-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/

-- | A Haskell-embedded DSL for monitoring hard real-time
--   distributed systems.
--   
--   The concrete syntax for Copilot.
--   
--   Copilot is a stream (i.e., infinite lists) domain-specific language
--   (DSL) in Haskell that compiles into embedded C. Copilot contains an
--   interpreter, multiple back-end compilers, and other verification
--   tools.
--   
--   A tutorial, examples, and other information are available at
--   https://copilot-language.github.io.
@package copilot-language
@version 4.7.1

-- | Reexports Prelude from package "base" hiding identifiers
--   redefined by Copilot.
module Copilot.Language.Prelude

-- | A fixed-precision integer type with at least the range [-2^29 ..
--   2^29-1]. The exact range for a given implementation can be
--   determined by using minBound and maxBound from the
--   Bounded class.
data Int

-- | Single-precision floating point numbers. It is desirable that this
--   type be at least equal in range and precision to the IEEE
--   single-precision type.
data Float

-- | The character type Char represents Unicode codespace and its
--   elements are code points as in definitions D9 and D10 of the
--   Unicode Standard.
--   
--   Character literals in Haskell are single-quoted: 'Q',
--   'Я' or 'Ω'. To represent a single quote itself use
--   '\'', and to represent a backslash use '\\'. The
--   full grammar can be found in the section 2.6 of the Haskell 2010
--   Language Report.
--   
--   To specify a character by its code point one can use decimal,
--   hexadecimal or octal notation: '\65', '\x41' and
--   '\o101' are all alternative forms of 'A'. The
--   largest code point is '\x10ffff'.
--   
--   There is a special escape syntax for ASCII control characters:
--   
--   TODO: table
--   
--   Data.Char provides utilities to work with Char.
data Char

-- | The Maybe type encapsulates an optional value. A value of type
--   Maybe a either contains a value of type a
--   (represented as Just a), or it is empty (represented
--   as Nothing). Using Maybe is a good way to deal with
--   errors or exceptional cases without resorting to drastic measures such
--   as error.
--   
--   The Maybe type is also a monad. It is a simple kind of error
--   monad, where all errors are represented by Nothing. A richer
--   error monad can be built using the Either type.
data Maybe a
Nothing :: Maybe a
Just :: a -> Maybe a

-- | A value of type IO a is a computation which, when
--   performed, does some I/O before returning a value of type a.
--   
--   There is really only one way to "perform" an I/O action: bind it to
--   Main.main in your program. When your program is run, the I/O
--   will be performed. It isn't possible to perform I/O from an arbitrary
--   function, unless that function is itself in the IO monad and
--   called at some point, directly or indirectly, from Main.main.
--   
--   IO is a monad, so IO actions can be combined using
--   either the do-notation or the >> and >>=
--   operations from the Monad class.
data IO a
data Bool
False :: Bool
True :: Bool

-- | Double-precision floating point numbers. It is desirable that this
--   type be at least equal in range and precision to the IEEE
--   double-precision type.
data Double

-- | A Word is an unsigned integral type, with the same size as
--   Int.
data Word
data Ordering
LT :: Ordering
EQ :: Ordering
GT :: Ordering

-- | Lifted, homogeneous equality. By lifted, we mean that it can be bogus
--   (deferred type error). By homogeneous, the two types a and
--   b must have the same kinds.
class a ~# b => (a :: k) ~ (b :: k)
infix 4 ~

-- | Arbitrary precision integers. In contrast with fixed-size integral
--   types such as Int, the Integer type represents the
--   entire infinite range of integers.
--   
--   Integers are stored in a kind of sign-magnitude form, hence do not
--   expect two's complement form when using bit operations.
--   
--   If the value is small (i.e., fits into an Int), the IS
--   constructor is used. Otherwise IP and IN constructors
--   are used to store a BigNat representing the positive or the
--   negative value magnitude, respectively.
--   
--   Invariant: IP and IN are used iff the value does not fit
--   in IS.
data Integer

-- | The Foldable class represents data structures that can be reduced to a
--   summary value one element at a time. Strict left-associative folds are
--   a good fit for space-efficient reduction, while lazy right-associative
--   folds are a good fit for corecursive iteration, or for folds that
--   short-circuit after processing an initial subsequence of the
--   structure's elements.
--   
--   Instances can be derived automatically by enabling the
--   DeriveFoldable extension. For example, a derived instance for
--   a binary tree might be:
--   
--   
--   {-# LANGUAGE DeriveFoldable #-}
--   data Tree a = Empty
--               | Leaf a
--               | Node (Tree a) a (Tree a)
--       deriving Foldable
--   
--   
--   A more detailed description can be found in the Overview
--   section of Data.Foldable#overview.
--   
--   For the class laws see the Laws section of
--   Data.Foldable#laws.
class Foldable (t :: Type -> Type)

-- | 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"
--   
foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

-- | Right-associative fold of a structure, lazy in the accumulator.
--   
--   In the case of lists, foldr, when applied to a binary operator,
--   a starting value (typically the right-identity of the operator), and a
--   list, reduces the list using the binary operator, from right to left:
--   
--   
--   foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
--   
--   
--   Note that since the head of the resulting expression is produced by an
--   application of the operator to the first element of the list, given an
--   operator lazy in its right argument, foldr can produce a
--   terminating expression from an unbounded list.
--   
--   For a general Foldable structure this should be semantically
--   identical to,
--   
--   
--   foldr f z = foldr f z . toList
--   
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> foldr (||) False [False, True, False]
--   True
--   
--   
--   
--   >>> foldr (||) False []
--   False
--   
--   
--   
--   >>> foldr (\c acc -> acc ++ [c]) "foo" ['a', 'b', 'c', 'd']
--   "foodcba"
--   
--   
--   
Infinite structures
--   
--   ⚠️ Applying foldr to infinite structures usually doesn't
--   terminate.
--   
--   It may still terminate under one of the following conditions:
--   
--   
--   
- the folding function is short-circuiting
--   
- the folding function is lazy on its second argument
--   
--   
--   
Short-circuiting
--   
--   (||) short-circuits on True values, so the
--   following terminates because there is a True value finitely far
--   from the left side:
--   
--   
--   >>> foldr (||) False (True : repeat False)
--   True
--   
--   
--   But the following doesn't terminate:
--   
--   
--   >>> foldr (||) False (repeat False ++ [True])
--   * Hangs forever *
--   
--   
--   
Laziness in the second argument
--   
--   Applying foldr to infinite structures terminates when the
--   operator is lazy in its second argument (the initial accumulator is
--   never used in this case, and so could be left undefined, but
--   [] is more clear):
--   
--   
--   >>> take 5 $ foldr (\i acc -> i : fmap (+3) acc) [] (repeat 1)
--   [1,4,7,10,13]
--   
foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b

-- | Left-associative fold of a structure, lazy in the accumulator. This is
--   rarely what you want, but can work well for structures with efficient
--   right-to-left sequencing and an operator that is lazy in its left
--   argument.
--   
--   In the case of lists, foldl, when applied to a binary operator,
--   a starting value (typically the left-identity of the operator), and a
--   list, reduces the list using the binary operator, from left to right:
--   
--   
--   foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
--   
--   
--   Note that to produce the outermost application of the operator the
--   entire input list must be traversed. Like all left-associative folds,
--   foldl will diverge if given an infinite list.
--   
--   If you want an efficient strict left-fold, you probably want to use
--   foldl' instead of foldl. The reason for this is that the
--   latter does not force the inner results (e.g. z `f` x1
--   in the above example) before applying them to the operator (e.g. to
--   (`f` x2)). This results in a thunk chain O(n) elements
--   long, which then must be evaluated from the outside-in.
--   
--   For a general Foldable structure this should be semantically
--   identical to:
--   
--   
--   foldl f z = foldl f z . toList
--   
--   
--   
Examples
--   
--   The first example is a strict fold, which in practice is best
--   performed with foldl'.
--   
--   
--   >>> foldl (+) 42 [1,2,3,4]
--   52
--   
--   
--   Though the result below is lazy, the input is reversed before
--   prepending it to the initial accumulator, so corecursion begins only
--   after traversing the entire input string.
--   
--   
--   >>> foldl (\acc c -> c : acc) "abcd" "efgh"
--   "hgfeabcd"
--   
--   
--   A left fold of a structure that is infinite on the right cannot
--   terminate, even when for any finite input the fold just returns the
--   initial accumulator:
--   
--   
--   >>> foldl (\a _ -> a) 0 $ repeat 1
--   * Hangs forever *
--   
--   
--   WARNING: When it comes to lists, you always want to use either
--   foldl' or foldr instead.
foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b

-- | Left-associative fold of a structure but with strict application of
--   the operator.
--   
--   This ensures that each step of the fold is forced to Weak Head Normal
--   Form before being applied, avoiding the collection of thunks that
--   would otherwise occur. This is often what you want to strictly reduce
--   a finite structure to a single strict result (e.g. sum).
--   
--   For a general Foldable structure this should be semantically
--   identical to,
--   
--   
--   foldl' f z = foldl' f z . toList
--   
foldl' :: Foldable t => (b -> a -> b) -> b -> t a -> b

-- | A variant of foldr that has no base case, and thus may only be
--   applied to non-empty structures.
--   
--   This function is non-total and will raise a runtime exception if the
--   structure happens to be empty.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> foldr1 (+) [1..4]
--   10
--   
--   
--   
--   >>> foldr1 (+) []
--   Exception: Prelude.foldr1: empty list
--   
--   
--   
--   >>> foldr1 (+) Nothing
--   *** Exception: foldr1: empty structure
--   
--   
--   
--   >>> foldr1 (-) [1..4]
--   -2
--   
--   
--   
--   >>> foldr1 (&&) [True, False, True, True]
--   False
--   
--   
--   
--   >>> foldr1 (||) [False, False, True, True]
--   True
--   
--   
--   
--   >>> foldr1 (+) [1..]
--   * Hangs forever *
--   
foldr1 :: Foldable t => (a -> a -> a) -> t a -> a

-- | A variant of foldl that has no base case, and thus may only be
--   applied to non-empty structures.
--   
--   This function is non-total and will raise a runtime exception if the
--   structure happens to be empty.
--   
--   
--   foldl1 f = foldl1 f . toList
--   
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> foldl1 (+) [1..4]
--   10
--   
--   
--   
--   >>> foldl1 (+) []
--   *** Exception: Prelude.foldl1: empty list
--   
--   
--   
--   >>> foldl1 (+) Nothing
--   *** Exception: foldl1: empty structure
--   
--   
--   
--   >>> foldl1 (-) [1..4]
--   -8
--   
--   
--   
--   >>> foldl1 (&&) [True, False, True, True]
--   False
--   
--   
--   
--   >>> foldl1 (||) [False, False, True, True]
--   True
--   
--   
--   
--   >>> foldl1 (+) [1..]
--   * Hangs forever *
--   
foldl1 :: Foldable t => (a -> a -> a) -> t a -> a

-- | Test whether the structure is empty. The default implementation is
--   Left-associative and lazy in both the initial element and the
--   accumulator. Thus optimised for structures where the first element can
--   be accessed in constant time. Structures where this is not the case
--   should have a non-default implementation.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> null []
--   True
--   
--   
--   
--   >>> null [1]
--   False
--   
--   
--   null is expected to terminate even for infinite structures. The
--   default implementation terminates provided the structure is bounded on
--   the left (there is a leftmost element).
--   
--   
--   >>> null [1..]
--   False
--   
null :: Foldable t => t a -> Bool

-- | Returns the size/length of a finite structure as an Int. The
--   default implementation just counts elements starting with the
--   leftmost. Instances for structures that can compute the element count
--   faster than via element-by-element counting, should provide a
--   specialised implementation.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> length []
--   0
--   
--   
--   
--   >>> length ['a', 'b', 'c']
--   3
--   
--   >>> length [1..]
--   * Hangs forever *
--   
length :: Foldable t => t a -> Int

-- | Does the element occur in the structure?
--   
--   Note: elem is often used in infix form.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> 3 `elem` []
--   False
--   
--   
--   
--   >>> 3 `elem` [1,2]
--   False
--   
--   
--   
--   >>> 3 `elem` [1,2,3,4,5]
--   True
--   
--   
--   For infinite structures, the default implementation of elem
--   terminates if the sought-after value exists at a finite distance from
--   the left side of the structure:
--   
--   
--   >>> 3 `elem` [1..]
--   True
--   
--   
--   
--   >>> 3 `elem` ([4..] ++ [3])
--   * Hangs forever *
--   
elem :: (Foldable t, Eq a) => a -> t a -> Bool

-- | The largest element of a non-empty structure.
--   
--   This function is non-total and will raise a runtime exception if the
--   structure happens to be empty. A structure that supports random access
--   and maintains its elements in order should provide a specialised
--   implementation to return the maximum in faster than linear time.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> maximum [1..10]
--   10
--   
--   
--   
--   >>> maximum []
--   *** Exception: Prelude.maximum: empty list
--   
--   
--   
--   >>> maximum Nothing
--   *** Exception: maximum: empty structure
--   
--   
--   WARNING: This function is partial for possibly-empty structures like
--   lists.
maximum :: (Foldable t, Ord a) => t a -> a

-- | The least element of a non-empty structure.
--   
--   This function is non-total and will raise a runtime exception if the
--   structure happens to be empty. A structure that supports random access
--   and maintains its elements in order should provide a specialised
--   implementation to return the minimum in faster than linear time.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> minimum [1..10]
--   1
--   
--   
--   
--   >>> minimum []
--   *** Exception: Prelude.minimum: empty list
--   
--   
--   
--   >>> minimum Nothing
--   *** Exception: minimum: empty structure
--   
--   
--   WARNING: This function is partial for possibly-empty structures like
--   lists.
minimum :: (Foldable t, Ord a) => t a -> a

-- | The product function computes the product of the numbers of a
--   structure.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> product []
--   1
--   
--   
--   
--   >>> product [42]
--   42
--   
--   
--   
--   >>> product [1..10]
--   3628800
--   
--   
--   
--   >>> product [4.1, 2.0, 1.7]
--   13.939999999999998
--   
--   
--   
--   >>> product [1..]
--   * Hangs forever *
--   
product :: (Foldable t, Num a) => t a -> a
infix 4 `elem`

-- | The value of seq a b is bottom if a is
--   bottom, and otherwise equal to b. In other words, it
--   evaluates the first argument a to weak head normal form
--   (WHNF). seq is usually introduced to improve performance by
--   avoiding unneeded laziness.
--   
--   A note on evaluation order: the expression seq a b
--   does not guarantee that a will be evaluated before
--   b. The only guarantee given by seq is that the both
--   a and b will be evaluated before seq returns
--   a value. In particular, this means that b may be evaluated
--   before a. If you need to guarantee a specific order of
--   evaluation, you must use the function pseq from the
--   "parallel" package.
seq :: a -> b -> b
infixr 0 `seq`

-- | The concatenation of all the elements of a container of lists.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> concat (Just [1, 2, 3])
--   [1,2,3]
--   
--   
--   
--   >>> concat (Left 42)
--   []
--   
--   
--   
--   >>> concat [[1, 2, 3], [4, 5], [6], []]
--   [1,2,3,4,5,6]
--   
concat :: Foldable t => t [a] -> [a]

-- | <math>. filter, applied to a predicate and a list,
--   returns the list of those elements that satisfy the predicate; i.e.,
--   
--   
--   filter p xs = [ x | x <- xs, p x]
--   
--   
--   
Examples
--   
--   
--   >>> filter odd [1, 2, 3]
--   [1,3]
--   
--   
--   
--   >>> filter (\l -> length l > 3) ["Hello", ", ", "World", "!"]
--   ["Hello","World"]
--   
--   
--   
--   >>> filter (/= 3) [1, 2, 3, 4, 3, 2, 1]
--   [1,2,4,2,1]
--   
filter :: (a -> Bool) -> [a] -> [a]

