-- 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 3.8 -- | Reexports Prelude from package "base" hiding identifiers -- redefined by Copilot. module Copilot.Language.Prelude -- | 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 :: forall (r :: RuntimeRep) a (b :: TYPE r). a -> b -> b infixr 0 `seq` -- | <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]
--

-- --

--   >>> filter odd [1, 2, 3]
--   [1,3]
--

filter :: (a -> Bool) -> [a] -> [a] -- | <math>. zip takes two lists and returns a list of -- corresponding pairs. -- --

--   zip [1, 2] ['a', 'b'] = [(1, 'a'), (2, 'b')]
--

-- -- If one input list is short, excess elements of the longer list are -- discarded: -- --

--   zip [1] ['a', 'b'] = [(1, 'a')]
--   zip [1, 2] ['a'] = [(1, 'a')]
--

-- -- zip is right-lazy: -- --

--   zip [] _|_ = []
--   zip _|_ [] = _|_
--

-- -- zip is capable of list fusion, but it is restricted to its -- first list argument and its resulting list. 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 () -- | Extract the first component of a pair. fst :: (a, b) -> a -- | Extract the second component of a pair. snd :: (a, b) -> b -- | 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, ...]
--

-- --

--   >>> map (+1) [1, 2, 3]
--

map :: (a -> b) -> [a] -> [b] -- | Application operator. This operator is redundant, since ordinary -- application (f x) means the same as (f $ x). -- However, $ has low, right-associative binding precedence, so it -- sometimes allows parentheses to be omitted; for example: -- --

--   f $ g $ h x  =  f (g (h x))
--

-- -- It is also useful in higher-order situations, such as map -- ($ 0) xs, or zipWith ($) fs xs. -- -- Note that ($) is levity-polymorphic in its result -- type, so that foo $ True where foo :: Bool -> -- Int# is well-typed. ($) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b infixr 0 $ -- | general coercion from integral types fromIntegral :: (Integral a, Num b) => a -> b -- | general coercion to fractional types realToFrac :: (Real a, Fractional b) => a -> b -- | 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 -- | 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 -- | the successor of a value. For numeric types, succ adds 1. succ :: Enum a => a -> a -- | the 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). For example: -- --

```
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 For example: -- --

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 = []. For -- example: -- --

```
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 = [] For example: -- --

enumFromThenTo 4 2 -6 :: [Integer] =
--   [4,2,0,-2,-4,-6]

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

enumFromThenTo :: Enum a => a -> a -> 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, == -- is customarily expected to implement an equivalence relationship where -- two values comparing equal are indistinguishable by "public" -- functions, with a "public" function being one not allowing to see -- implementation details. For example, for a type representing -- non-normalised natural numbers modulo 100, a "public" function doesn't -- make the difference between 1 and 201. It is expected to have the -- following properties: -- --

Reflexivity x == x = True
Symmetry x == y = y == x
Transitivity if x == y && y == z = -- True, then x == z = True
Substitutivity if x == y = True and -- f is a "public" function whose return type is an instance of -- Eq, then f x == f y = True
Negation x /= y = not (x == -- y)

-- -- Minimal complete definition: either == or /=. 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 ** -- | 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

-- -- 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 / -- | 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. class (Real a, Enum a) => Integral a -- | integer division truncated toward zero quot :: Integral a => a -> a -> a -- | integer remainder, satisfying -- --

--   (x `quot` y)*y + (x `rem` y) == x
--

rem :: Integral a => a -> a -> a -- | simultaneous quot and rem quotRem :: Integral a => a -> a -> (a, a) -- | simultaneous div and mod divMod :: Integral a => a -> a -> (a, a) -- | conversion to Integer toInteger :: Integral a => a -> Integer infixl 7 `rem` infixl 7 `quot` -- | 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 lists, 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
--

(>>=) :: 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
--

(>>) :: Monad m => m a -> m b -> m b -- | Inject a value into the monadic type. 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. class Functor (f :: Type -> Type) -- | Using ApplicativeDo: 'fmap f as' can be -- understood as the do expression -- --

--   do a <- as
--      pure (f a)
--

-- -- with an inferred Functor constraint. 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. -- -- Using ApplicativeDo: 'a <$ bs' can be -- understood as the do expression -- --

