    {-# OPTIONS_HADDOCK show-extensions #-}

    {-# LANGUAGE AutoDeriveTypeable #-}
    {-# LANGUAGE CPP #-}
    {-# LANGUAGE DataKinds #-}
    {-# LANGUAGE DeriveDataTypeable #-}
    {-# LANGUAGE DeriveGeneric #-}
    {-# LANGUAGE FlexibleContexts #-}
    {-# LANGUAGE FlexibleInstances #-}
    {-# LANGUAGE GeneralizedNewtypeDeriving #-}
    {-# LANGUAGE KindSignatures #-}
    {-# LANGUAGE MultiParamTypeClasses #-} -- for Vector instances only
    {-# LANGUAGE PatternGuards #-}
    {-# LANGUAGE RankNTypes #-}
    {-# LANGUAGE RoleAnnotations #-}
    {-# LANGUAGE ScopedTypeVariables #-}
    {-# LANGUAGE StandaloneDeriving #-}
    {-# LANGUAGE TypeFamilies #-}
    {-# LANGUAGE TypeOperators #-}

    {- |
       Copyright  : Copyright (C) 2006-2015 Bjorn Buckwalter
       License    : BSD3

       Maintainer : bjorn@buckwalter.se
       Stability  : Stable
       Portability: GHC only

    = Summary

    In this module we provide data types for performing arithmetic with
    physical quantities and units. Information about the physical
    dimensions of the quantities/units is embedded in their types and
    the validity of operations is verified by the type checker at compile
    time. The boxing and unboxing of numerical values as quantities is
    done by multiplication and division of units, of which an incomplete
    set is provided.

    We limit ourselves to \"Newtonian\" physics. We do not attempt to
    accommodate relativistic physics in which e.g. addition of length
    and time would be valid.

    As far as possible and/or practical the conventions and guidelines
    of NIST's "Guide for the Use of the International System of Units
    (SI)" <#note1 [1]> are followed. Occasionally we will reference specific
    sections from the guide and deviations will be explained.

    == Disclaimer

    Merely an engineer, the author doubtlessly uses a language and
    notation that makes mathematicians and physicist cringe. He does
    not mind constructive criticism (or pull requests).

    The sets of functions and units defined herein are incomplete and
    reflect only the author's needs to date. Again, patches are welcome.

    = Usage

    == Preliminaries

    This module requires GHC 7.8 or later. We utilize Data Kinds, TypeNats,
    Closed Type Families, etc. Clients of the module are generally not
    required to use these extensions.

    Clients probably will want to use the NegativeLiterals extension.

    == Examples

    We have defined operators and units that allow us to define and
    work with physical quantities. A physical quantity is defined by
    multiplying a number with a unit (the type signature is optional).

    > v :: Velocity Prelude.Double
    > v = 90 *~ (kilo meter / hour)

    It follows naturally that the numerical value of a quantity is
    obtained by division by a unit.

    > numval :: Prelude.Double
    > numval = v /~ (meter / second)

    The notion of a quantity as the product of a numerical value and a
    unit is supported by 7.1 "Value and numerical value of a quantity" of
    <#note1 [1]>. While the above syntax is fairly natural it is unfortunate that
    it must violate a number of the guidelines in <#note1 [1]>, in particular 9.3
    "Spelling unit names with prefixes", 9.4 "Spelling unit names obtained
    by multiplication", 9.5 "Spelling unit names obtained by division".

    As a more elaborate example of how to use the module we define a
    function for calculating the escape velocity of a celestial body
    <#note2 [2]>.

    > escapeVelocity :: (Floating a) => Mass a -> Length a -> Velocity a
    > escapeVelocity m r = sqrt (two * g * m / r)
    >   where
    >       two = 2 *~ one
    >       g = 6.6720e-11 *~ (newton * meter ^ pos2 / kilo gram ^ pos2)

    The following is an example GHC session where the above function
    is used to calculate the escape velocity of Earth in kilometer per
    second.