-- | <math>. zip takes two lists and returns a list of
--   corresponding pairs.
--   
--   zip is right-lazy:
--   
--   
--   >>> zip [] undefined
--   []
--   
--   >>> zip undefined []
--   *** Exception: Prelude.undefined
--   ...
--   
--   
--   zip is capable of list fusion, but it is restricted to its
--   first list argument and its resulting list.
--   
--   
Examples
--   
--   
--   >>> zip [1, 2, 3] ['a', 'b', 'c']
--   [(1,'a'),(2,'b'),(3,'c')]
--   
--   
--   If one input list is shorter than the other, excess elements of the
--   longer list are discarded, even if one of the lists is infinite:
--   
--   
--   >>> zip [1] ['a', 'b']
--   [(1,'a')]
--   
--   
--   
--   >>> zip [1, 2] ['a']
--   [(1,'a')]
--   
--   
--   
--   >>> zip [] [1..]
--   []
--   
--   
--   
--   >>> zip [1..] []
--   []
--   
zip :: [a] -> [b] -> [(a, b)]

-- | The print function outputs a value of any printable type to the
--   standard output device. Printable types are those that are instances
--   of class Show; print converts values to strings for
--   output using the show operation and adds a newline.
--   
--   For example, a program to print the first 20 integers and their powers
--   of 2 could be written as:
--   
--   
--   main = print ([(n, 2^n) | n <- [0..19]])
--   
print :: Show a => a -> IO ()

-- | otherwise is defined as the value True. It helps to make
--   guards more readable. eg.
--   
--   
--   f x | x < 0     = ...
--       | otherwise = ...
--   
otherwise :: Bool

-- | <math>. map f xs is the list obtained by
--   applying f to each element of xs, i.e.,
--   
--   
--   map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn]
--   map f [x1, x2, ...] == [f x1, f x2, ...]
--   
--   
--   this means that map id == id
--   
--   
Examples
--   
--   
--   >>> map (+1) [1, 2, 3]
--   [2,3,4]
--   
--   
--   
--   >>> map id [1, 2, 3]
--   [1,2,3]
--   
--   
--   
--   >>> map (\n -> 3 * n + 1) [1, 2, 3]
--   [4,7,10]
--   
map :: (a -> b) -> [a] -> [b]

-- | ($) is the function application operator.
--   
--   Applying ($) to a function f and an argument
--   x gives the same result as applying f to x
--   directly. The definition is akin to this:
--   
--   
--   ($) :: (a -> b) -> a -> b
--   ($) f x = f x
--   
--   
--   This is id specialized from a -> a to
--   (a -> b) -> (a -> b) which by the associativity of
--   (->) is the same as (a -> b) -> a -> b.
--   
--   On the face of it, this may appear pointless! But it's actually one of
--   the most useful and important operators in Haskell.
--   
--   The order of operations is very different between ($) and
--   normal function application. Normal function application has
--   precedence 10 - higher than any operator - and associates to the left.
--   So these two definitions are equivalent:
--   
--   
--   expr = min 5 1 + 5
--   expr = ((min 5) 1) + 5
--   
--   
--   ($) has precedence 0 (the lowest) and associates to the
--   right, so these are equivalent:
--   
--   
--   expr = min 5 $ 1 + 5
--   expr = (min 5) (1 + 5)
--   
--   
--   
Examples
--   
--   A common use cases of ($) is to avoid parentheses in complex
--   expressions.
--   
--   For example, instead of using nested parentheses in the following
--   Haskell function:
--   
--   
--   -- | Sum numbers in a string: strSum "100  5 -7" == 98
--   strSum :: String -> Int
--   strSum s = sum (mapMaybe readMaybe (words s))
--   
--   
--   we can deploy the function application operator:
--   
--   
--   -- | Sum numbers in a string: strSum "100  5 -7" == 98
--   strSum :: String -> Int
--   strSum s = sum $ mapMaybe readMaybe $ words s
--   
--   
--   ($) is also used as a section (a partially applied operator),
--   in order to indicate that we wish to apply some yet-unspecified
--   function to a given value. For example, to apply the argument
--   5 to a list of functions:
--   
--   
--   applyFive :: [Int]
--   applyFive = map ($ 5) [(+1), (2^)]
--   >>> [6, 32]
--   
--   
--   
Technical Remark (Representation Polymorphism)
--   
--   ($) is fully representation-polymorphic. This allows it to
--   also be used with arguments of unlifted and even unboxed kinds, such
--   as unboxed integers:
--   
--   
--   fastMod :: Int -> Int -> Int
--   fastMod (I# x) (I# m) = I# $ remInt# x m
--   
($) :: (a -> b) -> a -> b
infixr 0 $

-- | Basic numeric class.
--   
--   The Haskell Report defines no laws for Num. However,
--   (+) and (*) are customarily expected
--   to define a ring and have the following properties:
--   
--   
--   
- Associativity of (+) (x + y) +
--   z = x + (y + z)
--   
- Commutativity of (+) x + y
--   = y + x
--   
- fromInteger 0 is the additive
--   identity x + fromInteger 0 = x
--   
- negate gives the additive inverse x +
--   negate x = fromInteger 0
--   
- Associativity of (*) (x * y) *
--   z = x * (y * z)
--   
- fromInteger 1 is the multiplicative
--   identity x * fromInteger 1 = x and
--   fromInteger 1 * x = x
--   
- Distributivity of (*) with respect to
--   (+) a * (b + c) = (a * b) + (a *
--   c) and (b + c) * a = (b * a) + (c * a)
--   
- Coherence with toInteger if the type also
--   implements Integral, then fromInteger is a left inverse
--   for toInteger, i.e. fromInteger (toInteger i) ==
--   i
--   
--   
--   Note that it isn't customarily expected that a type instance of
--   both Num and Ord implement an ordered ring. Indeed, in
--   base only Integer and Rational do.
class Num a
(+) :: Num a => a -> a -> a
(-) :: Num a => a -> a -> a
(*) :: Num a => a -> a -> a

-- | Unary negation.
negate :: Num a => a -> a

-- | Absolute value.
abs :: Num a => a -> a

-- | Sign of a number. The functions abs and signum should
--   satisfy the law:
--   
--   
--   abs x * signum x == x
--   
--   
--   For real numbers, the signum is either -1 (negative),
--   0 (zero) or 1 (positive).
signum :: Num a => a -> a

-- | Conversion from an Integer. An integer literal represents the
--   application of the function fromInteger to the appropriate
--   value of type Integer, so such literals have type
--   (Num a) => a.
fromInteger :: Num a => Integer -> a
infixl 6 -
infixl 6 +
infixl 7 *

-- | Fractional numbers, supporting real division.
--   
--   The Haskell Report defines no laws for Fractional. However,
--   (+) and (*) are customarily expected
--   to define a division ring and have the following properties:
--   
--   
--   
- recip gives the multiplicative inverse x
--   * recip x = recip x * x = fromInteger 1
--   
- Totality of toRational toRational is
--   total
--   
- Coherence with toRational if the type also
--   implements Real, then fromRational is a left inverse for
--   toRational, i.e. fromRational (toRational i) = i
--   
--   
--   Note that it isn't customarily expected that a type instance of
--   Fractional implement a field. However, all instances in
--   base do.
class Num a => Fractional a

-- | Fractional division.
(/) :: Fractional a => a -> a -> a

-- | Reciprocal fraction.
recip :: Fractional a => a -> a

-- | Conversion from a Rational (that is Ratio
--   Integer). A floating literal stands for an application of
--   fromRational to a value of type Rational, so such
--   literals have type (Fractional a) => a.
fromRational :: Fractional a => Rational -> a
infixl 7 /

-- | Class Enum defines operations on sequentially ordered types.
--   
--   The enumFrom... methods are used in Haskell's translation of
--   arithmetic sequences.
--   
--   Instances of Enum may be derived for any enumeration type
--   (types whose constructors have no fields). The nullary constructors
--   are assumed to be numbered left-to-right by fromEnum from
--   0 through n-1. See Chapter 10 of the Haskell
--   Report for more details.
--   
--   For any type that is an instance of class Bounded as well as
--   Enum, the following should hold:
--   
--   
--   
- The calls succ maxBound and pred
--   minBound should result in a runtime error.
--   
- fromEnum and toEnum should give a runtime error if
--   the result value is not representable in the result type. For example,
--   toEnum 7 :: Bool is an error.
--   
- enumFrom and enumFromThen should be defined with an
--   implicit bound, thus:
--   
--   
--   
--   enumFrom     x   = enumFromTo     x maxBound
--   enumFromThen x y = enumFromThenTo x y bound
--     where
--       bound | fromEnum y >= fromEnum x = maxBound
--             | otherwise                = minBound
--   
class Enum a

-- | Successor of a value. For numeric types, succ adds 1.
succ :: Enum a => a -> a

-- | Predecessor of a value. For numeric types, pred subtracts 1.
pred :: Enum a => a -> a

-- | Convert from an Int.
toEnum :: Enum a => Int -> a

-- | Convert to an Int. It is implementation-dependent what
--   fromEnum returns when applied to a value that is too large to
--   fit in an Int.
fromEnum :: Enum a => a -> Int

-- | Used in Haskell's translation of [n..] with [n..] =
--   enumFrom n, a possible implementation being enumFrom n = n :
--   enumFrom (succ n).
--   
--   
Examples
--   
--   
--   
- 
enumFrom 4 :: [Integer] = [4,5,6,7,...]

--   
- 
enumFrom 6 :: [Int] = [6,7,8,9,...,maxBound ::
--   Int]

--   
enumFrom :: Enum a => a -> [a]

-- | Used in Haskell's translation of [n,n'..] with [n,n'..] =
--   enumFromThen n n', a possible implementation being
--   enumFromThen n n' = n : n' : worker (f x) (f x n'),
--   worker s v = v : worker s (s v), x = fromEnum n' -
--   fromEnum n and
--   
--   
--   f n y
--     | n > 0 = f (n - 1) (succ y)
--     | n < 0 = f (n + 1) (pred y)
--     | otherwise = y
--   
--   
--   
--   
Examples
--   
--   
--   
- 
enumFromThen 4 6 :: [Integer] = [4,6,8,10...]

--   
- 
enumFromThen 6 2 :: [Int] = [6,2,-2,-6,...,minBound ::
--   Int]

--   
enumFromThen :: Enum a => a -> a -> [a]

-- | Used in Haskell's translation of [n..m] with [n..m] =
--   enumFromTo n m, a possible implementation being
--   
--   
--   enumFromTo n m
--      | n <= m = n : enumFromTo (succ n) m
--      | otherwise = []
--   
--   
--   
--   
Examples
--   
--   
--   
- 
enumFromTo 6 10 :: [Int] = [6,7,8,9,10]

--   
- 
enumFromTo 42 1 :: [Integer] = []

--   
enumFromTo :: Enum a => a -> a -> [a]

-- | Used in Haskell's translation of [n,n'..m] with [n,n'..m]
--   = enumFromThenTo n n' m, a possible implementation being
--   enumFromThenTo n n' m = worker (f x) (c x) n m, x =
--   fromEnum n' - fromEnum n, c x = bool (>=) ((x
--   0)
--   
--   
--   f n y
--      | n > 0 = f (n - 1) (succ y)
--      | n < 0 = f (n + 1) (pred y)
--      | otherwise = y
--   
--   
--   
--   and
--   
--   
--   worker s c v m
--      | c v m = v : worker s c (s v) m
--      | otherwise = []
--   
--   
--   
--   
Examples
--   
--   
--   
- 
enumFromThenTo 4 2 -6 :: [Integer] =
--   [4,2,0,-2,-4,-6]

--   
- 
enumFromThenTo 6 8 2 :: [Int] = []

--   
enumFromThenTo :: Enum a => a -> a -> a -> [a]

-- | The Monad class defines the basic operations over a
--   monad, a concept from a branch of mathematics known as
--   category theory. From the perspective of a Haskell programmer,
--   however, it is best to think of a monad as an abstract datatype
--   of actions. Haskell's do expressions provide a convenient
--   syntax for writing monadic expressions.
--   
--   Instances of Monad should satisfy the following:
--   
--   
--   
- Left identity return a >>= k = k
--   a
--   
- Right identity m >>= return =
--   m
--   
- Associativity m >>= (\x -> k x
--   >>= h) = (m >>= k) >>=
--   h
--   
--   
--   Furthermore, the Monad and Applicative operations should
--   relate as follows:
--   
--   
--   
- 
pure = return

--   
- 
m1 <*> m2 = m1 >>= (\x1 -> m2
--   >>= (\x2 -> return (x1 x2)))

--   
--   
--   The above laws imply:
--   
--   
--   
- 
fmap f xs = xs >>= return .
--   f

--   
- 
(>>) = (*>)

--   
--   
--   and that pure and (<*>) satisfy the applicative
--   functor laws.
--   
--   The instances of Monad for List, Maybe and
--   IO defined in the Prelude satisfy these laws.
class Applicative m => Monad (m :: Type -> Type)

-- | Sequentially compose two actions, passing any value produced by the
--   first as an argument to the second.
--   
--   'as >>= bs' can be understood as the do
--   expression
--   
--   
--   do a <- as
--      bs a
--   
--   
--   An alternative name for this function is 'bind', but some people may
--   refer to it as 'flatMap', which results from it being equivialent to
--   
--   
--   \x f -> join (fmap f x) :: Monad m => m a -> (a -> m b) -> m b
--   
--   
--   which can be seen as mapping a value with Monad m => m a ->
--   m (m b) and then 'flattening' m (m b) to m b
--   using join.
(>>=) :: Monad m => m a -> (a -> m b) -> m b

-- | Sequentially compose two actions, discarding any value produced by the
--   first, like sequencing operators (such as the semicolon) in imperative
--   languages.
--   
--   'as >> bs' can be understood as the do
--   expression
--   
--   
--   do as
--      bs
--   
--   
--   or in terms of (>>=) as
--   
--   
--   as >>= const bs
--   
(>>) :: Monad m => m a -> m b -> m b

-- | Inject a value into the monadic type. This function should not
--   be different from its default implementation as pure. The
--   justification for the existence of this function is merely historic.
return :: Monad m => a -> m a
infixl 1 >>=
infixl 1 >>

-- | A type f is a Functor if it provides a function fmap
--   which, given any types a and b lets you apply any
--   function from (a -> b) to turn an f a into an
--   f b, preserving the structure of f. Furthermore
--   f needs to adhere to the following:
--   
--   
--   
- Identity fmap id == id
--   
- Composition fmap (f . g) == fmap f .
--   fmap g
--   
--   
--   Note, that the second law follows from the free theorem of the type
--   fmap and the first law, so you need only check that the former
--   condition holds. See these articles by School of Haskell or
--   David Luposchainsky for an explanation.
class Functor (f :: Type -> Type)

-- | 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)
--   
fmap :: Functor f => (a -> b) -> f a -> f b

-- | Replace all locations in the input with the same value. The default
--   definition is fmap . const, but this may be
--   overridden with a more efficient version.
--   
--   
Examples
--   
--   Perform a computation with Maybe and replace the result with a
--   constant value if it is Just:
--   
--   
--   >>> 'a' <$ Just 2
--   Just 'a'
--   
--   >>> 'a' <$ Nothing
--   Nothing
--   
(<$) :: Functor f => a -> f b -> f a
infixl 4 <$

-- | When a value is bound in do-notation, the pattern on the left
--   hand side of <- might not match. In this case, this class
--   provides a function to recover.
--   
--   A Monad without a MonadFail instance may only be used in
--   conjunction with pattern that always match, such as newtypes, tuples,
--   data types with only a single data constructor, and irrefutable
--   patterns (~pat).
--   
--   Instances of MonadFail should satisfy the following law:
--   fail s should be a left zero for >>=,
--   
--   
--   fail s >>= f  =  fail s
--   
--   
--   If your Monad is also MonadPlus, a popular definition is
--   
--   
--   fail _ = mzero
--   
--   
--   fail s should be an action that runs in the monad itself, not
--   an exception (except in instances of MonadIO). In particular,
--   fail should not be implemented in terms of error.
class Monad m => MonadFail (m :: Type -> Type)
fail :: MonadFail m => String -> m a

-- | General coercion from Integral types.
--   
--   WARNING: This function performs silent truncation if the result type
--   is not at least as big as the argument's type.
fromIntegral :: (Integral a, Num b) => a -> b

-- | General coercion to Fractional types.
--   
--   WARNING: This function goes through the Rational type, which
--   does not have values for NaN for example. This means it does
--   not round-trip.
--   
--   For Double it also behaves differently with or without -O0:
--   
--   
--   Prelude> realToFrac nan -- With -O0
--   -Infinity
--   Prelude> realToFrac nan
--   NaN
--   
realToFrac :: (Real a, Fractional b) => a -> b