--   do bs
--      pure a
--

-- -- with an inferred Functor constraint. (<$) :: Functor f => a -> f b -> f a infixl 4 <$ -- | 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)

-- -- 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 * -- | 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. -- -- The Haskell Report defines no laws for Ord. However, -- <= is customarily expected to implement a non-strict partial -- order and have the following properties: -- --

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

-- -- Note that 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 should be are expected to use -- double quotes, rather than square brackets. readList :: Read a => ReadS [a] class (Num a, Ord a) => Real a -- | the rational equivalent of its real argument with full precision toRational :: Real a => a -> Rational -- | 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 -- | 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
--

class Monad m => MonadFail (m :: Type -> Type) fail :: MonadFail m => String -> m 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. pure :: Applicative f => a -> f a -- | Sequential application. -- -- A few functors support an implementation of <*> that is -- more efficient than the default one. -- -- Using ApplicativeDo: 'fs <*> as' can be -- understood as the do expression -- --

--   do f <- fs
--      a <- as
--      pure (f a)
--

(<*>) :: Applicative f => f (a -> b) -> f a -> f b -- | Sequence actions, discarding the value of the first argument. -- -- 'as *> bs' can be understood as the do -- expression -- --

--   do as
--      bs
--

-- -- This is a tad complicated for our ApplicativeDo extension -- which will give it a Monad constraint. For an -- Applicative constraint we write it of the form -- --

--   do _ <- as
--      b <- bs
--      pure b
--

(*>) :: Applicative f => f a -> f b -> f b -- | Sequence actions, discarding the value of the second argument. -- -- Using ApplicativeDo: 'as <* bs' can be -- understood as the do expression -- --

--   do a <- as
--      bs
--      pure a
--

(<*) :: Applicative f => f a -> f b -> f a infixl 4 <* infixl 4 *> infixl 4 <*> -- | Data structures that can be folded. -- -- For example, given a data type -- --

--   data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
--

-- -- a suitable instance would be -- --

--   instance Foldable Tree where
--      foldMap f Empty = mempty
--      foldMap f (Leaf x) = f x
--      foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r
--

-- -- This is suitable even for abstract types, as the monoid is assumed to -- satisfy the monoid laws. Alternatively, one could define -- foldr: -- --

--   instance Foldable Tree where
--      foldr f z Empty = z
--      foldr f z (Leaf x) = f x z
--      foldr f z (Node l k r) = foldr f (f k (foldr f z r)) l
--

-- -- Foldable instances are expected to satisfy the following -- laws: -- --

--   foldr f z t = appEndo (foldMap (Endo . f) t ) z
--

-- --

--   foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z
--

-- --

--   fold = foldMap id
--

-- --

--   length = getSum . foldMap (Sum . const  1)
--

-- -- sum, product, maximum, and minimum -- should all be essentially equivalent to foldMap forms, such -- as -- --

--   sum = getSum . foldMap Sum
--

-- -- but may be less defined. -- -- If the type is also a Functor instance, it should satisfy -- --

--   foldMap f = fold . fmap f
--

-- -- which implies that -- --

--   foldMap f . fmap g = foldMap (f . g)
--

class Foldable (t :: Type -> Type) -- | Map each element of the structure to a monoid, and combine the -- results. foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m -- | Right-associative fold of a structure. -- -- 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, -- foldr can produce a terminating expression from an infinite -- list. -- -- For a general Foldable structure this should be semantically -- identical to, -- --

--   foldr f z = foldr f z . toList
--

foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b -- | Left-associative fold of a structure. -- -- 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. This means that foldl' -- will diverge if given an infinite list. -- -- Also note that if you want an efficient left-fold, you probably want -- to use foldl' instead of foldl. The reason for this is -- that 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 <math> -- 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
--

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. -- --

--   foldr1 f = foldr1 f . toList
--

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. -- --

--   foldl1 f = foldl1 f . toList
--

foldl1 :: Foldable t => (a -> a -> a) -> t a -> a -- | Test whether the structure is empty. The default implementation is -- optimized for structures that are similar to cons-lists, because there -- is no general way to do better. null :: Foldable t => t a -> Bool -- | Returns the size/length of a finite structure as an Int. The -- default implementation is optimized for structures that are similar to -- cons-lists, because there is no general way to do better. length :: Foldable t => t a -> Int -- | Does the element occur in the structure? elem :: (Foldable t, Eq a) => a -> t a -> Bool -- | The largest element of a non-empty structure. maximum :: (Foldable t, Ord a) => t a -> a -- | The least element of a non-empty structure. minimum :: (Foldable t, Ord a) => t a -> a -- | The product function computes the product of the numbers of a -- structure. product :: (Foldable t, Num a) => t a -> a infix 4 `elem` -- | Functors representing data structures that can be traversed from left -- to right. -- -- A definition of traverse must satisfy the following laws: -- --

