TheHoTTGame/Trinitarianism/Quest2.md

Pi Types

We will try to formulate and prove the statement

The sum of two even naturals is even. To do so we must define + on the naturals. Addition takes in two naturals and spits out a natural, so it should have type ℕ → ℕ → ℕ.

_+_ : ℕ → ℕ → ℕ
n + m = ?

Agda supports the notation _+_ (without spaces) which means from now on you can write 0 + 1 and so on (with spaces). Try coming up with a sensible definition. It may not look 'the same' as ours.

Hint

`n + 0` should be `n` and `n + (m + 1)` should be `(n + m) + 1`

Now we can make the statement:

SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
SumOfEven x y = ?

Tip: x .fst is another notation for fst x. This works for all sigma types. There are three ways to interpret this:

• For all even naturals x and for all even naturals y, their sum is even.
• isEven (x .fst + y .fst) is a construction depending on two recipes x and y. Given two recipes x and y of Σ ℕ isEven, we break them down into their first components, apply the conversion _+_, and form a recipe for isEven of the result.
• isEven (_ .fst + _ .fst) is a bundle over the categorical product Σ ℕ isEven × Σ ℕ isEven and SumOfEven is a section of the bundle.

More generally given A : Type and B : A → Type we can form the pi type (x : A) → B x : Type (in other languages Π (x : ℕ), isEven n). The notation suggests that these behave like functions, and indeed in the special case where the fiber is constant with respect to the base space a section is just a term of type A → B, i.e. a function. Hence pi types are also known as dependent function types.

We are now in a position to prove the statement. Have fun!

Important: Once you have proven the statement, check out our two ways of defining addition _+_ and _+'_ (in the solutions).

• Use C-c C-n to check that they compute the same values on different examples.
• Uncomment the code for Sum'OfEven in the solutions. It is just SumOfEven but with +s changed for +'s.
• Load the file. Does the proof still work?

Our proof SumOfEven relied on the explicit definition of _+_, which means if we wanted to use our proof on someone else's definition of addition, it might not work anymore.

But _+_ and _+'_ compute the same values. Are _+_ and _+'_ 'the same'? What is 'the same'?