-- | Integral numbers, supporting integer division.
--   
--   The Haskell Report defines no laws for Integral. However,
--   Integral instances are customarily expected to define a
--   Euclidean domain and have the following properties for the
--   div/mod and quot/rem pairs, given suitable
--   Euclidean functions f and g:
--   
--   
--   
- x = y * quot x y + rem x y with rem x y
--   = fromInteger 0 or g (rem x y) < g
--   y
--   
- x = y * div x y + mod x y with mod x y
--   = fromInteger 0 or f (mod x y) < f
--   y
--   
--   
--   An example of a suitable Euclidean function, for Integer's
--   instance, is abs.
--   
--   In addition, toInteger should be total, and
--   fromInteger should be a left inverse for it, i.e.
--   fromInteger (toInteger i) = i.
class (Real a, Enum a) => Integral a

-- | Integer division truncated toward zero.
--   
--   WARNING: This function is partial (because it throws when 0 is passed
--   as the divisor) for all the integer types in base.
quot :: Integral a => a -> a -> a

-- | Integer remainder, satisfying
--   
--   
--   (x `quot` y)*y + (x `rem` y) == x
--   
--   
--   WARNING: This function is partial (because it throws when 0 is passed
--   as the divisor) for all the integer types in base.
rem :: Integral a => a -> a -> a

-- | Simultaneous quot and rem.
--   
--   WARNING: This function is partial (because it throws when 0 is passed
--   as the divisor) for all the integer types in base.
quotRem :: Integral a => a -> a -> (a, a)

-- | simultaneous div and mod.
--   
--   WARNING: This function is partial (because it throws when 0 is passed
--   as the divisor) for all the integer types in base.
divMod :: Integral a => a -> a -> (a, a)

-- | Conversion to Integer.
toInteger :: Integral a => a -> Integer
infixl 7 `quot`
infixl 7 `rem`

-- | Real numbers.
--   
--   The Haskell report defines no laws for Real, however
--   Real instances are customarily expected to adhere to the
--   following law:
--   
--   
--   
- Coherence with fromRational if the type also
--   implements Fractional, then fromRational is a left
--   inverse for toRational, i.e. fromRational (toRational i) =
--   i
--   
--   
--   The law does not hold for Float, Double, CFloat,
--   CDouble, etc., because these types contain non-finite values,
--   which cannot be roundtripped through Rational.
class (Num a, Ord a) => Real a

-- | Rational equivalent of its real argument with full precision.
toRational :: Real a => a -> Rational

-- | The class of semigroups (types with an associative binary operation).
--   
--   Instances should satisfy the following:
--   
--   
--   
- Associativity x <> (y <> z) =
--   (x <> y) <> z
--   
--   
--   You can alternatively define sconcat instead of
--   (<>), in which case the laws are:
--   
--   
--   
- Unit sconcat (pure x) = x
--   
- Multiplication sconcat (join xss) =
--   sconcat (fmap sconcat xss)
--   
class Semigroup a

-- | An associative operation.
--   
--   
Examples
--   
--   
--   >>> [1,2,3] <> [4,5,6]
--   [1,2,3,4,5,6]
--   
--   
--   
--   >>> Just [1, 2, 3] <> Just [4, 5, 6]
--   Just [1,2,3,4,5,6]
--   
--   
--   
--   >>> putStr "Hello, " <> putStrLn "World!"
--   Hello, World!
--   
(<>) :: Semigroup a => a -> a -> a
infixr 6 <>

-- | The class of monoids (types with an associative binary operation that
--   has an identity). Instances should satisfy the following:
--   
--   
--   
- Right identity x <> mempty =
--   x
--   
- Left identity mempty <> x =
--   x
--   
- Associativity x <> (y <> z) =
--   (x <> y) <> z (Semigroup
--   law)
--   
- Concatenation mconcat = foldr
--   (<>) mempty
--   
--   
--   You can alternatively define mconcat instead of mempty,
--   in which case the laws are:
--   
--   
--   
- Unit mconcat (pure x) = x
--   
- Multiplication mconcat (join xss) =
--   mconcat (fmap mconcat xss)
--   
- Subclass mconcat (toList xs) =
--   sconcat xs
--   
--   
--   The method names refer to the monoid of lists under concatenation, but
--   there are many other instances.
--   
--   Some types can be viewed as a monoid in more than one way, e.g. both
--   addition and multiplication on numbers. In such cases we often define
--   newtypes and make those instances of Monoid, e.g.
--   Sum and Product.
--   
--   NOTE: Semigroup is a superclass of Monoid since
--   base-4.11.0.0.
class Semigroup a => Monoid a

-- | Identity of mappend
--   
--   
Examples
--   
--   
--   >>> "Hello world" <> mempty
--   "Hello world"
--   
--   
--   
--   >>> mempty <> [1, 2, 3]
--   [1,2,3]
--   
mempty :: Monoid a => a

-- | An associative operation
--   
--   NOTE: This method is redundant and has the default
--   implementation mappend = (<>) since
--   base-4.11.0.0. Should it be implemented manually, since
--   mappend is a synonym for (<>), it is expected that
--   the two functions are defined the same way. In a future GHC release
--   mappend will be removed from Monoid.
mappend :: Monoid a => a -> a -> a

-- | Fold a list using the monoid.
--   
--   For most types, the default definition for mconcat will be
--   used, but the function is included in the class definition so that an
--   optimized version can be provided for specific types.
--   
--   
--   >>> mconcat ["Hello", " ", "Haskell", "!"]
--   "Hello Haskell!"
--   
mconcat :: Monoid a => [a] -> a

-- | A functor with application, providing operations to
--   
--   
--   
- embed pure expressions (pure), and
--   
- sequence computations and combine their results (<*>
--   and liftA2).
--   
--   
--   A minimal complete definition must include implementations of
--   pure and of either <*> or liftA2. If it
--   defines both, then they must behave the same as their default
--   definitions:
--   
--   
--   (<*>) = liftA2 id
--   
--   
--   
--   liftA2 f x y = f <$> x <*> y
--   
--   
--   Further, any definition must satisfy the following:
--   
--   
--   
- Identity 
pure id <*> v =
--   v

--   
- Composition 
pure (.) <*> u
--   <*> v <*> w = u <*> (v
--   <*> w)

--   
- Homomorphism 
pure f <*>
--   pure x = pure (f x)

--   
- Interchange 
u <*> pure y =
--   pure ($ y) <*> u

--   
--   
--   The other methods have the following default definitions, which may be
--   overridden with equivalent specialized implementations:
--   
--   
--   
- 
u *> v = (id <$ u)
--   <*> v

--   
- 
u <* v = liftA2 const u v

--   
--   
--   As a consequence of these laws, the Functor instance for
--   f will satisfy
--   
--   
--   
- 
fmap f x = pure f <*> x

--   
--   
--   It may be useful to note that supposing
--   
--   
--   forall x y. p (q x y) = f x . g y
--   
--   
--   it follows from the above that
--   
--   
--   liftA2 p (liftA2 q u v) = liftA2 f u . liftA2 g v
--   
--   
--   If f is also a Monad, it should satisfy
--   
--   
--   
- 
pure = return

--   
- 
m1 <*> m2 = m1 >>= (\x1 -> m2
--   >>= (\x2 -> return (x1 x2)))

--   
- 
(*>) = (>>)

--   
--   
--   (which implies that pure and <*> satisfy the
--   applicative functor laws).
class Functor f => Applicative (f :: Type -> Type)

-- | Lift a value into the Structure.
--   
--   
Examples
--   
--   
--   >>> pure 1 :: Maybe Int
--   Just 1
--   
--   
--   
--   >>> pure 'z' :: [Char]
--   "z"
--   
--   
--   
--   >>> pure (pure ":D") :: Maybe [String]
--   Just [":D"]
--   
pure :: Applicative f => a -> f a

-- | Sequential application.
--   
--   A few functors support an implementation of <*> that is
--   more efficient than the default one.
--   
--   
Example
--   
--   Used in combination with (<$>),
--   (<*>) can be used to build a record.
--   
--   
--   >>> data MyState = MyState {arg1 :: Foo, arg2 :: Bar, arg3 :: Baz}
--   
--   
--   
--   >>> produceFoo :: Applicative f => f Foo
--   
--   >>> produceBar :: Applicative f => f Bar
--   
--   >>> produceBaz :: Applicative f => f Baz
--   
--   
--   
--   >>> mkState :: Applicative f => f MyState
--   
--   >>> mkState = MyState <$> produceFoo <*> produceBar <*> produceBaz
--   
(<*>) :: Applicative f => f (a -> b) -> f a -> f b

-- | Lift a binary function to actions.
--   
--   Some functors support an implementation of liftA2 that is more
--   efficient than the default one. In particular, if fmap is an
--   expensive operation, it is likely better to use liftA2 than to
--   fmap over the structure and then use <*>.
--   
--   This became a typeclass method in 4.10.0.0. Prior to that, it was a
--   function defined in terms of <*> and fmap.
--   
--   
Example
--   
--   
--   >>> liftA2 (,) (Just 3) (Just 5)
--   Just (3,5)
--   
--   
--   
--   >>> liftA2 (+) [1, 2, 3] [4, 5, 6]
--   [5,6,7,6,7,8,7,8,9]
--   
liftA2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c

-- | Sequence actions, discarding the value of the first argument.
--   
--   
Examples
--   
--   If used in conjunction with the Applicative instance for Maybe,
--   you can chain Maybe computations, with a possible "early return" in
--   case of Nothing.
--   
--   
--   >>> Just 2 *> Just 3
--   Just 3
--   
--   
--   
--   >>> Nothing *> Just 3
--   Nothing
--   
--   
--   Of course a more interesting use case would be to have effectful
--   computations instead of just returning pure values.
--   
--   
--   >>> import Data.Char
--   
--   >>> import GHC.Internal.Text.ParserCombinators.ReadP
--   
--   >>> let p = string "my name is " *> munch1 isAlpha <* eof
--   
--   >>> readP_to_S p "my name is Simon"
--   [("Simon","")]
--   
(*>) :: Applicative f => f a -> f b -> f b

-- | Sequence actions, discarding the value of the second argument.
(<*) :: Applicative f => f a -> f b -> f a
infixl 4 <*>
infixl 4 *>
infixl 4 <*

-- | The Bounded class is used to name the upper and lower limits of
--   a type. Ord is not a superclass of Bounded since types
--   that are not totally ordered may also have upper and lower bounds.
--   
--   The Bounded class may be derived for any enumeration type;
--   minBound is the first constructor listed in the data
--   declaration and maxBound is the last. Bounded may also
--   be derived for single-constructor datatypes whose constituent types
--   are in Bounded.
class Bounded a
minBound :: Bounded a => a
maxBound :: Bounded a => a

-- | The Eq class defines equality (==) and inequality
--   (/=). All the basic datatypes exported by the Prelude
--   are instances of Eq, and Eq may be derived for any
--   datatype whose constituents are also instances of Eq.
--   
--   The Haskell Report defines no laws for Eq. However, instances
--   are encouraged to follow these properties:
--   
--   
--   
- Reflexivity x == x = True
--   
- Symmetry x == y = y == x
--   
- Transitivity if x == y && y == z =
--   True, then x == z = True
--   
- Extensionality if x == y = True and
--   f is a function whose return type is an instance of
--   Eq, then f x == f y = True
--   
- Negation x /= y = not (x ==
--   y)
--   
class Eq a

-- | Trigonometric and hyperbolic functions and related functions.
--   
--   The Haskell Report defines no laws for Floating. However,
--   (+), (*) and exp are
--   customarily expected to define an exponential field and have the
--   following properties:
--   
--   
--   
- exp (a + b) = exp a * exp b
--   
- exp (fromInteger 0) = fromInteger 1
--   
class Fractional a => Floating a
pi :: Floating a => a
exp :: Floating a => a -> a
log :: Floating a => a -> a
sqrt :: Floating a => a -> a
(**) :: Floating a => a -> a -> a
logBase :: Floating a => a -> a -> a
sin :: Floating a => a -> a
cos :: Floating a => a -> a
tan :: Floating a => a -> a
asin :: Floating a => a -> a
acos :: Floating a => a -> a
atan :: Floating a => a -> a
sinh :: Floating a => a -> a
cosh :: Floating a => a -> a
tanh :: Floating a => a -> a
asinh :: Floating a => a -> a
acosh :: Floating a => a -> a
atanh :: Floating a => a -> a
infixr 8 **

-- | The Ord class is used for totally ordered datatypes.
--   
--   Instances of Ord can be derived for any user-defined datatype
--   whose constituent types are in Ord. The declared order of the
--   constructors in the data declaration determines the ordering in
--   derived Ord instances. The Ordering datatype allows a
--   single comparison to determine the precise ordering of two objects.
--   
--   Ord, as defined by the Haskell report, implements a total order
--   and has the following properties:
--   
--   
--   
- Comparability x <= y || y <= x =
--   True
--   
- Transitivity if x <= y && y <=
--   z = True, then x <= z = True
--   
- Reflexivity x <= x = True
--   
- Antisymmetry if x <= y && y <=
--   x = True, then x == y = True
--   
--   
--   The following operator interactions are expected to hold:
--   
--   
--   
- x >= y = y <= x
--   
- x < y = x <= y && x /= y
--   
- x > y = y < x
--   
- x < y = compare x y == LT
--   
- x > y = compare x y == GT
--   
- x == y = compare x y == EQ
--   
- min x y == if x <= y then x else y = True
--   
- max x y == if x >= y then x else y = True
--   
--   
--   Note that (7.) and (8.) do not require min and
--   max to return either of their arguments. The result is merely
--   required to equal one of the arguments in terms of (==).
--   
--   Minimal complete definition: either compare or <=.
--   Using compare can be more efficient for complex types.
class Eq a => Ord a
compare :: Ord a => a -> a -> Ordering

-- | Parsing of Strings, producing values.
--   
--   Derived instances of Read make the following assumptions, which
--   derived instances of Show obey:
--   
--   
--   
- If the constructor is defined to be an infix operator, then the
--   derived Read instance will parse only infix applications of the
--   constructor (not the prefix form).
--   
- Associativity is not used to reduce the occurrence of parentheses,
--   although precedence may be.
--   
- If the constructor is defined using record syntax, the derived
--   Read will parse only the record-syntax form, and furthermore,
--   the fields must be given in the same order as the original
--   declaration.
--   
- The derived Read instance allows arbitrary Haskell
--   whitespace between tokens of the input string. Extra parentheses are
--   also allowed.
--   
--   
--   For example, given the declarations
--   
--   
--   infixr 5 :^:
--   data Tree a =  Leaf a  |  Tree a :^: Tree a
--   
--   
--   the derived instance of Read in Haskell 2010 is equivalent to
--   
--   
--   instance (Read a) => Read (Tree a) where
--   
--           readsPrec d r =  readParen (d > app_prec)
--                            (\r -> [(Leaf m,t) |
--                                    ("Leaf",s) <- lex r,
--                                    (m,t) <- readsPrec (app_prec+1) s]) r
--   
--                         ++ readParen (d > up_prec)
--                            (\r -> [(u:^:v,w) |
--                                    (u,s) <- readsPrec (up_prec+1) r,
--                                    (":^:",t) <- lex s,
--                                    (v,w) <- readsPrec (up_prec+1) t]) r
--   
--             where app_prec = 10
--                   up_prec = 5
--   
--   
--   Note that right-associativity of :^: is unused.
--   
--   The derived instance in GHC is equivalent to
--   
--   
--   instance (Read a) => Read (Tree a) where
--   
--           readPrec = parens $ (prec app_prec $ do
--                                    Ident "Leaf" <- lexP
--                                    m <- step readPrec
--                                    return (Leaf m))
--   
--                        +++ (prec up_prec $ do
--                                    u <- step readPrec
--                                    Symbol ":^:" <- lexP
--                                    v <- step readPrec
--                                    return (u :^: v))
--   
--             where app_prec = 10
--                   up_prec = 5
--   
--           readListPrec = readListPrecDefault
--   
--   
--   Why do both readsPrec and readPrec exist, and why does
--   GHC opt to implement readPrec in derived Read instances
--   instead of readsPrec? The reason is that readsPrec is
--   based on the ReadS type, and although ReadS is mentioned
--   in the Haskell 2010 Report, it is not a very efficient parser data
--   structure.
--   
--   readPrec, on the other hand, is based on a much more efficient
--   ReadPrec datatype (a.k.a "new-style parsers"), but its
--   definition relies on the use of the RankNTypes language
--   extension. Therefore, readPrec (and its cousin,
--   readListPrec) are marked as GHC-only. Nevertheless, it is
--   recommended to use readPrec instead of readsPrec
--   whenever possible for the efficiency improvements it brings.
--   
--   As mentioned above, derived Read instances in GHC will
--   implement readPrec instead of readsPrec. The default
--   implementations of readsPrec (and its cousin, readList)
--   will simply use readPrec under the hood. If you are writing a
--   Read instance by hand, it is recommended to write it like so:
--   
--   
--   instance Read T where
--     readPrec     = ...
--     readListPrec = readListPrecDefault
--   
class Read a