    >>> :set +t
    >>> let me = 5.9742e24 *~ kilo gram -- Mass of Earth.
    me :: Quantity DMass GHC.Float.Double
    >>> let re = 6372.792 *~ kilo meter -- Mean radius of Earth.
    re :: Quantity DLength GHC.Float.Double
    >>> let ve = escapeVelocity me re   -- Escape velocity of Earth.
    ve :: Velocity GHC.Float.Double
    >>> ve /~ (kilo meter / second)
    11.184537332296259
    it :: GHC.Float.Double

    For completeness we should also show an example of the error messages
    we will get from GHC when performing invalid arithmetic. In the
    best case GHC will be able to use the type synonyms we have defined
    in its error messages.

    >>> x = 1 *~ meter + 1 *~ second
    Couldn't match type 'Numeric.NumType.DK.Integers.Zero
                   with 'Numeric.NumType.DK.Integers.Pos1
      Expected type: Unit 'Metric DLength a
        Actual type: Unit 'Metric DTime a
      In the second argument of `(*~)', namely `second'
      In the second argument of `(+)', namely `1 *~ second'

    In other cases the error messages aren't very friendly.

    >>> x = 1 *~ meter / (1 *~ second) + 1 *~ kilo gram
    Couldn't match type 'Numeric.NumType.DK.Integers.Zero
                   with 'Numeric.NumType.DK.Integers.Neg1
      Expected type: Quantity DMass a
        Actual type: Dimensional
                       ('Numeric.Units.Dimensional.Variants.DQuantity
                        Numeric.Units.Dimensional.Variants.* 'Numeric.Units.Dimensional.Variants.DQuantity)
                       (DLength / DTime)
                       a
      In the first argument of `(+)', namely `1 *~ meter / (1 *~ second)'
      In the expression: 1 *~ meter / (1 *~ second) + 1 *~ kilo gram
      In an equation for `x':
          x = 1 *~ meter / (1 *~ second) + 1 *~ kilo gram

    It is the author's experience that the usefullness of the compiler
    error messages is more often than not limited to pinpointing the
    location of errors.

    = Notes

    == Future work

    While there is an insane amount of units in use around the world
    it is reasonable to provide at least all SI units. Units outside
    of SI will most likely be added on an as-needed basis.

    There are also plenty of elementary functions to add. The 'Floating'
    class can be used as reference.

    Additional physics models could be implemented. See <#note3 [3]> for ideas.

    == Related work

    Henning Thielemann numeric prelude has a physical units library,
    however, checking of dimensions is dynamic rather than static.
    Aaron Denney has created a toy example of statically checked
    physical dimensions covering only length and time. HaskellWiki
    has pointers <#note4 [4]> to these.

    Also see Samuel Hoffstaetter's blog post <#note5 [5]> which uses techniques
    similar to this library.

    Libraries with similar functionality exist for other programming
    languages and may serve as inspiration. The author has found the
    Java library JScience <#note6 [6]> and the Fortress programming language <#note7 [7]>
    particularly noteworthy.

    == References

    1. #note1# http://physics.nist.gov/Pubs/SP811/
    2. #note2# http://en.wikipedia.org/wiki/Escape_velocity
    3. #note3# http://jscience.org/api/org/jscience/physics/models/package-summary.html
    4. #note4# http://www.haskell.org/haskellwiki/Physical_units
    5. #note5# http://liftm.wordpress.com/2007/06/03/scientificdimension-type-arithmetic-and-physical-units-in-haskell/
    6. #note6# http://jscience.org/
    7. #note7# http://research.sun.com/projects/plrg/fortress.pdf