Naturality t . traverse f = traverse (t . -- f) for every applicative transformation t
Identity traverse Identity = -- Identity
Composition traverse (Compose . -- fmap g . f) = Compose . fmap (traverse g) -- . traverse f

-- -- A definition of sequenceA must satisfy the following laws: -- --

Naturality t . sequenceA = sequenceA . -- fmap t for every applicative transformation -- t
Identity sequenceA . fmap Identity -- = Identity
Composition sequenceA . fmap -- Compose = Compose . fmap sequenceA . -- sequenceA

-- -- where an applicative transformation is a function -- --

--   t :: (Applicative f, Applicative g) => f a -> g a
--

-- -- preserving the Applicative operations, i.e. -- --

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

-- -- and the identity functor Identity and composition functors -- Compose are from Data.Functor.Identity and -- Data.Functor.Compose. -- -- A result of the naturality law is a purity law for traverse -- --

--   traverse pure = pure
--

-- -- (The naturality law is implied by parametricity and thus so is the -- purity law [1, p15].) -- -- Instances are similar to Functor, e.g. given a data type -- --

--   data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
--

-- -- a suitable instance would be -- --

--   instance Traversable Tree where
--      traverse f Empty = pure Empty
--      traverse f (Leaf x) = Leaf <$> f x
--      traverse f (Node l k r) = Node <$> traverse f l <*> f k <*> traverse f r
--

-- -- This is suitable even for abstract types, as the laws for -- <*> imply a form of associativity. -- -- The superclass instances should satisfy the following: -- --

In the Functor instance, fmap should be equivalent -- to traversal with the identity applicative functor -- (fmapDefault).
In the Foldable instance, foldMap should be -- equivalent to traversal with a constant applicative functor -- (foldMapDefault).

-- -- References: [1] The Essence of the Iterator Pattern, Jeremy Gibbons -- and Bruno C. d. S. Oliveira 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_. 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_. 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_. 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_. sequence :: (Traversable t, Monad m) => t (m a) -> m (t a) -- | The class of semigroups (types with an associative binary operation). -- -- Instances should satisfy the following: -- --

Associativity x <> (y <> z) = -- (x <> y) <> z

class Semigroup a -- | An associative operation. -- --

--   >>> [1,2,3] <> [4,5,6]
--   [1,2,3,4,5,6]
--

(<>) :: 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

-- -- 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 -- --

--   >>> "Hello world" <> mempty
--   "Hello world"
--

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 data Bool False :: Bool True :: Bool -- | The character type Char is an enumeration whose values -- represent Unicode (or equivalently ISO/IEC 10646) code points (i.e. -- characters, see http://www.unicode.org/ for details). This set -- extends the ISO 8859-1 (Latin-1) character set (the first 256 -- characters), which is itself an extension of the ASCII character set -- (the first 128 characters). A character literal in Haskell has type -- Char. -- -- To convert a Char to or from the corresponding Int value -- defined by Unicode, use toEnum and fromEnum from the -- Enum class respectively (or equivalently ord and -- chr). data Char -- | 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 -- | 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 -- | 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 -- | Arbitrary precision integers. In contrast with fixed-size integral -- types such as Int, the Integer type represents the -- entire infinite range of integers. -- -- For more information about this type's representation, see the -- comments in its implementation. data Integer -- | 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 data Ordering LT :: Ordering EQ :: Ordering GT :: Ordering -- | 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 -- | 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 -- | A Word is an unsigned integral type, with the same size as -- Int. data Word -- | 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 -- | Identity function. -- --