-- | attempts to parse a value from the front of the string, returning a
--   list of (parsed value, remaining string) pairs. If there is no
--   successful parse, the returned list is empty.
--   
--   Derived instances of Read and Show satisfy the
--   following:
--   
--   
--   
- (x,"") is an element of (readsPrec d
--   (showsPrec d x "")).
--   
--   
--   That is, readsPrec parses the string produced by
--   showsPrec, and delivers the value that showsPrec started
--   with.
readsPrec :: Read a => Int -> ReadS a

-- | The method readList is provided to allow the programmer to give
--   a specialised way of parsing lists of values. For example, this is
--   used by the predefined Read instance of the Char type,
--   where values of type String are expected to use double quotes,
--   rather than square brackets.
readList :: Read a => ReadS [a]

-- | Efficient, machine-independent access to the components of a
--   floating-point number.
class (RealFrac a, Floating a) => RealFloat a

-- | a constant function, returning the radix of the representation (often
--   2)
floatRadix :: RealFloat a => a -> Integer

-- | a constant function, returning the number of digits of
--   floatRadix in the significand
floatDigits :: RealFloat a => a -> Int

-- | a constant function, returning the lowest and highest values the
--   exponent may assume
floatRange :: RealFloat a => a -> (Int, Int)

-- | The function decodeFloat applied to a real floating-point
--   number returns the significand expressed as an Integer and an
--   appropriately scaled exponent (an Int). If
--   decodeFloat x yields (m,n), then x
--   is equal in value to m*b^^n, where b is the
--   floating-point radix, and furthermore, either m and
--   n are both zero or else b^(d-1) <= abs m <
--   b^d, where d is the value of floatDigits
--   x. In particular, decodeFloat 0 = (0,0). If the
--   type contains a negative zero, also decodeFloat (-0.0) =
--   (0,0). The result of decodeFloat x is
--   unspecified if either of isNaN x or
--   isInfinite x is True.
decodeFloat :: RealFloat a => a -> (Integer, Int)

-- | encodeFloat performs the inverse of decodeFloat in the
--   sense that for finite x with the exception of -0.0,
--   uncurry encodeFloat (decodeFloat x) = x.
--   encodeFloat m n is one of the two closest
--   representable floating-point numbers to m*b^^n (or
--   ±Infinity if overflow occurs); usually the closer, but if
--   m contains too many bits, the result may be rounded in the
--   wrong direction.
encodeFloat :: RealFloat a => Integer -> Int -> a

-- | exponent corresponds to the second component of
--   decodeFloat. exponent 0 = 0 and for finite
--   nonzero x, exponent x = snd (decodeFloat x)
--   + floatDigits x. If x is a finite floating-point
--   number, it is equal in value to significand x * b ^^
--   exponent x, where b is the floating-point radix.
--   The behaviour is unspecified on infinite or NaN values.
exponent :: RealFloat a => a -> Int

-- | The first component of decodeFloat, scaled to lie in the open
--   interval (-1,1), either 0.0 or of absolute
--   value >= 1/b, where b is the floating-point
--   radix. The behaviour is unspecified on infinite or NaN
--   values.
significand :: RealFloat a => a -> a

-- | multiplies a floating-point number by an integer power of the radix
scaleFloat :: RealFloat a => Int -> a -> a

-- | True if the argument is an IEEE "not-a-number" (NaN) value
isNaN :: RealFloat a => a -> Bool

-- | True if the argument is an IEEE infinity or negative infinity
isInfinite :: RealFloat a => a -> Bool

-- | True if the argument is too small to be represented in
--   normalized format
isDenormalized :: RealFloat a => a -> Bool

-- | True if the argument is an IEEE negative zero
isNegativeZero :: RealFloat a => a -> Bool

-- | True if the argument is an IEEE floating point number
isIEEE :: RealFloat a => a -> Bool

-- | a version of arctangent taking two real floating-point arguments. For
--   real floating x and y, atan2 y x
--   computes the angle (from the positive x-axis) of the vector from the
--   origin to the point (x,y). atan2 y x returns
--   a value in the range [-pi, pi]. It follows the
--   Common Lisp semantics for the origin when signed zeroes are supported.
--   atan2 y 1, with y in a type that is
--   RealFloat, should return the same value as atan
--   y. A default definition of atan2 is provided, but
--   implementors can provide a more accurate implementation.
atan2 :: RealFloat a => a -> a -> a

-- | Extracting components of fractions.
class (Real a, Fractional a) => RealFrac a

-- | The function properFraction takes a real fractional number
--   x and returns a pair (n,f) such that x =
--   n+f, and:
--   
--   
--   
- n is an integral number with the same sign as x;
--   and
--   
- f is a fraction with the same type and sign as
--   x, and with absolute value less than 1.
--   
--   
--   The default definitions of the ceiling, floor,
--   truncate and round functions are in terms of
--   properFraction.
properFraction :: (RealFrac a, Integral b) => a -> (b, a)

-- | truncate x returns the integer nearest x
--   between zero and x
truncate :: (RealFrac a, Integral b) => a -> b

-- | round x returns the nearest integer to x; the
--   even integer if x is equidistant between two integers
round :: (RealFrac a, Integral b) => a -> b

-- | ceiling x returns the least integer not less than
--   x
ceiling :: (RealFrac a, Integral b) => a -> b

-- | floor x returns the greatest integer not greater than
--   x
floor :: (RealFrac a, Integral b) => a -> b

-- | Conversion of values to readable Strings.
--   
--   Derived instances of Show have the following properties, which
--   are compatible with derived instances of Read:
--   
--   
--   
- The result of show is a syntactically correct Haskell
--   expression containing only constants, given the fixity declarations in
--   force at the point where the type is declared. It contains only the
--   constructor names defined in the data type, parentheses, and spaces.
--   When labelled constructor fields are used, braces, commas, field
--   names, and equal signs are also used.
--   
- If the constructor is defined to be an infix operator, then
--   showsPrec will produce infix applications of the
--   constructor.
--   
- the representation will be enclosed in parentheses if the
--   precedence of the top-level constructor in x is less than
--   d (associativity is ignored). Thus, if d is
--   0 then the result is never surrounded in parentheses; if
--   d is 11 it is always surrounded in parentheses,
--   unless it is an atomic expression.
--   
- If the constructor is defined using record syntax, then
--   show will produce the record-syntax form, with the fields given
--   in the same order as the original declaration.
--   
--   
--   For example, given the declarations
--   
--   
--   infixr 5 :^:
--   data Tree a =  Leaf a  |  Tree a :^: Tree a
--   
--   
--   the derived instance of Show is equivalent to
--   
--   
--   instance (Show a) => Show (Tree a) where
--   
--          showsPrec d (Leaf m) = showParen (d > app_prec) $
--               showString "Leaf " . showsPrec (app_prec+1) m
--            where app_prec = 10
--   
--          showsPrec d (u :^: v) = showParen (d > up_prec) $
--               showsPrec (up_prec+1) u .
--               showString " :^: "      .
--               showsPrec (up_prec+1) v
--            where up_prec = 5
--   
--   
--   Note that right-associativity of :^: is ignored. For example,
--   
--   
--   
- show (Leaf 1 :^: Leaf 2 :^: Leaf 3) produces the
--   string "Leaf 1 :^: (Leaf 2 :^: Leaf 3)".
--   
class Show a

-- | Convert a value to a readable String.
--   
--   showsPrec should satisfy the law
--   
--   
--   showsPrec d x r ++ s  ==  showsPrec d x (r ++ s)
--   
--   
--   Derived instances of Read and Show satisfy the
--   following:
--   
--   
--   
- (x,"") is an element of (readsPrec d
--   (showsPrec d x "")).
--   
--   
--   That is, readsPrec parses the string produced by
--   showsPrec, and delivers the value that showsPrec started
--   with.
showsPrec :: Show a => Int -> a -> ShowS

-- | A specialised variant of showsPrec, using precedence context
--   zero, and returning an ordinary String.
show :: Show a => a -> String

-- | The method showList is provided to allow the programmer to give
--   a specialised way of showing lists of values. For example, this is
--   used by the predefined Show instance of the Char type,
--   where values of type String should be shown in double quotes,
--   rather than between square brackets.
showList :: Show a => [a] -> ShowS

-- | Functors representing data structures that can be transformed to
--   structures of the same shape by performing an
--   Applicative (or, therefore, Monad) action on each
--   element from left to right.
--   
--   A more detailed description of what same shape means, the
--   various methods, how traversals are constructed, and example advanced
--   use-cases can be found in the Overview section of
--   Data.Traversable#overview.
--   
--   For the class laws see the Laws section of
--   Data.Traversable#laws.
class (Functor t, Foldable t) => Traversable (t :: Type -> Type)

-- | Map each element of a structure to an action, evaluate these actions
--   from left to right, and collect the results. For a version that
--   ignores the results see traverse_.
--   
--   
Examples
--   
--   Basic usage:
--   
--   In the first two examples we show each evaluated action mapping to the
--   output structure.
--   
--   
--   >>> traverse Just [1,2,3,4]
--   Just [1,2,3,4]
--   
--   
--   
--   >>> traverse id [Right 1, Right 2, Right 3, Right 4]
--   Right [1,2,3,4]
--   
--   
--   In the next examples, we show that Nothing and Left
--   values short circuit the created structure.
--   
--   
--   >>> traverse (const Nothing) [1,2,3,4]
--   Nothing
--   
--   
--   
--   >>> traverse (\x -> if odd x then Just x else Nothing)  [1,2,3,4]
--   Nothing
--   
--   
--   
--   >>> traverse id [Right 1, Right 2, Right 3, Right 4, Left 0]
--   Left 0
--   
traverse :: (Traversable t, Applicative f) => (a -> f b) -> t a -> f (t b)

-- | Evaluate each action in the structure from left to right, and collect
--   the results. For a version that ignores the results see
--   sequenceA_.
--   
--   
Examples
--   
--   Basic usage:
--   
--   For the first two examples we show sequenceA fully evaluating a a
--   structure and collecting the results.
--   
--   
--   >>> sequenceA [Just 1, Just 2, Just 3]
--   Just [1,2,3]
--   
--   
--   
--   >>> sequenceA [Right 1, Right 2, Right 3]
--   Right [1,2,3]
--   
--   
--   The next two example show Nothing and Just will short
--   circuit the resulting structure if present in the input. For more
--   context, check the Traversable instances for Either and
--   Maybe.
--   
--   
--   >>> sequenceA [Just 1, Just 2, Just 3, Nothing]
--   Nothing
--   
--   
--   
--   >>> sequenceA [Right 1, Right 2, Right 3, Left 4]
--   Left 4
--   
sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a)

-- | Map each element of a structure to a monadic action, evaluate these
--   actions from left to right, and collect the results. For a version
--   that ignores the results see mapM_.
--   
--   
Examples
--   
--   mapM is literally a traverse with a type signature
--   restricted to Monad. Its implementation may be more efficient
--   due to additional power of Monad.
mapM :: (Traversable t, Monad m) => (a -> m b) -> t a -> m (t b)

-- | Evaluate each monadic action in the structure from left to right, and
--   collect the results. For a version that ignores the results see
--   sequence_.
--   
--   
Examples
--   
--   Basic usage:
--   
--   The first two examples are instances where the input and and output of
--   sequence are isomorphic.
--   
--   
--   >>> sequence $ Right [1,2,3,4]
--   [Right 1,Right 2,Right 3,Right 4]
--   
--   
--   
--   >>> sequence $ [Right 1,Right 2,Right 3,Right 4]
--   Right [1,2,3,4]
--   
--   
--   The following examples demonstrate short circuit behavior for
--   sequence.
--   
--   
--   >>> sequence $ Left [1,2,3,4]
--   Left [1,2,3,4]
--   
--   
--   
--   >>> sequence $ [Left 0, Right 1,Right 2,Right 3,Right 4]
--   Left 0
--   
sequence :: (Traversable t, Monad m) => t (m a) -> m (t a)

-- | String is an alias for a list of characters.
--   
--   String constants in Haskell are values of type String. That
--   means if you write a string literal like "hello world", it
--   will have the type [Char], which is the same as
--   String.
--   
--   Note: You can ask the compiler to automatically infer different
--   types with the -XOverloadedStrings language extension, for
--   example "hello world" :: Text. See IsString for more
--   information.
--   
--   Because String is just a list of characters, you can use
--   normal list functions to do basic string manipulation. See
--   Data.List for operations on lists.
--   
--   
Performance considerations
--   
--   [Char] is a relatively memory-inefficient type. It is a
--   linked list of boxed word-size characters, internally it looks
--   something like:
--   
--   
--   ╭─────┬───┬──╮  ╭─────┬───┬──╮  ╭─────┬───┬──╮  ╭────╮
--   │ (:) │   │ ─┼─>│ (:) │   │ ─┼─>│ (:) │   │ ─┼─>│ [] │
--   ╰─────┴─┼─┴──╯  ╰─────┴─┼─┴──╯  ╰─────┴─┼─┴──╯  ╰────╯
--           v               v               v
--          'a'             'b'             'c'
--   
--   
--   The String "abc" will use 5*3+1 = 16 (in general
--   5n+1) words of space in memory.
--   
--   Furthermore, operations like (++) (string concatenation) are
--   O(n) (in the left argument).
--   
--   For historical reasons, the base library uses String
--   in a lot of places for the conceptual simplicity, but library code
--   dealing with user-data should use the text package for Unicode
--   text, or the the bytestring package for binary data.
type String = [Char]

-- | Arbitrary-precision rational numbers, represented as a ratio of two
--   Integer values. A rational number may be constructed using the
--   % operator.
type Rational = Ratio Integer

-- | The Either type represents values with two possibilities: a
--   value of type Either a b is either Left
--   a or Right b.
--   
--   The Either type is sometimes used to represent a value which is
--   either correct or an error; by convention, the Left constructor
--   is used to hold an error value and the Right constructor is
--   used to hold a correct value (mnemonic: "right" also means "correct").
--   
--   
Examples
--   
--   The type Either String Int is the type
--   of values which can be either a String or an Int. The
--   Left constructor can be used only on Strings, and the
--   Right constructor can be used only on Ints:
--   
--   
--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> s
--   Left "foo"
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> n
--   Right 3
--   
--   >>> :type s
--   s :: Either String Int
--   
--   >>> :type n
--   n :: Either String Int
--   
--   
--   The fmap from our Functor instance will ignore
--   Left values, but will apply the supplied function to values
--   contained in a Right:
--   
--   
--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> fmap (*2) s
--   Left "foo"
--   
--   >>> fmap (*2) n
--   Right 6
--   
--   
--   The Monad instance for Either allows us to chain
--   together multiple actions which may fail, and fail overall if any of
--   the individual steps failed. First we'll write a function that can
--   either parse an Int from a Char, or fail.
--   
--   
--   >>> import Data.Char ( digitToInt, isDigit )
--   
--   >>> :{
--       let parseEither :: Char -> Either String Int
--           parseEither c
--             | isDigit c = Right (digitToInt c)
--             | otherwise = Left "parse error"
--   
--   >>> :}
--   
--   
--   The following should work, since both '1' and '2'
--   can be parsed as Ints.
--   
--   
--   >>> :{
--       let parseMultiple :: Either String Int
--           parseMultiple = do
--             x <- parseEither '1'
--             y <- parseEither '2'
--             return (x + y)
--   
--   >>> :}
--   
--   
--   
--   >>> parseMultiple
--   Right 3
--   
--   
--   But the following should fail overall, since the first operation where
--   we attempt to parse 'm' as an Int will fail:
--   
--   
--   >>> :{
--       let parseMultiple :: Either String Int
--           parseMultiple = do
--             x <- parseEither 'm'
--             y <- parseEither '2'
--             return (x + y)
--   
--   >>> :}
--   
--   
--   
--   >>> parseMultiple
--   Left "parse error"
--   
data Either a b
Left :: a -> Either a b
Right :: b -> Either a b