    -}

    module Numeric.Units.Dimensional
      (
        -- * Types
        -- $types
        Dimensional,
        Unit, Quantity,
        Metricality(..),
        -- * Physical Dimensions
        -- $dimensions
        Dimension (Dim),
        -- ** Dimension Arithmetic
        -- $dimension-arithmetic
        type (*), type (/), type (^), Root, Recip,
        -- ** Term Level Representation of Dimensions
        -- $dimension-terms
        Dimension' (Dim'), HasDimension(..), KnownDimension,
        -- * Dimensional Arithmetic
        (*~), (/~),
        (^), (^/), (**), (*), (/), (+), (-),
        negate, abs, nroot, sqrt, cbrt,
        -- ** Transcendental Functions
        exp, log, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, atan2,
        -- ** Operations on Collections
        -- $collections
        (*~~), (/~~), sum, mean, dimensionlessLength, nFromTo,
        -- * Dimension Synonyms
        -- $dimension-synonyms
        DOne, DLength, DMass, DTime, DElectricCurrent, DThermodynamicTemperature, DAmountOfSubstance, DLuminousIntensity,
        -- * Quantity Synonyms
        -- $quantity-synonyms
        Dimensionless, Length, Mass, Time, ElectricCurrent, ThermodynamicTemperature, AmountOfSubstance, LuminousIntensity,
        -- * Constants
        -- $constants
        _0, _1, _2, _3, _4, _5, _6, _7, _8, _9, pi, tau,
        -- * Constructing Units
        siUnit, one, mkUnitR, mkUnitQ, mkUnitZ,
        -- * Unit Metadata
        name, exactValue, weaken, strengthen, exactify,
        -- * Pretty Printing
        showIn,
        -- * On 'Functor', and Conversion Between Number Representations
        -- $functor
        KnownVariant(dmap), changeRep, changeRepApproximate
      )
      where

    import Prelude
      ( Eq(..), Num, Fractional, Floating, Real, RealFloat, Functor, fmap
      , (.), flip, (++), fromIntegral, fromInteger, fromRational, error, max, succ
      , Int, Integer, Integral, ($), uncurry, realToFrac, otherwise
      )
    import qualified Prelude
    import Numeric.NumType.DK.Integers
      ( TypeInt (Pos2, Pos3)
      , pos2, pos3
      , KnownTypeInt, toNum
      )
    import Data.Data
    import Data.ExactPi
    import Data.Foldable (Foldable(foldr, foldl'))
    import Data.Maybe
    import Data.Ratio
    import Numeric.Units.Dimensional.Dimensions
    import Numeric.Units.Dimensional.Internal
    import Numeric.Units.Dimensional.UnitNames hiding ((*), (/), (^), weaken, strengthen)
    import qualified Numeric.Units.Dimensional.UnitNames.Internal as Name
    import Numeric.Units.Dimensional.Variants hiding (type (*))
    import qualified Numeric.Units.Dimensional.Variants as V

    {-
    We will reuse the operators and function names from the Prelude.
    To prevent unpleasant surprises we give operators the same fixity
    as the Prelude.
    -}

    infixr 8  ^, ^/, **
    infixl 7  *, /
    infixl 6  +, -

    {- $types
    Our primary objective is to define a data type that can be used to
    represent (while still differentiating between) units and quantities.
    There are two reasons for consolidating units and quantities in one
    data type. The first being to allow code reuse as they are largely
    subject to the same operations. The second being that it allows
    reuse of operators (and functions) between the two without resorting
    to occasionally cumbersome type classes.

    The relationship between (the value of) a 'Quantity', its numerical
    value and its 'Unit' is described in 7.1 "Value and numerical value
    of a quantity" of <#note1 [1]>. In short a 'Quantity' is the product of a
    number and a 'Unit'. We define the '*~' operator as a convenient
    way to declare quantities as such a product.

    -}

    -- | Extracts the 'UnitName' of a 'Unit'.
    name :: Unit m d a -> UnitName m
    name (Unit n _ _) = n

    -- | Extracts the exact value of a 'Unit', expressed in terms of the SI coherent derived unit (see 'siUnit') of the same 'Dimension'.
    --
    -- Note that the actual value may in some cases be approximate, for example if the unit is defined by experiment.
    exactValue :: Unit m d a -> ExactPi
    exactValue (Unit _ e _) = e

    -- | Discards potentially unwanted type level information about a 'Unit'.
    weaken :: Unit m d a -> Unit 'NonMetric d a
    weaken (Unit n e v) = Unit (Name.weaken n) e v

    -- | Attempts to convert a 'Unit' which may or may not be 'Metric' to one
    -- which is certainly 'Metric'.
    strengthen :: Unit m d a -> Maybe (Unit 'Metric d a)
    strengthen (Unit n e v) | Just n' <- Name.strengthen n = Just $ Unit n' e v
                            | otherwise                    = Nothing

    -- | Forms the exact version of a 'Unit'.
    exactify :: Unit m d a -> Unit m d ExactPi
    exactify (Unit n e _) = Unit n e e

