104 lines
3.3 KiB
Markdown
104 lines
3.3 KiB
Markdown
# Pi Types
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We will try to formulate and prove the statement
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> The sum of two even naturals is even.
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## Defining Addition
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To do so we must define `+` on the naturals.
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Addition takes in two naturals and spits out a natural,
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so it should have type `ℕ → ℕ → ℕ`.
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```agda
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_+_ : ℕ → ℕ → ℕ
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n + m = ?
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```
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Try coming up with a sensible definition.
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It may not look 'the same' as ours.
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<p>
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<details>
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<summary>Hint</summary>
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`n + 0` should be `n` and `n + (m + 1)` should be `(n + m) + 1`
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</details>
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</p>
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## The Statement
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Now we can make the statement:
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```agda
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SumOfEven : (x : Σ ℕ isEven) → (y : Σ ℕ isEven) → isEven (x .fst + y .fst)
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SumOfEven x y = ?
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```
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> Tip: `x .fst` is another notation for `fst x`.
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> This works for all sigma types.
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There are three ways to interpret this:
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- For all even naturals `x` and `y`,
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their sum is even.
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- `isEven (x .fst + y .fst)` is a construction depending on two recipes
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`x` and `y`.
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Given two recipes `x` and `y` of `Σ ℕ isEven`,
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we break them down into their first components,
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apply the conversion `_+_`,
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and form a recipe for `isEven` of the result.
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- `isEven (_ .fst + _ .fst)` is a bundle over the categorical product
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`Σ ℕ isEven × Σ ℕ isEven` and `SumOfEven` is a _section_ of the bundle.
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More generally given `A : Type` and `B : A → Type`
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we can form the _pi type_ `(x : A) → B x : Type`
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(in other languages `Π (x : ℕ), isEven n`).
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The notation suggests that these behave like functions,
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and indeed in the special case where the fiber is constant
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with respect to the base space
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a section is just a term of type `A → B`, i.e. a function.
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Hence pi types are also known as _dependent function types_.
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We are now in a position to prove the statement. Have fun!
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## Remarks
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_Important_: Once you have proven the statement,
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check out our two ways of defining addition `_+_` and `_+'_`
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(in the solutions).
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- Use `C-c C-n` to check that they compute the same values
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on different examples.
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- Uncomment the code for `Sum'OfEven` in the solutions.
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It is just `SumOfEven` but with `+`s changed for `+'`s.
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- Load the file. Does the proof still work?
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Our proof `SumOfEven` relied on
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the explicit definition of `_+_`,
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which means if we wanted to use our proof on
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someone else's definition of addition,
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it might not work anymore.
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> But `_+_` and `_+'_` compute the same values.
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> Are `_+_` and `_+'_` 'the same'? What is 'the same'?
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## Another Task : Decidability of `isEven`
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As the final task of the Quest,
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try to express and prove in agda the statement
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> For any natural number it is even or is is not even.
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We will make a summary of what is needed:
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- a definition of the type `A ⊕ B` (input `\oplus`),
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which has three interpretations
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- the proposition '`A` or `B`'
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- the construction with two ways of making recipes
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`left : A → A ⊕ B`
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and `right : B → A ⊕ B`.
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- the coproduct of two objects `A` and `B`.
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The type needs to take in parameters `A : Type` and `B : Type`
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```agda
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data _⊕_ (A : Type) (B : Type) : Type where
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???
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```
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- a definition of negation. One can motivate it by the following
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- Define `A ↔ B : Type` for two types `A : Type` and `B : Type`.
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- Show that for any `A : Type` we have `(A ↔ ⊥) ↔ (A → ⊥)`
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- Define `¬ : Type → Type` to be `λ A → (A → ⊥)`.
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- a formulation and proof of the statement above
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