-- | Extract the first component of a pair.
fst :: (a, b) -> a

-- | Extract the second component of a pair.
snd :: (a, b) -> b

-- | Convert an uncurried function to a curried function.
--   
--   
Examples
--   
--   
--   >>> curry fst 1 2
--   1
--   
curry :: ((a, b) -> c) -> a -> b -> c

-- | uncurry converts a curried function to a function on pairs.
--   
--   
Examples
--   
--   
--   >>> uncurry (+) (1,2)
--   3
--   
--   
--   
--   >>> uncurry ($) (show, 1)
--   "1"
--   
--   
--   
--   >>> map (uncurry max) [(1,2), (3,4), (6,8)]
--   [2,4,8]
--   
uncurry :: (a -> b -> c) -> (a, b) -> c

-- | error stops execution and displays an error message.
error :: HasCallStack => [Char] -> a

-- | A variant of error that does not produce a stack trace.
errorWithoutStackTrace :: [Char] -> a

-- | A special case of error. It is expected that compilers will
--   recognize this and insert error messages which are more appropriate to
--   the context in which undefined appears.
undefined :: HasCallStack => a

-- | Same as >>=, but with the arguments interchanged.
--   
--   
--   as >>= f == f =<< as
--   
(=<<) :: Monad m => (a -> m b) -> m a -> m b
infixr 1 =<<

-- | Identity function.
--   
--   
--   id x = x
--   
--   
--   This function might seem useless at first glance, but it can be very
--   useful in a higher order context.
--   
--   
Examples
--   
--   
--   >>> length $ filter id [True, True, False, True]
--   3
--   
--   
--   
--   >>> Just (Just 3) >>= id
--   Just 3
--   
--   
--   
--   >>> foldr id 0 [(^3), (*5), (+2)]
--   1000
--   
id :: a -> a

-- | Right to left function composition.
--   
--   
--   (f . g) x = f (g x)
--   
--   
--   
--   f . id = f = id . f
--   
--   
--   
Examples
--   
--   
--   >>> map ((*2) . length) [[], [0, 1, 2], [0]]
--   [0,6,2]
--   
--   
--   
--   >>> foldr (.) id [(+1), (*3), (^3)] 2
--   25
--   
--   
--   
--   >>> let (...) = (.).(.) in ((*2)...(+)) 5 10
--   30
--   
(.) :: (b -> c) -> (a -> b) -> a -> c
infixr 9 .

-- | flip f takes its (first) two arguments in the reverse
--   order of f.
--   
--   
--   flip f x y = f y x
--   
--   
--   
--   flip . flip = id
--   
--   
--   
Examples
--   
--   
--   >>> flip (++) "hello" "world"
--   "worldhello"
--   
--   
--   
--   >>> let (.>) = flip (.) in (+1) .> show $ 5
--   "6"
--   
flip :: (a -> b -> c) -> b -> a -> c

-- | Strict (call-by-value) application operator. It takes a function and
--   an argument, evaluates the argument to weak head normal form (WHNF),
--   then calls the function with that value.
($!) :: (a -> b) -> a -> b
infixr 0 $!

-- | asTypeOf is a type-restricted version of const. It is
--   usually used as an infix operator, and its typing forces its first
--   argument (which is usually overloaded) to have the same type as the
--   second.
asTypeOf :: a -> a -> a

-- | the same as flip (-).
--   
--   Because - is treated specially in the Haskell grammar,
--   (- e) is not a section, but an application of
--   prefix negation. However, (subtract
--   exp) is equivalent to the disallowed section.
subtract :: Num a => a -> a -> a

-- | The maybe function takes a default value, a function, and a
--   Maybe value. If the Maybe value is Nothing, the
--   function returns the default value. Otherwise, it applies the function
--   to the value inside the Just and returns the result.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> maybe False odd (Just 3)
--   True
--   
--   
--   
--   >>> maybe False odd Nothing
--   False
--   
--   
--   Read an integer from a string using readMaybe. If we succeed,
--   return twice the integer; that is, apply (*2) to it. If
--   instead we fail to parse an integer, return 0 by default:
--   
--   
--   >>> import GHC.Internal.Text.Read ( readMaybe )
--   
--   >>> maybe 0 (*2) (readMaybe "5")
--   10
--   
--   >>> maybe 0 (*2) (readMaybe "")
--   0
--   
--   
--   Apply show to a Maybe Int. If we have Just n,
--   we want to show the underlying Int n. But if we have
--   Nothing, we return the empty string instead of (for example)
--   "Nothing":
--   
--   
--   >>> maybe "" show (Just 5)
--   "5"
--   
--   >>> maybe "" show Nothing
--   ""
--   
maybe :: b -> (a -> b) -> Maybe a -> b

-- | <math>. Extract the first element of a list, which must be
--   non-empty.
--   
--   To disable the warning about partiality put {-# OPTIONS_GHC
--   -Wno-x-partial -Wno-unrecognised-warning-flags #-} at the top of
--   the file. To disable it throughout a package put the same options into
--   ghc-options section of Cabal file. To disable it in GHCi put
--   :set -Wno-x-partial -Wno-unrecognised-warning-flags into
--   ~/.ghci config file. See also the migration guide.
--   
--   
Examples
--   
--   
--   >>> head [1, 2, 3]
--   1
--   
--   
--   
--   >>> head [1..]
--   1
--   
--   
--   
--   >>> head []
--   *** Exception: Prelude.head: empty list
--   
head :: HasCallStack => [a] -> a

-- | <math>. Extract the elements after the head of a list, which
--   must be non-empty.
--   
--   To disable the warning about partiality put {-# OPTIONS_GHC
--   -Wno-x-partial -Wno-unrecognised-warning-flags #-} at the top of
--   the file. To disable it throughout a package put the same options into
--   ghc-options section of Cabal file. To disable it in GHCi put
--   :set -Wno-x-partial -Wno-unrecognised-warning-flags into
--   ~/.ghci config file. See also the migration guide.
--   
--   
Examples
--   
--   
--   >>> tail [1, 2, 3]
--   [2,3]
--   
--   
--   
--   >>> tail [1]
--   []
--   
--   
--   
--   >>> tail []
--   *** Exception: Prelude.tail: empty list
--   
tail :: HasCallStack => [a] -> [a]

-- | <math>. Extract the last element of a list, which must be finite
--   and non-empty.
--   
--   WARNING: This function is partial. Consider using unsnoc
--   instead.
--   
--   
Examples
--   
--   
--   >>> last [1, 2, 3]
--   3
--   
--   
--   
--   >>> last [1..]
--   * Hangs forever *
--   
--   
--   
--   >>> last []
--   *** Exception: Prelude.last: empty list
--   
last :: HasCallStack => [a] -> a

-- | <math>. Return all the elements of a list except the last one.
--   The list must be non-empty.
--   
--   WARNING: This function is partial. Consider using unsnoc
--   instead.
--   
--   
Examples
--   
--   
--   >>> init [1, 2, 3]
--   [1,2]
--   
--   
--   
--   >>> init [1]
--   []
--   
--   
--   
--   >>> init []
--   *** Exception: Prelude.init: empty list
--   
init :: HasCallStack => [a] -> [a]

-- | <math>. scanl is similar to foldl, but returns a
--   list of successive reduced values from the left:
--   
--   
--   scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]
--   
--   
--   Note that
--   
--   
--   last (scanl f z xs) == foldl f z xs
--   
--   
--   
Examples
--   
--   
--   >>> scanl (+) 0 [1..4]
--   [0,1,3,6,10]
--   
--   
--   
--   >>> scanl (+) 42 []
--   [42]
--   
--   
--   
--   >>> scanl (-) 100 [1..4]
--   [100,99,97,94,90]
--   
--   
--   
--   >>> scanl (\reversedString nextChar -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
--   ["foo","afoo","bafoo","cbafoo","dcbafoo"]
--   
--   
--   
--   >>> take 10 (scanl (+) 0 [1..])
--   [0,1,3,6,10,15,21,28,36,45]
--   
--   
--   
--   >>> take 1 (scanl undefined 'a' undefined)
--   "a"
--   
scanl :: (b -> a -> b) -> b -> [a] -> [b]

-- | <math>. scanl1 is a variant of scanl that has no
--   starting value argument:
--   
--   
--   scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...]
--   
--   
--   
Examples
--   
--   
--   >>> scanl1 (+) [1..4]
--   [1,3,6,10]
--   
--   
--   
--   >>> scanl1 (+) []
--   []
--   
--   
--   
--   >>> scanl1 (-) [1..4]
--   [1,-1,-4,-8]
--   
--   
--   
--   >>> scanl1 (&&) [True, False, True, True]
--   [True,False,False,False]
--   
--   
--   
--   >>> scanl1 (||) [False, False, True, True]
--   [False,False,True,True]
--   
--   
--   
--   >>> take 10 (scanl1 (+) [1..])
--   [1,3,6,10,15,21,28,36,45,55]
--   
--   
--   
--   >>> take 1 (scanl1 undefined ('a' : undefined))
--   "a"
--   
scanl1 :: (a -> a -> a) -> [a] -> [a]

-- | <math>. scanr is the right-to-left dual of scanl.
--   Note that the order of parameters on the accumulating function are
--   reversed compared to scanl. Also note that
--   
--   
--   head (scanr f z xs) == foldr f z xs.
--   
--   
--   
Examples
--   
--   
--   >>> scanr (+) 0 [1..4]
--   [10,9,7,4,0]
--   
--   
--   
--   >>> scanr (+) 42 []
--   [42]
--   
--   
--   
--   >>> scanr (-) 100 [1..4]
--   [98,-97,99,-96,100]
--   
--   
--   
--   >>> scanr (\nextChar reversedString -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
--   ["abcdfoo","bcdfoo","cdfoo","dfoo","foo"]
--   
--   
--   
--   >>> force $ scanr (+) 0 [1..]
--   *** Exception: stack overflow
--   
scanr :: (a -> b -> b) -> b -> [a] -> [b]

-- | <math>. scanr1 is a variant of scanr that has no
--   starting value argument.
--   
--   
Examples
--   
--   
--   >>> scanr1 (+) [1..4]
--   [10,9,7,4]
--   
--   
--   
--   >>> scanr1 (+) []
--   []
--   
--   
--   
--   >>> scanr1 (-) [1..4]
--   [-2,3,-1,4]
--   
--   
--   
--   >>> scanr1 (&&) [True, False, True, True]
--   [False,False,True,True]
--   
--   
--   
--   >>> scanr1 (||) [True, True, False, False]
--   [True,True,False,False]
--   
--   
--   
--   >>> force $ scanr1 (+) [1..]
--   *** Exception: stack overflow
--   
scanr1 :: (a -> a -> a) -> [a] -> [a]

-- | iterate f x returns an infinite list of repeated
--   applications of f to x:
--   
--   
--   iterate f x == [x, f x, f (f x), ...]
--   
--   
--   
Laziness
--   
--   Note that iterate is lazy, potentially leading to thunk
--   build-up if the consumer doesn't force each iterate. See
--   iterate' for a strict variant of this function.
--   
--   
--   >>> take 1 $ iterate undefined 42
--   [42]
--   
--   
--   
Examples
--   
--   
--   >>> take 10 $ iterate not True
--   [True,False,True,False,True,False,True,False,True,False]
--   
--   
--   
--   >>> take 10 $ iterate (+3) 42
--   [42,45,48,51,54,57,60,63,66,69]
--   
--   
--   iterate id == repeat:
--   
--   
--   >>> take 10 $ iterate id 1
--   [1,1,1,1,1,1,1,1,1,1]
--   
iterate :: (a -> a) -> a -> [a]

-- | repeat x is an infinite list, with x the
--   value of every element.
--   
--   
Examples
--   
--   
--   >>> take 10 $ repeat 17
--   [17,17,17,17,17,17,17,17,17, 17]
--   
--   
--   
--   >>> repeat undefined
--   [*** Exception: Prelude.undefined
--   
repeat :: a -> [a]

-- | replicate n x is a list of length n with
--   x the value of every element. It is an instance of the more
--   general genericReplicate, in which n may be of any
--   integral type.
--   
--   
Examples
--   
--   
--   >>> replicate 0 True
--   []
--   
--   
--   
--   >>> replicate (-1) True
--   []
--   
--   
--   
--   >>> replicate 4 True
--   [True,True,True,True]
--   
replicate :: Int -> a -> [a]

-- | takeWhile, applied to a predicate p and a list
--   xs, returns the longest prefix (possibly empty) of
--   xs of elements that satisfy p.
--   
--   
Laziness
--   
--   
--   >>> takeWhile (const False) undefined
--   *** Exception: Prelude.undefined
--   
--   
--   
--   >>> takeWhile (const False) (undefined : undefined)
--   []
--   
--   
--   
--   >>> take 1 (takeWhile (const True) (1 : undefined))
--   [1]
--   
--   
--   
Examples
--   
--   
--   >>> takeWhile (< 3) [1,2,3,4,1,2,3,4]
--   [1,2]
--   
--   
--   
--   >>> takeWhile (< 9) [1,2,3]
--   [1,2,3]
--   
--   
--   
--   >>> takeWhile (< 0) [1,2,3]
--   []
--   
takeWhile :: (a -> Bool) -> [a] -> [a]

-- | dropWhile p xs returns the suffix remaining after
--   takeWhile p xs.
--   
--   
Examples
--   
--   
--   >>> dropWhile (< 3) [1,2,3,4,5,1,2,3]
--   [3,4,5,1,2,3]
--   
--   
--   
--   >>> dropWhile (< 9) [1,2,3]
--   []
--   
--   
--   
--   >>> dropWhile (< 0) [1,2,3]
--   [1,2,3]
--   
dropWhile :: (a -> Bool) -> [a] -> [a]

-- | splitAt n xs returns a tuple where first element is
--   xs prefix of length n and second element is the
--   remainder of the list:
--   
--   splitAt is an instance of the more general
--   genericSplitAt, in which n may be of any integral
--   type.
--   
--   
Laziness
--   
--   It is equivalent to (take n xs, drop n xs)
--   unless n is _|_: splitAt _|_ xs = _|_, not
--   (_|_, _|_)).
--   
--   The first component of the tuple is produced lazily:
--   
--   
--   >>> fst (splitAt 0 undefined)
--   []
--   
--   
--   
--   >>> take 1 (fst (splitAt 10 (1 : undefined)))
--   [1]
--   
--   
--   
Examples
--   
--   
--   >>> splitAt 6 "Hello World!"
--   ("Hello ","World!")
--   
--   
--   
--   >>> splitAt 3 [1,2,3,4,5]
--   ([1,2,3],[4,5])
--   
--   
--   
--   >>> splitAt 1 [1,2,3]
--   ([1],[2,3])
--   
--   
--   
--   >>> splitAt 3 [1,2,3]
--   ([1,2,3],[])
--   
--   
--   
--   >>> splitAt 4 [1,2,3]
--   ([1,2,3],[])
--   
--   
--   
--   >>> splitAt 0 [1,2,3]
--   ([],[1,2,3])
--   
--   
--   
--   >>> splitAt (-1) [1,2,3]
--   ([],[1,2,3])
--   
splitAt :: Int -> [a] -> ([a], [a])

-- | span, applied to a predicate p and a list xs,
--   returns a tuple where first element is the longest prefix (possibly
--   empty) of xs of elements that satisfy p and second
--   element is the remainder of the list:
--   
--   span p xs is equivalent to (takeWhile p xs,
--   dropWhile p xs), even if p is _|_.
--   
--   
Laziness
--   
--   
--   >>> span undefined []
--   ([],[])
--   
--   >>> fst (span (const False) undefined)
--   *** Exception: Prelude.undefined
--   
--   >>> fst (span (const False) (undefined : undefined))
--   []
--   
--   >>> take 1 (fst (span (const True) (1 : undefined)))
--   [1]
--   
--   
--   span produces the first component of the tuple lazily:
--   
--   
--   >>> take 10 (fst (span (const True) [1..]))
--   [1,2,3,4,5,6,7,8,9,10]
--   
--   
--   
Examples
--   
--   
--   >>> span (< 3) [1,2,3,4,1,2,3,4]
--   ([1,2],[3,4,1,2,3,4])
--   
--   
--   
--   >>> span (< 9) [1,2,3]
--   ([1,2,3],[])
--   
--   
--   
--   >>> span (< 0) [1,2,3]
--   ([],[1,2,3])
--   
span :: (a -> Bool) -> [a] -> ([a], [a])