--   id x = x
--

id :: a -> 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 -- | A String is a list of characters. String constants in Haskell -- are values of type String. -- -- See Data.List for operations on lists. type String = [Char] -- | 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 -- | 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 -- | The readLn function combines getLine and readIO. readLn :: Read a => IO a -- | 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 computation writeFile file str function writes the -- string str, to the file file. writeFile :: FilePath -> 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 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 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 -- | Read a line from the standard input device (same as hGetLine -- stdin). getLine :: IO String -- | Read a character from the standard input device (same as -- hGetChar stdin). getChar :: IO Char -- | The same as putStr, but adds a newline character. putStrLn :: String -> IO () -- | Write a string to the standard output device (same as hPutStr -- stdout). putStr :: String -> IO () -- | Write a character to the standard output device (same as -- hPutChar stdout). putChar :: Char -> IO () -- | Raise an IOError in the IO monad. ioError :: IOError -> IO a -- | 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 -- | 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 -- | 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 -- | notElem is the negation of elem. notElem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 `notElem` -- | Determines whether all elements of the structure satisfy the -- predicate. all :: Foldable t => (a -> Bool) -> t a -> Bool -- | Determines whether any element of the structure satisfies the -- predicate. any :: Foldable t => (a -> Bool) -> t a -> 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. or :: Foldable t => t Bool -> Bool -- | 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. and :: Foldable t => t Bool -> Bool -- | Map a function over all the elements of a container and concatenate -- the resulting lists. concatMap :: Foldable t => (a -> [b]) -> t a -> [b] -- | The concatenation of all the elements of a container of lists. concat :: Foldable t => t [a] -> [a] -- | 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. -- -- As of base 4.8.0.0, sequence_ is just sequenceA_, -- specialized to Monad. sequence_ :: (Foldable t, Monad m) => t (m a) -> m () -- | 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. -- -- As of base 4.8.0.0, mapM_ is just traverse_, specialized -- to Monad. mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m () -- | unwords is an inverse operation to words. It joins words -- with separating spaces. -- --

--   >>> unwords ["Lorem", "ipsum", "dolor"]
--   "Lorem ipsum dolor"
--

unwords :: [String] -> String -- | words breaks a string up into a list of words, which were -- delimited by white space. -- --

--   >>> words "Lorem ipsum\ndolor"
--   ["Lorem","ipsum","dolor"]
--

words :: String -> [String] -- | unlines is an inverse operation to lines. It joins -- lines, after appending a terminating newline to each. -- --

--   >>> unlines ["Hello", "World", "!"]
--   "Hello\nWorld\n!\n"
--

unlines :: [String] -> String -- | lines breaks a string up into a list of strings at newline -- characters. The resulting strings do not contain newlines. -- -- Note that after splitting the string at newline characters, the last -- part of the string is considered a line even if it doesn't end with a -- newline. For example, -- --

--   >>> lines ""
--   []
--

-- --

--   >>> lines "\n"
--   [""]
--

-- --

--   >>> lines "one"
--   ["one"]
--

-- --

--   >>> lines "one\n"
--   ["one"]
--

-- --

--   >>> lines "one\n\n"
--   ["one",""]
--

-- --

--   >>> lines "one\ntwo"
--   ["one","two"]
--

-- --

--   >>> lines "one\ntwo\n"
--   ["one","two"]
--

-- -- Thus lines s contains at least as many elements as -- newlines in s. lines :: String -> [String] -- | equivalent to readsPrec with a precedence of 0. reads :: Read a => ReadS a -- | 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 -- | 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)] -- | lcm x y is the smallest positive integer that both -- x and y divide. lcm :: Integral a => a -> a -> a -- | 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 -- | raise a number to an integral power (^^) :: (Fractional a, Integral b) => a -> b -> a infixr 8 ^^ odd :: Integral a => a -> Bool even :: Integral a => a -> Bool -- | utility function that surrounds the inner show function with -- parentheses when the Bool parameter is True. showParen :: Bool -> ShowS -> ShowS -- | utility function converting a String to a show function that -- simply prepends the string unchanged. showString :: String -> ShowS -- | utility function converting a Char to a show function that -- simply prepends the character unchanged. showChar :: Char -> ShowS -- | equivalent to showsPrec with a precedence of 0. shows :: Show a => a -> ShowS -- | 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 -- | The unzip3 function takes a list of triples and returns three -- lists, analogous to unzip. unzip3 :: [(a, b, c)] -> ([a], [b], [c]) -- | unzip transforms a list of pairs into a list of first -- components and a list of second components. unzip :: [(a, b)] -> ([a], [b]) -- | The zipWith3 function takes a function which combines three -- elements, as well as three lists and returns a list of their -- point-wise combination, analogous to zipWith. It is capable of -- list fusion, but it is restricted to its first list argument and its -- resulting list. zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d] -- | <math>. zipWith generalises zip by zipping with -- the function given as the first argument, instead of a tupling -- function. For example, zipWith (+) is applied to two -- lists to produce the list of corresponding sums: -- --

