in the total space of a bundle fst x is the point that x is over in the base space and snd x is the point in the fiber that x represents. In particular you can interpret fst as projection from the total space to the base space, collapsing fibers.

For example to define a map that takes an even natural and divides it by two we can do

div2 : Σ  isEven  
div2 x = {!!}
  • Load the file, go to the hole and case on x. You might want to rename fst₁ and snd₁.

    div2 : Σ  isEven  
    div2 (fst₁ , snd₁) = {!!}
    
  • Case on fst₁ and tell agda what to give for 0 , *, i.e. what “zero divided by two” ought to be.

    div2 : Σ  isEven  
    div2 (zero , snd₁) = {!!}
    div2 (suc fst₁ , snd₁) = {!!}
    
  • Navigate to the second hole and case on fst₁ again. Notice that agda knows there is no term looking like 1 , * so it has skipped that case for us.

    div2 : Σ  isEven  
    div2 (zero , snd₁) = 0
    div2 (suc (suc fst₁) , snd₁) = {!!}
    
  • (n + 2) / 2 should just be n/2 + 1 so try writing in suc and refining the goal

  • How do you write down n/2? Hint: we are in the “inductive step”.

    Try dividing some terms by 2:

  • Use C-c C-n and write div2 (2 , tt) for example.

  • Try dividing 36 by 2.

Important observation : the two proofs 2 , tt and 36 , tt of the statement “there exists an even natural” are not “the same” in any sense, since if they were div2 (2 , tt) would be “the same” div2 (36/2 , tt), and hence 1 would be “the same” as 18.

“The same”

Are they “the same”? What is “the same”?