-- | break, applied to a predicate p and a list
--   xs, returns a tuple where first element is longest prefix
--   (possibly empty) of xs of elements that do not satisfy
--   p and second element is the remainder of the list:
--   
--   break p is equivalent to span (not .
--   p) and consequently to (takeWhile (not . p) xs,
--   dropWhile (not . p) xs), even if p is
--   _|_.
--   
--   
Laziness
--   
--   
--   >>> break undefined []
--   ([],[])
--   
--   
--   
--   >>> fst (break (const True) undefined)
--   *** Exception: Prelude.undefined
--   
--   
--   
--   >>> fst (break (const True) (undefined : undefined))
--   []
--   
--   
--   
--   >>> take 1 (fst (break (const False) (1 : undefined)))
--   [1]
--   
--   
--   break produces the first component of the tuple lazily:
--   
--   
--   >>> take 10 (fst (break (const False) [1..]))
--   [1,2,3,4,5,6,7,8,9,10]
--   
--   
--   
Examples
--   
--   
--   >>> break (> 3) [1,2,3,4,1,2,3,4]
--   ([1,2,3],[4,1,2,3,4])
--   
--   
--   
--   >>> break (< 9) [1,2,3]
--   ([],[1,2,3])
--   
--   
--   
--   >>> break (> 9) [1,2,3]
--   ([1,2,3],[])
--   
break :: (a -> Bool) -> [a] -> ([a], [a])

-- | <math>. reverse xs returns the elements of
--   xs in reverse order. xs must be finite.
--   
--   
Laziness
--   
--   reverse is lazy in its elements.
--   
--   
--   >>> head (reverse [undefined, 1])
--   1
--   
--   
--   
--   >>> reverse (1 : 2 : undefined)
--   *** Exception: Prelude.undefined
--   
--   
--   
Examples
--   
--   
--   >>> reverse []
--   []
--   
--   
--   
--   >>> reverse [42]
--   [42]
--   
--   
--   
--   >>> reverse [2,5,7]
--   [7,5,2]
--   
--   
--   
--   >>> reverse [1..]
--   * Hangs forever *
--   
reverse :: [a] -> [a]

-- | and returns the conjunction of a container of Bools. For the
--   result to be True, the container must be finite; False,
--   however, results from a False value finitely far from the left
--   end.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> and []
--   True
--   
--   
--   
--   >>> and [True]
--   True
--   
--   
--   
--   >>> and [False]
--   False
--   
--   
--   
--   >>> and [True, True, False]
--   False
--   
--   
--   
--   >>> and (False : repeat True) -- Infinite list [False,True,True,True,...
--   False
--   
--   
--   
--   >>> and (repeat True)
--   * Hangs forever *
--   
and :: Foldable t => t Bool -> Bool

-- | or returns the disjunction of a container of Bools. For the
--   result to be False, the container must be finite; True,
--   however, results from a True value finitely far from the left
--   end.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> or []
--   False
--   
--   
--   
--   >>> or [True]
--   True
--   
--   
--   
--   >>> or [False]
--   False
--   
--   
--   
--   >>> or [True, True, False]
--   True
--   
--   
--   
--   >>> or (True : repeat False) -- Infinite list [True,False,False,False,...
--   True
--   
--   
--   
--   >>> or (repeat False)
--   * Hangs forever *
--   
or :: Foldable t => t Bool -> Bool

-- | Determines whether any element of the structure satisfies the
--   predicate.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> any (> 3) []
--   False
--   
--   
--   
--   >>> any (> 3) [1,2]
--   False
--   
--   
--   
--   >>> any (> 3) [1,2,3,4,5]
--   True
--   
--   
--   
--   >>> any (> 3) [1..]
--   True
--   
--   
--   
--   >>> any (> 3) [0, -1..]
--   * Hangs forever *
--   
any :: Foldable t => (a -> Bool) -> t a -> Bool

-- | Determines whether all elements of the structure satisfy the
--   predicate.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> all (> 3) []
--   True
--   
--   
--   
--   >>> all (> 3) [1,2]
--   False
--   
--   
--   
--   >>> all (> 3) [1,2,3,4,5]
--   False
--   
--   
--   
--   >>> all (> 3) [1..]
--   False
--   
--   
--   
--   >>> all (> 3) [4..]
--   * Hangs forever *
--   
all :: Foldable t => (a -> Bool) -> t a -> Bool

-- | notElem is the negation of elem.
--   
--   
Examples
--   
--   Basic usage:
--   
--   
--   >>> 3 `notElem` []
--   True
--   
--   
--   
--   >>> 3 `notElem` [1,2]
--   True
--   
--   
--   
--   >>> 3 `notElem` [1,2,3,4,5]
--   False
--   
--   
--   For infinite structures, notElem terminates if the value exists
--   at a finite distance from the left side of the structure:
--   
--   
--   >>> 3 `notElem` [1..]
--   False
--   
--   
--   
--   >>> 3 `notElem` ([4..] ++ [3])
--   * Hangs forever *
--   
notElem :: (Foldable t, Eq a) => a -> t a -> Bool
infix 4 `notElem`

-- | <math>. lookup key assocs looks up a key in an
--   association list. For the result to be Nothing, the list must
--   be finite.
--   
--   
Examples
--   
--   
--   >>> lookup 2 []
--   Nothing
--   
--   
--   
--   >>> lookup 2 [(1, "first")]
--   Nothing
--   
--   
--   
--   >>> lookup 2 [(1, "first"), (2, "second"), (3, "third")]
--   Just "second"
--   
lookup :: Eq a => a -> [(a, b)] -> Maybe b

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

-- | zip3 takes three lists and returns a list of triples, analogous
--   to zip. It is capable of list fusion, but it is restricted to
--   its first list argument and its resulting list.
zip3 :: [a] -> [b] -> [c] -> [(a, b, c)]

-- | <math>. zipWith generalises zip by zipping with
--   the function given as the first argument, instead of a tupling
--   function.
--   
--   
--   zipWith (,) xs ys == zip xs ys
--   zipWith f [x1,x2,x3..] [y1,y2,y3..] == [f x1 y1, f x2 y2, f x3 y3..]
--   
--   
--   zipWith is right-lazy:
--   
--   
--   >>> let f = undefined
--   
--   >>> zipWith f [] undefined
--   []
--   
--   
--   zipWith is capable of list fusion, but it is restricted to its
--   first list argument and its resulting list.
--   
--   
Examples
--   
--   zipWith (+) can be applied to two lists to
--   produce the list of corresponding sums:
--   
--   
--   >>> zipWith (+) [1, 2, 3] [4, 5, 6]
--   [5,7,9]
--   
--   
--   
--   >>> zipWith (++) ["hello ", "foo"] ["world!", "bar"]
--   ["hello world!","foobar"]
--   
zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]

-- | <math>. The zipWith3 function takes a function which
--   combines three elements, as well as three lists and returns a list of
--   the function applied to corresponding elements, analogous to
--   zipWith. It is capable of list fusion, but it is restricted to
--   its first list argument and its resulting list.
--   
--   
--   zipWith3 (,,) xs ys zs == zip3 xs ys zs
--   zipWith3 f [x1,x2,x3..] [y1,y2,y3..] [z1,z2,z3..] == [f x1 y1 z1, f x2 y2 z2, f x3 y3 z3..]
--   
--   
--   
Examples
--   
--   
--   >>> zipWith3 (\x y z -> [x, y, z]) "123" "abc" "xyz"
--   ["1ax","2by","3cz"]
--   
--   
--   
--   >>> zipWith3 (\x y z -> (x * y) + z) [1, 2, 3] [4, 5, 6] [7, 8, 9]
--   [11,18,27]
--   
zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d]

-- | unzip transforms a list of pairs into a list of first
--   components and a list of second components.
--   
--   
Examples
--   
--   
--   >>> unzip []
--   ([],[])
--   
--   
--   
--   >>> unzip [(1, 'a'), (2, 'b')]
--   ([1,2],"ab")
--   
unzip :: [(a, b)] -> ([a], [b])

-- | The unzip3 function takes a list of triples and returns three
--   lists of the respective components, analogous to unzip.
--   
--   
Examples
--   
--   
--   >>> unzip3 []
--   ([],[],[])
--   
--   
--   
--   >>> unzip3 [(1, 'a', True), (2, 'b', False)]
--   ([1,2],"ab",[True,False])
--   
unzip3 :: [(a, b, c)] -> ([a], [b], [c])

-- | The shows functions return a function that prepends the
--   output String to an existing String. This allows
--   constant-time concatenation of results using function composition.
type ShowS = String -> String

-- | equivalent to showsPrec with a precedence of 0.
shows :: Show a => a -> ShowS

-- | utility function converting a Char to a show function that
--   simply prepends the character unchanged.
showChar :: Char -> ShowS

-- | utility function converting a String to a show function that
--   simply prepends the string unchanged.
showString :: String -> ShowS

-- | utility function that surrounds the inner show function with
--   parentheses when the Bool parameter is True.
showParen :: Bool -> ShowS -> ShowS
even :: Integral a => a -> Bool
odd :: Integral a => a -> Bool

-- | raise a number to an integral power
(^^) :: (Fractional a, Integral b) => a -> b -> a
infixr 8 ^^

-- | gcd x y is the non-negative factor of both x
--   and y of which every common factor of x and
--   y is also a factor; for example gcd 4 2 = 2,
--   gcd (-4) 6 = 2, gcd 0 4 = 4.
--   gcd 0 0 = 0. (That is, the common divisor
--   that is "greatest" in the divisibility preordering.)
--   
--   Note: Since for signed fixed-width integer types, abs
--   minBound < 0, the result may be negative if one of the
--   arguments is minBound (and necessarily is if the other
--   is 0 or minBound) for such types.
gcd :: Integral a => a -> a -> a

-- | lcm x y is the smallest positive integer that both
--   x and y divide.
lcm :: Integral a => a -> a -> a

-- | An infix synonym for fmap.
--   
--   The name of this operator is an allusion to $. Note the
--   similarities between their types:
--   
--   
--    ($)  ::              (a -> b) ->   a ->   b
--   (<$>) :: Functor f => (a -> b) -> f a -> f b
--   
--   
--   Whereas $ is function application, <$> is function
--   application lifted over a Functor.
--   
--   
Examples
--   
--   Convert from a Maybe Int to a Maybe
--   String using show:
--   
--   
--   >>> show <$> Nothing
--   Nothing
--   
--   
--   
--   >>> show <$> Just 3
--   Just "3"
--   
--   
--   Convert from an Either Int Int to an
--   Either Int String using show:
--   
--   
--   >>> show <$> Left 17
--   Left 17
--   
--   
--   
--   >>> show <$> Right 17
--   Right "17"
--   
--   
--   Double each element of a list:
--   
--   
--   >>> (*2) <$> [1,2,3]
--   [2,4,6]
--   
--   
--   Apply even to the second element of a pair:
--   
--   
--   >>> even <$> (2,2)
--   (2,True)
--   
(<$>) :: Functor f => (a -> b) -> f a -> f b
infixl 4 <$>

-- | A parser for a type a, represented as a function that takes a
--   String and returns a list of possible parses as
--   (a,String) pairs.
--   
--   Note that this kind of backtracking parser is very inefficient;
--   reading a large structure may be quite slow (cf ReadP).
type ReadS a = String -> [(a, String)]

-- | The lex function reads a single lexeme from the input,
--   discarding initial white space, and returning the characters that
--   constitute the lexeme. If the input string contains only white space,
--   lex returns a single successful `lexeme' consisting of the
--   empty string. (Thus lex "" = [("","")].) If there is
--   no legal lexeme at the beginning of the input string, lex fails
--   (i.e. returns []).
--   
--   This lexer is not completely faithful to the Haskell lexical syntax in
--   the following respects:
--   
--   
--   
- Qualified names are not handled properly
--   
- Octal and hexadecimal numerics are not recognized as a single
--   token
--   
- Comments are not treated properly
--   
lex :: ReadS String

-- | readParen True p parses what p parses,
--   but surrounded with parentheses.
--   
--   readParen False p parses what p
--   parses, but optionally surrounded with parentheses.
readParen :: Bool -> ReadS a -> ReadS a

-- | Case analysis for the Either type. If the value is
--   Left a, apply the first function to a; if it
--   is Right b, apply the second function to b.
--   
--   
Examples
--   
--   We create two values of type Either String
--   Int, one using the Left constructor and another
--   using the Right constructor. Then we apply "either" the
--   length function (if we have a String) or the "times-two"
--   function (if we have an Int):
--   
--   
--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> either length (*2) s
--   3
--   
--   >>> either length (*2) n
--   6
--   
either :: (a -> c) -> (b -> c) -> Either a b -> c

-- | equivalent to readsPrec with a precedence of 0.
reads :: Read a => ReadS a

-- | The read function reads input from a string, which must be
--   completely consumed by the input process. read fails with an
--   error if the parse is unsuccessful, and it is therefore
--   discouraged from being used in real applications. Use readMaybe
--   or readEither for safe alternatives.
--   
--   
--   >>> read "123" :: Int
--   123
--   
--   
--   
--   >>> read "hello" :: Int
--   *** Exception: Prelude.read: no parse
--   
read :: Read a => String -> a

-- | Map each element of a structure to a monadic action, evaluate these
--   actions from left to right, and ignore the results. For a version that
--   doesn't ignore the results see mapM.
--   
--   mapM_ is just like traverse_, but specialised to monadic
--   actions.
mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()

-- | Evaluate each monadic action in the structure from left to right, and
--   ignore the results. For a version that doesn't ignore the results see
--   sequence.
--   
--   sequence_ is just like sequenceA_, but specialised to
--   monadic actions.
sequence_ :: (Foldable t, Monad m) => t (m a) -> m ()

-- | Splits the argument into a list of lines stripped of their
--   terminating \n characters. The \n terminator is
--   optional in a final non-empty line of the argument string.
--   
--   When the argument string is empty, or ends in a \n character,
--   it can be recovered by passing the result of lines to the
--   unlines function. Otherwise, unlines appends the missing
--   terminating \n. This makes unlines . lines
--   idempotent:
--   
--   
--   (unlines . lines) . (unlines . lines) = (unlines . lines)
--   
--   
--   
Examples
--   
--   
--   >>> lines ""           -- empty input contains no lines
--   []
--   
--   
--   
--   >>> lines "\n"         -- single empty line
--   [""]
--   
--   
--   
--   >>> lines "one"        -- single unterminated line
--   ["one"]
--   
--   
--   
--   >>> lines "one\n"      -- single non-empty line
--   ["one"]
--   
--   
--   
--   >>> lines "one\n\n"    -- second line is empty
--   ["one",""]
--   
--   
--   
--   >>> lines "one\ntwo"   -- second line is unterminated
--   ["one","two"]
--   
--   
--   
--   >>> lines "one\ntwo\n" -- two non-empty lines
--   ["one","two"]
--   
lines :: String -> [String]

-- | Appends a \n character to each input string, then
--   concatenates the results. Equivalent to foldMap (s ->
--   s ++ "\n").
--   
--   
Examples
--   
--   
--   >>> unlines ["Hello", "World", "!"]
--   "Hello\nWorld\n!\n"
--   
--   
--   Note that unlines . lines /=
--   id when the input is not \n-terminated:
--   
--   
--   >>> unlines . lines $ "foo\nbar"
--   "foo\nbar\n"
--   
unlines :: [String] -> String

-- | words breaks a string up into a list of words, which were
--   delimited by white space (as defined by isSpace). This function
--   trims any white spaces at the beginning and at the end.
--   
--   
Examples
--   
--   
--   >>> words "Lorem ipsum\ndolor"
--   ["Lorem","ipsum","dolor"]
--   
--   
--   
--   >>> words " foo bar "
--   ["foo","bar"]
--   
words :: String -> [String]