--   >>> zipWith (+) [1, 2, 3] [4, 5, 6]
--   [5,7,9]
--

-- -- zipWith is right-lazy: -- --

--   zipWith f [] _|_ = []
--

-- -- zipWith is capable of list fusion, but it is restricted to its -- first list argument and its resulting list. zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] -- | 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>. lookup key assocs looks up a key in an -- association list. -- --

--   >>> lookup 2 [(1, "first"), (2, "second"), (3, "third")]
--   Just "second"
--

lookup :: Eq a => a -> [(a, b)] -> Maybe b -- | reverse xs returns the elements of xs in -- reverse order. xs must be finite. reverse :: [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 (> 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 p is equivalent to span (not . -- p). break :: (a -> Bool) -> [a] -> ([a], [a]) -- | span, 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 satisfy p and second element -- is the remainder of the list: -- --

--   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 p xs is equivalent to (takeWhile p xs, -- dropWhile p xs) span :: (a -> Bool) -> [a] -> ([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 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])
--

-- -- It is equivalent to (take n xs, drop n xs) when -- n is not _|_ (splitAt _|_ xs = _|_). -- splitAt is an instance of the more general -- genericSplitAt, in which n may be of any integral -- type. splitAt :: Int -> [a] -> ([a], [a]) -- | dropWhile p xs returns the suffix remaining after -- takeWhile p xs: -- --

--   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] -- | takeWhile, applied to a predicate p and a list -- xs, returns the longest prefix (possibly empty) of -- xs of elements that satisfy p: -- --

--   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] -- | 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. replicate :: Int -> a -> [a] -- | repeat x is an infinite list, with x the -- value of every element. repeat :: 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), ...]
--

-- -- 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. iterate :: (a -> a) -> a -> [a] -- | <math>. scanr1 is a variant of scanr that has no -- starting value argument. scanr1 :: (a -> a -> a) -> [a] -> [a] -- | <math>. scanr is the right-to-left dual of scanl. -- Note that -- --

--   head (scanr f z xs) == foldr f z xs.
--

scanr :: (a -> b -> 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, ...]
--

scanl1 :: (a -> a -> a) -> [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.
--

scanl :: (b -> a -> b) -> b -> [a] -> [b] -- | <math>. Return all the elements of a list except the last one. -- The list must be non-empty. init :: [a] -> [a] -- | <math>. Extract the last element of a list, which must be finite -- and non-empty. last :: [a] -> a -- | <math>. Extract the elements after the head of a list, which -- must be non-empty. tail :: [a] -> [a] -- | <math>. Extract the first element of a list, which must be -- non-empty. head :: [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 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 -- | 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)
--

(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4 <$> -- | 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 -- | curry converts an uncurried function to a curried function. -- --

Examples

-- --

--   >>> curry fst 1 2
--   1
--

curry :: ((a, b) -> c) -> a -> b -> c -- | 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 -- | 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 -- | 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. ($!) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b infixr 0 $! -- | flip f takes its (first) two arguments in the reverse -- order of f. -- --

