Added definition, exercise, and solution for Nat to Section 6.1 (#89)

* added definition, exercise, and solution for Nat to Section 6.1

* renamed type and data constructors for Nat introduction

* cleaned up parenthetical remark

Author: Liable5607

src/Solutions/Dependent.idr

@@ -11,29 +11,34 @@ data Vect : (len : Nat) -> Type -> Type where
   (::) : a -> Vect n a -> Vect (S n) a
 
 -- 1
+len : List a -> Nat
+len Nil = Z
+len (_ :: xs) = S (len xs)
+
+-- 2
 head : Vect (S n) a -> a
 head (x :: _) = x
 head Nil impossible
 
--- 2
+-- 3
 tail : Vect (S n) a -> Vect n a
 tail (_ :: xs) = xs
 tail Nil impossible
 
--- 3
+-- 4
 zipWith3 : (a -> b -> c -> d) -> Vect n a -> Vect n b -> Vect n c -> Vect n d
 zipWith3 f []        []        []        = []
 zipWith3 f (x :: xs) (y :: ys) (z :: zs) = f x y z :: zipWith3 f xs ys zs
 
--- 4
+-- 5
 -- Since we only have a `Semigroup` constraint, we can't conjure
 -- a value of type `a` out of nothing in case of an empty list.
 -- We therefore have to return a `Nothing` in case of an empty list.
 foldSemi : Semigroup a => List a -> Maybe a
 foldSemi []        = Nothing
 foldSemi (x :: xs) = Just . maybe x (x <+>) $ foldSemi xs
 
--- 5
+-- 6
 -- the `Nil` case is impossible here, so unlike in Exercise 4,
 -- we don't need to wrap the result in a `Maybe`.
 -- However, we need to pattern match on the tail of the Vect to
@@ -42,24 +47,24 @@ foldSemiVect : Semigroup a => Vect (S n) a -> a
 foldSemiVect (x :: [])         = x
 foldSemiVect (x :: t@(_ :: _)) = x <+> foldSemiVect t
 
--- 6
+-- 7
 iterate : (n : Nat) -> (a -> a) -> a -> Vect n a
 iterate 0     _ _ = Nil
 iterate (S k) f v = v :: iterate k f (f v)
 
--- 7
+-- 8
 generate : (n : Nat) -> (s -> (s,a)) -> s -> Vect n a
 generate 0     _ _ = Nil
 generate (S k) f v =
   let (v', va) = f v
    in va :: generate k f v'
 
--- 8
+-- 9
 fromList : (as : List a) -> Vect (length as) a
 fromList []        = []
 fromList (x :: xs) = x :: fromList xs
 
--- 9
+-- 10
 -- Lookup the type and implementation of functions `maybe` `const` and
 -- try figuring out, what's going on here. An alternative implementation
 -- would of course just pattern match on the argument.

src/Tutorial/Dependent.md

@@ -36,8 +36,31 @@ How could we even describe such a precondition?
 ## Length-Indexed Lists
 
 The answer to the issues described above is of course: Dependent types.
-And the most common introductory example is the *vector*: A list indexed
-by its length:
+Before we proceed to our example, first consider how Idris recursively
+defines the natural numbers (here affixed with apostrophes to avoid introducing
+a conflict with the actual definition of `Nat`, which you can find
+[here](https://github.com/idris-lang/Idris-dev/blob/master/libs/prelude/Prelude/Nat.idr)
+for reference)):
+
+```idris
+data Nat' : Type where
+  Z' : Nat'
+  S' : Nat' -> Nat'
+```
+
+In this scheme, 0 is represented by `Z`, 1 is represented by `S Z`, 2 is
+represented by `S (S Z)`, and so on. Idris does this automatically so if you
+enter `Z` or `S Z` into the REPL, it will return `0` or `1`. Note that the
+only function inherently available to act on a value of type `Nat` is our data
+constructor `S`, which represents the successor function, i.e. adding 1.
+
+Also note that in Idris, every `Nat` can be represented as either a `Z` or
+an `S n` where `n` is another `Nat`. Much as every `List a` can be represented
+as either a `Nil` or an `x :: xs` (where `x` is an `a` and `xs` is a `List a`),
+this informs our pattern matching when solving problems.
+
+Now we can consider the textbook introductory example of dependent types,
+the *vector*, which is a list indexed by its length:
 
 ```idris
 data Vect : (len : Nat) -> (a : Type) -> Type where
@@ -443,7 +466,10 @@ this, which constructor(s) of the type family to use.
 
 ### Exercises part 1
 
-1. Implement function `head` for non-empty vectors:
+1. Implement a function `len : List a -> Nat` for calculating the
+   length of a `List`. For example, `len [1, 1, 1]` produces `3`.
+
+2. Implement function `head` for non-empty vectors:
 
    ```idris
    head : Vect (S n) a -> a
@@ -455,27 +481,27 @@ this, which constructor(s) of the type family to use.
    a `Maybe`! Make sure to add an `impossible` clause for the `Nil`
    case (although this is not strictly necessary here).
 
-2. Using `head` as a reference, declare and implement function `tail`
+3. Using `head` as a reference, declare and implement function `tail`
    for non-empty vectors. The types should reflect that the output
    is exactly one element shorter than the input.
 
-3. Implement `zipWith3`. If possible, try to doing so without looking at
+4. Implement `zipWith3`. If possible, try to doing so without looking at
    the implementation of `zipWith`:
 
    ```idris
    zipWith3 : (a -> b -> c -> d) -> Vect n a -> Vect n b -> Vect n c -> Vect n d
    ```
 
-4. Declare and implement a function `foldSemi`
+5. Declare and implement a function `foldSemi`
    for accumulating the values stored
    in a `List` through `Semigroup`s append operator (`(<+>)`).
    (Make sure to only use a `Semigroup` constraint, as opposed to
    a `Monoid` constraint.)
 
-5. Do the same as in Exercise 4, but for non-empty vectors. How
+6. Do the same as in Exercise 4, but for non-empty vectors. How
    does a vector's non-emptiness affect the output type?
 
-6. Given an initial value of type `a` and a function `a -> a`,
+7. Given an initial value of type `a` and a function `a -> a`,
    we'd like to generate `Vect`s of `a`s, the first value of
    which is `a`, the second value being `f a`, the third
    being `f (f a)` and so on.
@@ -487,7 +513,7 @@ this, which constructor(s) of the type family to use.
    encapsulate this behavior. Get some inspiration from `replicate`
    if you don't know where to start.
 
-7. Given an initial value of a state type `s` and
+8. Given an initial value of a state type `s` and
    a function `fun : s -> (s,a)`,
    we'd like to generate `Vect`s of `a`s. Declare and implement
    function `generate`, which should encapsulate this behavior. Make sure to use
@@ -501,7 +527,7 @@ this, which constructor(s) of the type family to use.
    [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
    ```
 
-8. Implement function `fromList`, which converts a list of
+9. Implement function `fromList`, which converts a list of
    values to a `Vect` of the same length. Use holes if you
    get stuck:
 
@@ -513,7 +539,7 @@ this, which constructor(s) of the type family to use.
    length of the resulting vector by passing the list argument
    to function *length*.
 
-9. Consider the following declarations:
+10. Consider the following declarations:
 
    ```idris
    maybeSize : Maybe a -> Nat