-- | unwords joins words with separating spaces (U+0020 SPACE).
--   
--   unwords is neither left nor right inverse of words:
--   
--   
--   >>> words (unwords [" "])
--   []
--   
--   >>> unwords (words "foo\nbar")
--   "foo bar"
--   
--   
--   
Examples
--   
--   
--   >>> unwords ["Lorem", "ipsum", "dolor"]
--   "Lorem ipsum dolor"
--   
--   
--   
--   >>> unwords ["foo", "bar", "", "baz"]
--   "foo bar  baz"
--   
unwords :: [String] -> String

-- | The Haskell 2010 type for exceptions in the IO monad. Any I/O
--   operation may raise an IOError instead of returning a result.
--   For a more general type of exception, including also those that arise
--   in pure code, see Exception.
--   
--   In Haskell 2010, this is an opaque type.
type IOError = IOException

-- | Construct an IOError value with a string describing the error.
--   The fail method of the IO instance of the Monad
--   class raises a userError, thus:
--   
--   
--   instance Monad IO where
--     ...
--     fail s = ioError (userError s)
--   
userError :: String -> IOError

-- | File and directory names are values of type String, whose
--   precise meaning is operating system dependent. Files can be opened,
--   yielding a handle which can then be used to operate on the contents of
--   that file.
type FilePath = String

-- | Raise an IOError in the IO monad.
ioError :: IOError -> IO a

-- | Write a character to the standard output device (same as
--   hPutChar stdout).
putChar :: Char -> IO ()

-- | Write a string to the standard output device (same as hPutStr
--   stdout).
putStr :: String -> IO ()

-- | The same as putStr, but adds a newline character.
putStrLn :: String -> IO ()

-- | Read a character from the standard input device (same as
--   hGetChar stdin).
getChar :: IO Char

-- | Read a line from the standard input device (same as hGetLine
--   stdin).
getLine :: IO String

-- | The getContents operation returns all user input as a single
--   string, which is read lazily as it is needed (same as
--   hGetContents stdin).
getContents :: IO String

-- | The interact function takes a function of type
--   String->String as its argument. The entire input from the
--   standard input device is passed to this function as its argument, and
--   the resulting string is output on the standard output device.
interact :: (String -> String) -> IO ()

-- | The readFile function reads a file and returns the contents of
--   the file as a string. The file is read lazily, on demand, as with
--   getContents.
readFile :: FilePath -> IO String

-- | The computation writeFile file str function writes the
--   string str, to the file file.
writeFile :: FilePath -> String -> IO ()

-- | The computation appendFile file str function appends
--   the string str, to the file file.
--   
--   Note that writeFile and appendFile write a literal
--   string to a file. To write a value of any printable type, as with
--   print, use the show function to convert the value to a
--   string first.
--   
--   
--   main = appendFile "squares" (show [(x,x*x) | x <- [0,0.1..2]])
--   
appendFile :: FilePath -> String -> IO ()

-- | The readLn function combines getLine and readIO.
readLn :: Read a => IO a

-- | The readIO function is similar to read except that it
--   signals parse failure to the IO monad instead of terminating
--   the program.
readIO :: Read a => String -> IO a

-- | Abstract syntax for streams and operators.
module Copilot.Language.Stream

-- | A stream in Copilot is an infinite succession of values of the same
--   type.
--   
--   Streams can be built using simple primities (e.g., Const), by
--   applying step-wise (e.g., Op1) or temporal transformations
--   (e.g., Append, Drop) to streams, or by combining
--   existing streams to form new streams (e.g., Op2, Op3).
data Stream a
[Append] :: forall a. Typed a => [a] -> Maybe (Stream Bool) -> Stream a -> Stream a
[Const] :: forall a. Typed a => a -> Stream a
[Drop] :: forall a. Typed a => Int -> Stream a -> Stream a
[Extern] :: forall a. Typed a => String -> Maybe [a] -> Stream a
[Local] :: forall a1 a. (Typed a1, Typed a) => Stream a1 -> (Stream a1 -> Stream a) -> Stream a
[Var] :: forall a. Typed a => String -> Stream a
[Op1] :: forall a1 a. (Typed a1, Typed a) => Op1 a1 a -> Stream a1 -> Stream a
[Op2] :: forall a1 b a. (Typed a1, Typed b, Typed a) => Op2 a1 b a -> Stream a1 -> Stream b -> Stream a
[Op3] :: forall a1 b c a. (Typed a1, Typed b, Typed c, Typed a) => Op3 a1 b c a -> Stream a1 -> Stream b -> Stream c -> Stream a
[Label] :: forall a. Typed a => String -> Stream a -> Stream a

-- | Point-wise application of ceiling to a stream.
--   
--   Unlike the Haskell variant of this function, this variant takes and
--   returns two streams of the same type. Use a casting function to
--   convert the result to an intergral type of your choice.
--   
--   Note that the result can be too big (or, if negative, too small) for
--   that type (see the man page of ceil for details), so you must
--   check that the value fits in the desired integral type before casting
--   it.
--   
--   This definition clashes with one in RealFrac in Haskell's
--   Prelude, re-exported from Language.Copilot, so you need to
--   import this module qualified to use this function.
ceiling :: (Typed a, RealFrac a) => Stream a -> Stream a

-- | Point-wise application of floor to a stream.
--   
--   Unlike the Haskell variant of this function, this variant takes and
--   returns two streams of the same type. Use a casting function to
--   convert the result to an intergral type of your choice.
--   
--   Note that the result can be too big (or, if negative, too small) for
--   that type (see the man page of floor for details), so you
--   must check that the value fits in the desired integral type before
--   casting it.
--   
--   This definition clashes with one in RealFrac in Haskell's
--   Prelude, re-exported from Language.Copilot, so you need to
--   import this module qualified to use this function.
floor :: (Typed a, RealFrac a) => Stream a -> Stream a

-- | Point-wise application of atan2 to the values of two streams.
--   
--   For each pair of real floating-point samples x and
--   y, one from each stream, atan2 computes the angle of
--   the vector from (0, 0) to the point (x, y).
--   
--   This definition clashes with one in RealFloat in Haskell's
--   Prelude, re-exported from Language.Copilot, so you need to
--   import this module qualified to use this function.
atan2 :: (Typed a, RealFloat a) => Stream a -> Stream a -> Stream a
instance GHC.Classes.Eq (Copilot.Language.Stream.Stream a)
instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Internal.Float.Floating a) => GHC.Internal.Float.Floating (Copilot.Language.Stream.Stream a)
instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Internal.Real.Fractional a) => GHC.Internal.Real.Fractional (Copilot.Language.Stream.Stream a)
instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Internal.Num.Num a) => GHC.Internal.Num.Num (Copilot.Language.Stream.Stream a)
instance GHC.Internal.Show.Show (Copilot.Language.Stream.Stream a)

-- | Copilot specifications consistute the main declaration of Copilot
--   modules.
--   
--   A specification normally contains the association between streams to
--   monitor and their handling functions, or streams to observe, or a
--   theorem that must be proved.
--   
--   In order to be executed, Specs must be turned into Copilot Core
--   (see Reify) and either simulated or converted into C99 code
--   to be executed.
module Copilot.Language.Spec

-- | A specification is a list of declarations of triggers, observers,
--   properties and theorems.
--   
--   Specifications are normally declared in monadic style, for example:
--   
--   
--   monitor1 :: Stream Bool
--   monitor1 = [False] ++ not monitor1
--   
--   counter :: Stream Int32
--   counter = [0] ++ not counter
--   
--   spec :: Spec
--   spec = do
--     trigger "handler_1" monitor1 []
--     trigger "handler_2" (counter > 10) [arg counter]
--   
type Spec = Writer [SpecItem] ()

-- | An action in a specification (e.g., a declaration) that returns a
--   value that can be used in subsequent actions.
type Spec' a = Writer [SpecItem] a

-- | Return the complete list of declarations inside a Spec or
--   Spec'.
--   
--   The word run in this function is unrelated to running the underlying
--   specifications or monitors, and is merely related to the monad defined
--   by a Spec
runSpec :: Spec' a -> [SpecItem]

-- | An item of a Copilot specification.
data SpecItem

-- | An observer, representing a stream that we observe during execution at
--   every sample.
data Observer
[Observer] :: forall a. Typed a => String -> Stream a -> Observer

-- | Define a new observer as part of a specification. This allows someone
--   to print the value at every iteration during interpretation. Observers
--   do not have any functionality outside the interpreter.
observer :: Typed a => String -> Stream a -> Spec

-- | Filter a list of spec items to keep only the observers.
observers :: [SpecItem] -> [Observer]

-- | A trigger, representing a function we execute when a boolean stream
--   becomes true at a sample.
data Trigger
[Trigger] :: Name -> Stream Bool -> [Arg] -> Trigger

-- | Define a new trigger as part of a specification. A trigger declares
--   which external function, or handler, will be called when a guard
--   defined by a boolean stream becomes true.
trigger :: String -> Stream Bool -> [Arg] -> Spec

-- | Filter a list of spec items to keep only the triggers.
triggers :: [SpecItem] -> [Trigger]

-- | Construct a function argument from a stream.
--   
--   Args can be used to pass arguments to handlers or trigger
--   functions, to provide additional information to monitor handlers in
--   order to address property violations. At any given point (e.g., when
--   the trigger must be called due to a violation), the arguments passed
--   using arg will contain the current samples of the given
--   streams.
arg :: Typed a => Stream a -> Arg

-- | Wrapper to use Streams as arguments to triggers.
data Arg
[Arg] :: forall a. Typed a => Stream a -> Arg

-- | A property, representing a boolean stream that is existentially or
--   universally quantified over time.
data Property
[Property] :: forall a. String -> Prop a -> Property

-- | A proposition, representing the quantification of a boolean streams
--   over time.
data Prop a
[Forall] :: Stream Bool -> Prop Universal
[Exists] :: Stream Bool -> Prop Existential

-- | A proposition, representing a boolean stream that is existentially or
--   universally quantified over time, as part of a specification.
--   
--   This function returns, in the monadic context, a reference to the
--   proposition.
prop :: String -> Prop a -> Writer [SpecItem] (PropRef a)

-- | Filter a list of spec items to keep only the properties.
properties :: [SpecItem] -> [Property]

-- | A theorem, or proposition together with a proof.
--   
--   This function returns, in the monadic context, a reference to the
--   proposition.
theorem :: String -> Prop a -> Proof a -> Writer [SpecItem] (PropRef a)

-- | Filter a list of spec items to keep only the theorems.
theorems :: [SpecItem] -> [(Property, UProof)]

-- | Universal quantification of boolean streams over time.
forAll :: Stream Bool -> Prop Universal

-- | Existential quantification of boolean streams over time.
exists :: Stream Bool -> Prop Existential

-- | Extract the underlying stream from a quantified proposition.
--   
--   Think carefully before using this function, as this function will
--   remove the quantifier from a proposition. Universally quantified
--   streams usually require separate treatment from existentially
--   quantified ones, so carelessly using this function to remove
--   quantifiers can result in hard-to-spot soundness bugs.
extractProp :: Prop a -> Stream Bool

-- | Empty datatype to mark proofs of universally quantified predicates.
data Universal

-- | Empty datatype to mark proofs of existentially quantified predicates.
data Existential

-- | Temporal stream transformations.
module Copilot.Language.Operators.Temporal

-- | Prepend a fixed number of samples to a stream.
--   
--   The elements to be appended at the beginning of the stream must be
--   limited, that is, the list must have finite length.
--   
--   Prepending elements to a stream may increase the memory requirements
--   of the generated programs (which now must hold the same number of
--   elements in memory for future processing).
(++) :: Typed a => [a] -> Stream a -> Stream a
infixr 1 ++

-- | Drop a number of samples from a stream.
--   
--   The elements must be realizable at the present time to be able to drop
--   elements. For most kinds of streams, you cannot drop elements without
--   prepending an equal or greater number of elements to them first, as it
--   could result in undefined samples.
drop :: Typed a => Int -> Stream a -> Stream a

-- | Interface to access portions of a larger data structure.
--   
--   Default operations to access elements from structs and arrays (e.g., a
--   field of a struct, an element of an array) allow obtaining the values
--   of those elements, but not modifying.
--   
--   This module defines a common interface to manipulate portions of a
--   larger data structure. We define the interface in a generic way, using
--   a type class with two operations: one to set the value of the
--   sub-element, and one to update the value of such element applying a
--   stream function.
module Copilot.Language.Operators.Projection

-- | Common interface to manipulate portions of a larger data structure.
--   
--   A projectable d s t means that it is possible to manipulate a
--   sub-element s of type t carried in a stream of type d.
class Projectable d s t | d s -> t where {
    
    -- | Unapplied projection or element access on a type.
    data Projection d s t;
}

-- | Modify the value of a sub-element of a type in a stream of elements of
--   that type.
(=:) :: Projectable d s t => Projection d s t -> Stream t -> Stream d

-- | Update the value of a sub-element of a type in a stream of elements of
--   that type, by applying a function on streams.
(=$) :: Projectable d s t => Projection d s t -> (Stream t -> Stream t) -> Stream d
infixl 8 =:
infixl 8 =$

-- | Combinators to deal with streams carrying structs.
--   
--   We support two kinds of operations on structs: reading the fields of
--   structs and modifying the fields of structs.
--   
--   To obtain the values of field x of a struct s, you
--   can just write:
--   
--   
--   expr = s # x
--   
--   
--   If you want to update it, use instead a double hash to refer to the
--   field. You can either update the field:
--   
--   
--   expr = s ## x =: 75
--   
--   
--   To update it by applying a function to it, for example, the function
--   that updates a stream by one unit, just do:
--   
--   
--   expr = s ## x =$ (+1)
--   
module Copilot.Language.Operators.Struct

-- | Common interface to manipulate portions of a larger data structure.
--   
--   A projectable d s t means that it is possible to manipulate a
--   sub-element s of type t carried in a stream of type d.
class Projectable d s t | d s -> t where {
    
    -- | Unapplied projection or element access on a type.
    data Projection d s t;
}

-- | Modify the value of a sub-element of a type in a stream of elements of
--   that type.
(=:) :: Projectable d s t => Projection d s t -> Stream t -> Stream d

-- | Update the value of a sub-element of a type in a stream of elements of
--   that type, by applying a function on streams.
(=$) :: Projectable d s t => Projection d s t -> (Stream t -> Stream t) -> Stream d
infixl 8 =:
infixl 8 =$

-- | Create a stream that carries a field of a struct in another stream.
--   
--   This function implements a projection of a field of a struct over
--   time. For example, if a struct of type T has two fields,
--   t1 of type Int and t2 of type
--   Word8, and s is a stream of type Stream T,
--   then s # t2 has type Stream Word8 and contains the
--   values of the t2 field of the structs in s at any
--   point in time.
(#) :: forall (f :: Symbol) t s. (KnownSymbol f, Typed t, Typed s, Struct s) => Stream s -> (s -> Field f t) -> Stream t

-- | Pair a stream with a field accessor, without applying it to obtain the
--   value of the field.
--   
--   This function is needed to refer to a field accessor when the goal is
--   to update the field value, not just to read it.
(##) :: forall (f :: Symbol) t s. (KnownSymbol f, Typed t, Typed s, Struct s) => Stream s -> (s -> Field f t) -> Projection s (s -> Field f t) t
instance (GHC.Internal.TypeLits.KnownSymbol f, Copilot.Core.Type.Typed s, Copilot.Core.Type.Typed t, Copilot.Core.Type.Struct s) => Copilot.Language.Operators.Projection.Projectable s (s -> Copilot.Core.Type.Field f t) t

-- | Comparison operators applied point-wise on streams.
module Copilot.Language.Operators.Ord

-- | Compare two streams point-wise for order.
--   
--   The output stream contains the value True at a point in time if the
--   element in the first stream is smaller or equal than the element in
--   the second stream at that point in time, and False otherwise.
(<=) :: (Ord a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Compare two streams point-wise for order.
--   
--   The output stream contains the value True at a point in time if the
--   element in the first stream is greater or equal than the element in
--   the second stream at that point in time, and False otherwise.
(>=) :: (Ord a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Compare two streams point-wise for order.
--   
--   The output stream contains the value True at a point in time if the
--   element in the first stream is smaller than the element in the second
--   stream at that point in time, and False otherwise.
(<) :: (Ord a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Compare two streams point-wise for order.
--   
--   The output stream contains the value True at a point in time if the
--   element in the first stream is greater than the element in the second
--   stream at that point in time, and False otherwise.
(>) :: (Ord a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Pick values from one of two streams, depending whether a condition is
--   true or false.
module Copilot.Language.Operators.Mux