--   >>> flip (++) "hello" "world"
--   "worldhello"
--

flip :: (a -> b -> c) -> b -> a -> c -- | Function composition. (.) :: (b -> c) -> (a -> b) -> a -> c infixr 9 . -- | Same as >>=, but with the arguments interchanged. (=<<) :: Monad m => (a -> m b) -> m a -> m b infixr 1 =<< -- | 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 :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a -- | A variant of error that does not produce a stack trace. errorWithoutStackTrace :: forall (r :: RuntimeRep) (a :: TYPE r). [Char] -> a -- | error stops execution and displays an error message. error :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => [Char] -> 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 :: * -> * [Append] :: Typed a => [a] -> Maybe (Stream Bool) -> Stream a -> Stream a [Const] :: Typed a => a -> Stream a [Drop] :: Typed a => Int -> Stream a -> Stream a [Extern] :: Typed a => String -> Maybe [a] -> Stream a [Local] :: (Typed a, Typed b) => Stream a -> (Stream a -> Stream b) -> Stream b [Var] :: Typed a => String -> Stream a [Op1] :: (Typed a, Typed b) => Op1 a b -> Stream a -> Stream b [Op2] :: (Typed a, Typed b, Typed c) => Op2 a b c -> Stream a -> Stream b -> Stream c [Op3] :: (Typed a, Typed b, Typed c, Typed d) => Op3 a b c d -> Stream a -> Stream b -> Stream c -> Stream d [Label] :: Typed a => String -> Stream a -> Stream a -- | Wrapper to use Streams as arguments to triggers. data Arg [Arg] :: Typed a => Stream a -> Arg -- | 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.Show.Show (Copilot.Language.Stream.Stream a) instance GHC.Classes.Eq (Copilot.Language.Stream.Stream a) instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Num.Num a) => GHC.Num.Num (Copilot.Language.Stream.Stream a) instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Real.Fractional a) => GHC.Real.Fractional (Copilot.Language.Stream.Stream a) instance (Copilot.Core.Type.Typed a, GHC.Classes.Eq a, GHC.Float.Floating a) => GHC.Float.Floating (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] :: 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 -- | A property, representing a boolean stream that is existentially or -- universally quantified over time. data Property [Property] :: String -> Stream Bool -> 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. 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 -- | Combinators to deal with streams carrying structs. module Copilot.Language.Operators.Struct -- | 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. (#) :: (KnownSymbol s, Typed t, Typed a, Struct a) => Stream a -> (a -> Field s t) -> Stream 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, Typed a, Typed b) => Stream a -> Stream b -- | Perform an unsafe cast from Stream a to Stream b. unsafeCast :: (UnsafeCast a b, Typed a, Typed 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.UnsafeCast GHC.Word.Word64 GHC.Word.Word32 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word64 GHC.Word.Word16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word64 GHC.Word.Word8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word32 GHC.Word.Word16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word32 GHC.Word.Word8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word16 GHC.Word.Word8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Int.Int8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int32 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int32 GHC.Int.Int8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int16 GHC.Int.Int8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int32 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int16 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int8 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int32 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int16 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int8 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word64 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word32 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word16 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word8 GHC.Types.Float instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word64 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word32 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word16 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word8 GHC.Types.Double instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word64 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word32 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word16 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Word.Word8 GHC.Int.Int8 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int64 GHC.Word.Word64 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int32 GHC.Word.Word32 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int16 GHC.Word.Word16 instance Copilot.Language.Operators.Cast.UnsafeCast GHC.Int.Int8 GHC.Word.Word8 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Types.Bool instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Word.Word8 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Word.Word16 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Word.Word32 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Word.Word64 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Int.Int8 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Int.Int16 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Types.Bool GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Word.Word8 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Word.Word16 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Word.Word32 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Word.Word64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word8 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word16 GHC.Word.Word16 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word16 GHC.Word.Word32 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word16 GHC.Word.Word64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word16 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word16 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word32 GHC.Word.Word32 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word32 GHC.Word.Word64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word32 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Word.Word64 GHC.Word.Word64 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int8 GHC.Int.Int8 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int8 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int8 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int8 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int16 GHC.Int.Int16 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int16 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int16 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int32 GHC.Int.Int32 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int32 GHC.Int.Int64 instance Copilot.Language.Operators.Cast.Cast GHC.Int.Int64 GHC.Int.Int64 -- | 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 .|. -- | See xor. (.^.) :: Bits a => a -> a -> a -- | 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, Data.Bits.Bits a) => Data.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. (.!!) :: (KnownNat n, t' ~ InnerType t, Flatten t t', Typed t, Typed t') => Stream (Array n t) -> Stream Word32 -> Stream 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. It also provides a default pretty printer that prints a -- specification to stdout. 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 name of a trigger, an external variable, or an external function. type Name = String -- | 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 -- | Report an error due to a bug in Copilot. impossible :: String -> String -> a -- | Report an error due to an error detected by Copilot (e.g., user -- error). badUsage :: String -> a -- | 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 () -- | 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 () -- | 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 :: * -> * -- | 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 -- | Transform a high-level Copilot Language specification into a low-level -- Copilot Core specification and pretty-print it to stdout. prettyPrint :: Spec -> IO ()