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1. fmap :: Functor f => (a -> b) -> f a -> f b

   numeric-prelude	NumericPrelude.Base

   fmap is used to apply a function of type (a -> b) to a value of type f a, where f is a functor, to produce a value of type f b. Note that for any type constructor with more than one parameter (e.g., Either), only the last type parameter can be modified with fmap (e.g., b in `Either a b`). Some type constructors with two parameters or more have a Bifunctor instance that allows both the last and the penultimate parameters to be mapped over.

   Examples

   Convert from a Maybe Int to a Maybe String using show:

   >>> fmap show Nothing
   Nothing
   
   >>> fmap show (Just 3)
   Just "3"
   

   Convert from an Either Int Int to an Either Int String using show:

   >>> fmap show (Left 17)
   Left 17
   
   >>> fmap show (Right 17)
   Right "17"
   

   Double each element of a list:

   >>> fmap (*2) [1,2,3]
   [2,4,6]
   

   Apply even to the second element of a pair:

   >>> fmap even (2,2)
   (2,True)
   

   It may seem surprising that the function is only applied to the last element of the tuple compared to the list example above which applies it to every element in the list. To understand, remember that tuples are type constructors with multiple type parameters: a tuple of 3 elements (a,b,c) can also be written (,,) a b c and its Functor instance is defined for Functor ((,,) a b) (i.e., only the third parameter is free to be mapped over with fmap). It explains why fmap can be used with tuples containing values of different types as in the following example:

   >>> fmap even ("hello", 1.0, 4)
   ("hello",1.0,True)
   

2. concatMap :: Foldable t => (a -> [b]) -> t a -> [b]

   numhask	NumHask.Prelude

   Map a function over all the elements of a container and concatenate the resulting lists.

   Examples

   Basic usage:

   >>> concatMap (take 3) [[1..], [10..], [100..], [1000..]]
   [1,2,3,10,11,12,100,101,102,1000,1001,1002]
   

   >>> concatMap (take 3) (Just [1..])
   [1,2,3]
   

3. concatMap :: Foldable t => (a -> [b]) -> t a -> [b]

   numhask	NumHask.Prelude

   Map a function over all the elements of a container and concatenate the resulting lists.

   Examples

   Basic usage:

   >>> concatMap (take 3) [[1..], [10..], [100..], [1000..]]
   [1,2,3,10,11,12,100,101,102,1000,1001,1002]
   

   >>> concatMap (take 3) (Just [1..])
   [1,2,3]
   

4. fmap :: Functor f => (a -> b) -> f a -> f b

   numhask	NumHask.Prelude

   fmap is used to apply a function of type (a -> b) to a value of type f a, where f is a functor, to produce a value of type f b. Note that for any type constructor with more than one parameter (e.g., Either), only the last type parameter can be modified with fmap (e.g., b in `Either a b`). Some type constructors with two parameters or more have a Bifunctor instance that allows both the last and the penultimate parameters to be mapped over.

   Examples

   Convert from a Maybe Int to a Maybe String using show:

   >>> fmap show Nothing
   Nothing
   
   >>> fmap show (Just 3)
   Just "3"
   

   Convert from an Either Int Int to an Either Int String using show:

   >>> fmap show (Left 17)
   Left 17
   
   >>> fmap show (Right 17)
   Right "17"
   

   Double each element of a list:

   >>> fmap (*2) [1,2,3]
   [2,4,6]
   

   Apply even to the second element of a pair:

   >>> fmap even (2,2)
   (2,True)
   

   It may seem surprising that the function is only applied to the last element of the tuple compared to the list example above which applies it to every element in the list. To understand, remember that tuples are type constructors with multiple type parameters: a tuple of 3 elements (a,b,c) can also be written (,,) a b c and its Functor instance is defined for Functor ((,,) a b) (i.e., only the third parameter is free to be mapped over with fmap). It explains why fmap can be used with tuples containing values of different types as in the following example:

   >>> fmap even ("hello", 1.0, 4)
   ("hello",1.0,True)
   

5. foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

   numhask	NumHask.Prelude

   Map each element of the structure into a monoid, and combine the results with (<>). This fold is right-associative and lazy in the accumulator. For strict left-associative folds consider foldMap' instead.

   Examples

   Basic usage:

   >>> foldMap Sum [1, 3, 5]
   Sum {getSum = 9}
   

   >>> foldMap Product [1, 3, 5]
   Product {getProduct = 15}
   

   >>> foldMap (replicate 3) [1, 2, 3]
   [1,1,1,2,2,2,3,3,3]
   

   When a Monoid's (<>) is lazy in its second argument, foldMap can return a result even from an unbounded structure. For example, lazy accumulation enables Data.ByteString.Builder to efficiently serialise large data structures and produce the output incrementally:

   >>> import qualified Data.ByteString.Lazy as L
   
   >>> import qualified Data.ByteString.Builder as B
   
   >>> let bld :: Int -> B.Builder; bld i = B.intDec i <> B.word8 0x20
   
   >>> let lbs = B.toLazyByteString $ foldMap bld [0..]
   