-- | Convenient synonym for ifThenElse.
mux :: Typed a => Stream Bool -> Stream a -> Stream a -> Stream a

-- | If-then-else applied point-wise to three streams (a condition stream,
--   a then-branch stream, and an else-branch stream).
--   
--   Produce a stream that, at any point in time, if the value of the first
--   stream at that point is true, contains the value in the second stream
--   at that time, otherwise it contains the value in the third stream.
ifThenElse :: Typed a => Stream Bool -> Stream a -> Stream a -> Stream a

-- | Let expressions.
--   
--   Although Copilot is a DSL embedded in Haskell and Haskell does support
--   let expressions, we want Copilot to be able to implement sharing, to
--   detect when the same stream is being used in multiple places in a
--   specification and avoid recomputing it unnecessarily.
module Copilot.Language.Operators.Local

-- | Let expressions.
--   
--   Create a stream that results from applying a stream to a function on
--   streams. Standard usage would be similar to Haskell's let. See the
--   following example, where stream1, stream2 and
--   s are all streams carrying values of some numeric type:
--   
--   
--   expression = local (stream1 + stream2) $ \s ->
--                (s >= 0 && s <= 10)
--   
--   
local :: (Typed a, Typed b) => Stream a -> (Stream a -> Stream b) -> Stream b

-- | Label a stream with additional information.
module Copilot.Language.Operators.Label

-- | This function allows you to label a stream with a tag, which can be
--   used by different backends to provide additional information either in
--   error messages or in the generated code (e.g., for traceability
--   purposes).
--   
--   Semantically, a labelled stream is just the stream inside it. The use
--   of label should not affect the observable behavior of the monitor, and
--   how it is used in the code generated is a decision specific to each
--   backend.
label :: Typed a => String -> Stream a -> Stream a

-- | Primitives to build streams connected to external variables.
module Copilot.Language.Operators.Extern

-- | Create a stream populated by an external global variable.
--   
--   The Copilot compiler does not check that the type is correct. If the
--   list given as second argument does not constrain the type of the
--   values carried by the stream, this primitive stream building function
--   will match any stream of any type, which is potentially dangerous if
--   the global variable mentioned has a different type. We rely on the
--   compiler used with the generated code to detect type errors of this
--   kind.
extern :: Typed a => String -> Maybe [a] -> Stream a

-- | Create a stream carrying values of type Bool, populated by an external
--   global variable.
externB :: String -> Maybe [Bool] -> Stream Bool

-- | Create a stream carrying values of type Word8, populated by an
--   external global variable.
externW8 :: String -> Maybe [Word8] -> Stream Word8

-- | Create a stream carrying values of type Word16, populated by an
--   external global variable.
externW16 :: String -> Maybe [Word16] -> Stream Word16

-- | Create a stream carrying values of type Word32, populated by an
--   external global variable.
externW32 :: String -> Maybe [Word32] -> Stream Word32

-- | Create a stream carrying values of type Word64, populated by an
--   external global variable.
externW64 :: String -> Maybe [Word64] -> Stream Word64

-- | Create a stream carrying values of type Int8, populated by an external
--   global variable.
externI8 :: String -> Maybe [Int8] -> Stream Int8

-- | Create a stream carrying values of type Int16, populated by an
--   external global variable.
externI16 :: String -> Maybe [Int16] -> Stream Int16

-- | Create a stream carrying values of type Int32, populated by an
--   external global variable.
externI32 :: String -> Maybe [Int32] -> Stream Int32

-- | Create a stream carrying values of type Int64, populated by an
--   external global variable.
externI64 :: String -> Maybe [Int64] -> Stream Int64

-- | Create a stream carrying values of type Float, populated by an
--   external global variable.
externF :: String -> Maybe [Float] -> Stream Float

-- | Create a stream carrying values of type Double, populated by an
--   external global variable.
externD :: String -> Maybe [Double] -> Stream Double

-- | Equality applied point-wise on streams.
module Copilot.Language.Operators.Eq

-- | Compare two streams point-wise for equality.
--   
--   The output stream contains the value True at a point in time if both
--   argument streams contain the same value at that point in time.
(==) :: (Eq a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Compare two streams point-wise for inequality.
--   
--   The output stream contains the value True at a point in time if both
--   argument streams contain different values at that point in time.
(/=) :: (Eq a, Typed a) => Stream a -> Stream a -> Stream Bool

-- | Primitives to build constant streams.
module Copilot.Language.Operators.Constant

-- | Create a constant stream that is equal to the given argument, at any
--   point in time.
constant :: Typed a => a -> Stream a

-- | Create a constant stream carrying values of type Bool that is
--   equal to the given argument, at any point in time.
constB :: Bool -> Stream Bool

-- | Create a constant stream carrying values of type Word8 that is
--   equal to the given argument, at any point in time.
constW8 :: Word8 -> Stream Word8

-- | Create a constant stream carrying values of type Word16 that is
--   equal to the given argument, at any point in time.
constW16 :: Word16 -> Stream Word16

-- | Create a constant stream carrying values of type Word32 that is
--   equal to the given argument, at any point in time.
constW32 :: Word32 -> Stream Word32

-- | Create a constant stream carrying values of type Word64 that is
--   equal to the given argument, at any point in time.
constW64 :: Word64 -> Stream Word64

-- | Create a constant stream carrying values of type Int8 that is
--   equal to the given argument, at any point in time.
constI8 :: Int8 -> Stream Int8

-- | Create a constant stream carrying values of type Int16 that is
--   equal to the given argument, at any point in time.
constI16 :: Int16 -> Stream Int16

-- | Create a constant stream carrying values of type Int32 that is
--   equal to the given argument, at any point in time.
constI32 :: Int32 -> Stream Int32

-- | Create a constant stream carrying values of type Int64 that is
--   equal to the given argument, at any point in time.
constI64 :: Int64 -> Stream Int64

-- | Create a constant stream carrying values of type Float that is
--   equal to the given argument, at any point in time.
constF :: Float -> Stream Float

-- | Create a constant stream carrying values of type Double that is
--   equal to the given argument, at any point in time.
constD :: Double -> Stream Double

-- | Type-safe casting operators.
module Copilot.Language.Operators.Cast

-- | Perform a safe cast from Stream a to Stream b.
cast :: Cast a b => Stream a -> Stream b

-- | Perform an unsafe cast from Stream a to Stream b.
unsafeCast :: UnsafeCast a b => Stream a -> Stream b

-- | Class to capture casting between types for which it can be performed
--   safely.
class Cast a b

-- | Class to capture casting between types for which casting may be unsafe
--   and/or result in a loss of precision or information.
class UnsafeCast a b
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Types.Bool
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Int.Int8
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int16 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int16 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int16 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int32 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int32 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int64 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int8 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int8 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int8 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Int.Int8 GHC.Internal.Int.Int8
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word16 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word16 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word16 GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word16 GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word16 GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word32 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word32 GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word32 GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word64 GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.Cast GHC.Internal.Word.Word8 GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int16 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int16 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int16 GHC.Internal.Int.Int8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int16 GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int32 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int32 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int32 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int32 GHC.Internal.Int.Int8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int32 GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Internal.Int.Int8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int64 GHC.Internal.Word.Word64
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int8 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int8 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Int.Int8 GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word16 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word16 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word16 GHC.Internal.Int.Int16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word16 GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word32 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word32 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word32 GHC.Internal.Int.Int32
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word32 GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word32 GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Internal.Int.Int64
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Internal.Word.Word16
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Internal.Word.Word32
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word64 GHC.Internal.Word.Word8
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word8 GHC.Types.Double
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word8 GHC.Types.Float
instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Internal.Word.Word8 GHC.Internal.Int.Int8

-- | Boolean operators applied point-wise to streams.
module Copilot.Language.Operators.Boolean

-- | Apply the and (&&) operator to two boolean streams,
--   point-wise.
(&&) :: Stream Bool -> Stream Bool -> Stream Bool
infixr 4 &&

-- | Apply the or (||) operator to two boolean streams, point-wise.
(||) :: Stream Bool -> Stream Bool -> Stream Bool
infixr 4 ||

-- | Negate all the values in a boolean stream.
not :: Stream Bool -> Stream Bool

-- | A stream that contains the constant value True.
true :: Stream Bool

-- | A stream that contains the constant value False.
false :: Stream Bool

-- | Apply the exclusive-or (xor) operator to two boolean streams,
--   point-wise.
xor :: Stream Bool -> Stream Bool -> Stream Bool

-- | Apply the implication (==>) operator to two boolean streams,
--   point-wise.
(==>) :: Stream Bool -> Stream Bool -> Stream Bool

-- | Implement negation over quantified extensions of boolean streams.
--   
--   For details, see Prop.
module Copilot.Language.Operators.Propositional

-- | Negate a proposition.
not :: Negatable a b => a -> b
instance Copilot.Language.Operators.Propositional.Negatable (Copilot.Language.Spec.Prop Copilot.Theorem.Prove.Existential) (Copilot.Language.Spec.Prop Copilot.Theorem.Prove.Universal)
instance Copilot.Language.Operators.Propositional.Negatable (Copilot.Language.Spec.Prop Copilot.Theorem.Prove.Universal) (Copilot.Language.Spec.Prop Copilot.Theorem.Prove.Existential)

-- | Bitwise operators applied on streams, pointwise.
module Copilot.Language.Operators.BitWise

-- | The Bits class defines bitwise operations over integral types.
--   
--   
--   
- Bits are numbered from 0 with bit 0 being the least significant
--   bit.
--   
class Eq a => Bits a

-- | Bitwise "and"
(.&.) :: Bits a => a -> a -> a

-- | Bitwise "or"
(.|.) :: Bits a => a -> a -> a

-- | Reverse all the bits in the argument
complement :: Bits a => a -> a
infixl 7 .&.
infixl 5 .|.

-- | Infix version of xor.
(.^.) :: Bits a => a -> a -> a
infixl 6 .^.

-- | Shifting values of a stream to the left.
(.<<.) :: (Bits a, Typed a, Typed b, Integral b) => Stream a -> Stream b -> Stream a

-- | Shifting values of a stream to the right.
(.>>.) :: (Bits a, Typed a, Typed b, Integral b) => Stream a -> Stream b -> Stream a
instance (Copilot.Core.Type.Typed a, GHC.Internal.Bits.Bits a) => GHC.Internal.Bits.Bits (Copilot.Language.Stream.Stream a)

-- | Integral class operators applied point-wise on streams.
module Copilot.Language.Operators.Integral

-- | Apply the div operation to two streams, point-wise.
div :: (Typed a, Integral a) => Stream a -> Stream a -> Stream a

-- | Apply the mod operation to two streams, point-wise.
mod :: (Typed a, Integral a) => Stream a -> Stream a -> Stream a

-- | Apply a limited form of exponentiation (^) to two streams,
--   point-wise.
--   
--   Either the first stream must be the constant 2, or the second must be
--   a constant stream.
(^) :: (Typed a, Typed b, Num a, Bits a, Integral b) => Stream a -> Stream b -> Stream a

-- | Combinators to deal with streams carrying arrays.
module Copilot.Language.Operators.Array

-- | Create a stream that carries an element of an array in another stream.
--   
--   This function implements a projection of the element of an array at a
--   given position, over time. For example, if s is a stream of
--   type Stream (Array '5 Word8), then s ! 3 has type
--   Stream Word8 and contains the 3rd element (starting from
--   zero) of the arrays in s at any point in time.
(!) :: forall (n :: Nat) t. (KnownNat n, Typed t) => Stream (Array n t) -> Stream Word32 -> Stream t

-- | Pair a stream with an element accessor, without applying it to obtain
--   the value of the element.
--   
--   This function is needed to refer to an element accessor when the goal
--   is to update the element value, not just to read it.
(!!) :: forall (n :: Nat) t. Stream (Array n t) -> Stream Word32 -> Projection (Array n t) (Stream Word32) t

-- | Modify the value of a sub-element of a type in a stream of elements of
--   that type.
(=:) :: Projectable d s t => Projection d s t -> Stream t -> Stream d
infixl 8 =:

-- | Update the value of a sub-element of a type in a stream of elements of
--   that type, by applying a function on streams.
(=$) :: Projectable d s t => Projection d s t -> (Stream t -> Stream t) -> Stream d
infixl 8 =$
instance (GHC.Internal.TypeNats.KnownNat n, Copilot.Core.Type.Typed t) => Copilot.Language.Operators.Projection.Projectable (Copilot.Core.Type.Array.Array n t) (Copilot.Language.Stream.Stream GHC.Internal.Word.Word32) t

-- | Transform a Copilot Language specification into a Copilot Core
--   specification.
module Copilot.Language.Reify

-- | Transform a Copilot Language specification into a Copilot Core
--   specification.
reify :: Spec' a -> IO Spec

-- | Main Copilot language export file.
--   
--   This is mainly a meta-module that re-exports most definitions in this
--   library.
module Copilot.Language

-- | 8-bit signed integer type
data Int8

-- | 16-bit signed integer type
data Int16

-- | 32-bit signed integer type
data Int32

-- | 64-bit signed integer type
data Int64

-- | A typed expression, from which we can obtain the two type
--   representations used by Copilot: Type and SimpleType.
class (Show a, Typeable a) => Typed a

-- | A name of a trigger, an external variable, or an external function.
type Name = String

-- | Report an error due to an error detected by Copilot (e.g., user
--   error).
badUsage :: String -> a

-- | Report an error due to a bug in Copilot.
impossible :: String -> String -> a

-- | Simulate a number of steps of a given specification, printing the
--   results in a table in readable format.
--   
--   Compared to csv, this function is slower but the output may be
--   more readable.
interpret :: Integer -> Spec -> IO ()

-- | Simulate a number of steps of a given specification, printing the
--   results in a table in comma-separated value (CSV) format.
csv :: Integer -> Spec -> IO ()

-- | A specification is a list of declarations of triggers, observers,
--   properties and theorems.
--   
--   Specifications are normally declared in monadic style, for example:
--   
--   
--   monitor1 :: Stream Bool
--   monitor1 = [False] ++ not monitor1
--   
--   counter :: Stream Int32
--   counter = [0] ++ not counter
--   
--   spec :: Spec
--   spec = do
--     trigger "handler_1" monitor1 []
--     trigger "handler_2" (counter > 10) [arg counter]
--   
type Spec = Writer [SpecItem] ()

-- | A stream in Copilot is an infinite succession of values of the same
--   type.
--   
--   Streams can be built using simple primities (e.g., Const), by
--   applying step-wise (e.g., Op1) or temporal transformations
--   (e.g., Append, Drop) to streams, or by combining
--   existing streams to form new streams (e.g., Op2, Op3).
data Stream a

-- | Define a new observer as part of a specification. This allows someone
--   to print the value at every iteration during interpretation. Observers
--   do not have any functionality outside the interpreter.
observer :: Typed a => String -> Stream a -> Spec

-- | Define a new trigger as part of a specification. A trigger declares
--   which external function, or handler, will be called when a guard
--   defined by a boolean stream becomes true.
trigger :: String -> Stream Bool -> [Arg] -> Spec

-- | Construct a function argument from a stream.
--   
--   Args can be used to pass arguments to handlers or trigger
--   functions, to provide additional information to monitor handlers in
--   order to address property violations. At any given point (e.g., when
--   the trigger must be called due to a violation), the arguments passed
--   using arg will contain the current samples of the given
--   streams.
arg :: Typed a => Stream a -> Arg

-- | A proposition, representing a boolean stream that is existentially or
--   universally quantified over time, as part of a specification.
--   
--   This function returns, in the monadic context, a reference to the
--   proposition.
prop :: String -> Prop a -> Writer [SpecItem] (PropRef a)

-- | A theorem, or proposition together with a proof.
--   
--   This function returns, in the monadic context, a reference to the
--   proposition.
theorem :: String -> Prop a -> Proof a -> Writer [SpecItem] (PropRef a)

-- | Universal quantification of boolean streams over time.
forAll :: Stream Bool -> Prop Universal

-- | Existential quantification of boolean streams over time.
exists :: Stream Bool -> Prop Existential