   >>> L.take 64 lbs
   "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24"
   

6. foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

   numhask	NumHask.Prelude

   Map each element of the structure into a monoid, and combine the results with (<>). This fold is right-associative and lazy in the accumulator. For strict left-associative folds consider foldMap' instead.

   Examples

   Basic usage:

   >>> foldMap Sum [1, 3, 5]
   Sum {getSum = 9}
   

   >>> foldMap Product [1, 3, 5]
   Product {getProduct = 15}
   

   >>> foldMap (replicate 3) [1, 2, 3]
   [1,1,1,2,2,2,3,3,3]
   

   When a Monoid's (<>) is lazy in its second argument, foldMap can return a result even from an unbounded structure. For example, lazy accumulation enables Data.ByteString.Builder to efficiently serialise large data structures and produce the output incrementally:

   >>> import qualified Data.ByteString.Lazy as L
   
   >>> import qualified Data.ByteString.Builder as B
   
   >>> let bld :: Int -> B.Builder; bld i = B.intDec i <> B.word8 0x20
   
   >>> let lbs = B.toLazyByteString $ foldMap bld [0..]
   
   >>> L.take 64 lbs
   "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24"
   

7. foldMap' :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

   numhask	NumHask.Prelude

   A left-associative variant of foldMap that is strict in the accumulator. Use this method for strict reduction when partial results are merged via (<>).

   Examples

   Define a Monoid over finite bit strings under xor. Use it to strictly compute the xor of a list of Int values.

   >>> :set -XGeneralizedNewtypeDeriving
   
   >>> import Data.Bits (Bits, FiniteBits, xor, zeroBits)
   
   >>> import Data.Foldable (foldMap')
   
   >>> import Numeric (showHex)
   
   >>> 
   
   >>> newtype X a = X a deriving (Eq, Bounded, Enum, Bits, FiniteBits)
   
   >>> instance Bits a => Semigroup (X a) where X a <> X b = X (a `xor` b)
   
   >>> instance Bits a => Monoid    (X a) where mempty     = X zeroBits
   
   >>> 
   
   >>> let bits :: [Int]; bits = [0xcafe, 0xfeed, 0xdeaf, 0xbeef, 0x5411]
   
   >>> (\ (X a) -> showString "0x" . showHex a $ "") $ foldMap' X bits
   "0x42"
   

8. module Data.Parameterized.FinMap

   FinMap n a conceptually (see NOTE) a map with Fin n keys, implying a maximum size of n. Here's how FinMap compares to other map-like types:

   • FinMap n a is conceptually isomorphic to a Vector n (Maybe a), but can be more space-efficient especially if n is large and the vector is populated with a small number of Just values.
   • FinMap is less general than Map, because it has a fixed key type (Fin n).
   • FinMap n a is similar to IntMap a, but it provides a static guarantee of a maximum size, and its operations (such as size) allow the recovery of more type-level information.
   • FinMap is dissimilar from Data.Parameterized.Map.MapF in that neither the key nor value type of FinMap is parameterized.
   The type parameter n doesn't track the current number of key-value pairs in a FinMap n (i.e., the size of the map), but rather an upper bound. insert and delete don't alter n, whereas incMax does - despite the fact that it has no effect on the keys and values in the FinMap. The FinMap interface has two implementations:
   • The implementation in Data.Parameterized.FinMap.Unsafe is backed by an IntMap, and must have a size of at most maxBound :: Int. This module uses unsafe operations like unsafeCoerce internally for the sake of time and space efficiency.
   • The implementation in Data.Parameterized.FinMap.Safe is backed by an Map (Fin n). All of its functions are implemented using safe operations.
   The implementation in FinMap is property tested against that in FinMap to ensure they have the same behavior. In this documentation, W is used in big-O notations the same way as in the Data.IntMap documentation. NOTE: Where the word "conceptually" is used, it implies that this correspondence is not literally true, but is true modulo some details such as differences between bounded types like Int and unbounded types like Integer. Several of the functions in both implementations are marked INLINE or INLINABLE. There are three reasons for this:
   • Some of these just have very small definitions and so inlining is likely more beneficial than harmful.
   • Some participate in RULES relevant to functions used in their implementations.
   • They are thin wrappers (often just newtype wrappers) around functions marked INLINE, which should therefore also be inlined.

9. data FinMap (n :: Nat) a

   parameterized-utils	Data.Parameterized.FinMap.Safe

   FinMap n a is a map with Fin n keys and a values.

10. data FinMap (n :: Nat) a

    parameterized-utils	Data.Parameterized.FinMap.Unsafe

    FinMap n a is a map with Fin n keys and a values.

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