    -- | Forms a 'Quantity' by multipliying a number and a unit.
    (*~) :: Num a => a -> Unit m d a -> Quantity d a
    x *~ (Unit _ _ y) = Quantity (x Prelude.* y)

    -- | Divides a 'Quantity' by a 'Unit' of the same physical dimension, obtaining the
    -- numerical value of the quantity expressed in that unit.
    (/~) :: Fractional a => Quantity d a -> Unit m d a -> a
    (Quantity x) /~ (Unit _ _ y) = (x Prelude./ y)

    {-
    We give '*~' and '/~' the same fixity as '*' and '/' defined below.
    Note that this necessitates the use of parenthesis when composing
    units using '*' and '/', e.g. "1 *~ (meter / second)".
    -}

    infixl 7  *~, /~

    {- $dimensions
    The phantom type variable d encompasses the physical dimension of
    a 'Dimensional'. As detailed in <#note5 [5]> there are seven base dimensions,
    which can be combined in integer powers to a given physical dimension.
    We represent physical dimensions as the powers of the seven base
    dimensions that make up the given dimension. The powers are represented
    using NumTypes. For convenience we collect all seven base dimensions
    in a data kind 'Dimension'.

    We could have chosen to provide type variables for the seven base
    dimensions in 'Dimensional' instead of creating a new data kind
    'Dimension'. However, that would have made any type signatures involving
    'Dimensional' very cumbersome.  By encompassing the physical dimension
    in a single type variable we can "hide" the cumbersome type arithmetic
    behind convenient type classes as will be seen later.

    -}

    {- $dimension-synonyms
    Using our 'Dimension' data kind we define some type synonyms for convenience.
    We start with the base dimensions, others can be found in "Numeric.Units.Dimensional.Quantities".

    -}

    {- $quantity-synonyms
    Using the above type synonyms we can define type synonyms for
    quantities of particular physical dimensions.

    Again we limit ourselves to the base dimensions, others can be found in "Numeric.Units.Dimensional.Quantities".

    -}

    type Dimensionless            = Quantity DOne
    type Length                   = Quantity DLength
    type Mass                     = Quantity DMass
    type Time                     = Quantity DTime
    type ElectricCurrent          = Quantity DElectricCurrent
    type ThermodynamicTemperature = Quantity DThermodynamicTemperature
    type AmountOfSubstance        = Quantity DAmountOfSubstance
    type LuminousIntensity        = Quantity DLuminousIntensity

    {- $dimension-arithmetic
    When performing arithmetic on units and quantities the arithmetics
    must be applied to both the numerical values of the Dimensionals
    but also to their physical dimensions. The type level arithmetic
    on physical dimensions is governed by closed type families expressed
    as type operators.

    We could provide the 'Mul' and 'Div' classes with full functional
    dependencies but that would be of limited utility as there is no
    limited use for "backwards" type inference. Efforts are underway to
    develop a type-checker plugin that does enable these scenarios, e.g.
    for linear algebra.

    -}

    {-
    = Arithmetic on units and quantities =

    Thanks to the arithmetic on physical dimensions having been sorted
    out separately a lot of the arithmetic on Dimensionals is straight
    forward. In particular the type signatures are much simplified.

    Multiplication, division and powers apply to both units and quantities.
    -}

    -- | Multiplies two 'Quantity's or two 'Unit's.
    --
    -- The intimidating type signature captures the similarity between these operations
    -- and ensures that composite 'Unit's are 'NonMetric'.
    (*) :: (KnownVariant v1, KnownVariant v2, KnownVariant (v1 V.* v2), Num a) => Dimensional v1 d1 a -> Dimensional v2 d2 a -> Dimensional (v1 V.* v2) (d1 * d2) a
    (*) = liftD2 (Prelude.*) (Prelude.*) (Name.*)

    -- | Divides one 'Quantity' by another or one 'Unit' by another.
    --
    -- The intimidating type signature captures the similarity between these operations
    -- and ensures that composite 'Unit's are 'NotPrefixable'.
    (/) :: (KnownVariant v1, KnownVariant v2, KnownVariant (v1 V.* v2), Fractional a) => Dimensional v1 d1 a -> Dimensional v2 d2 a -> Dimensional (v1 V.* v2) (d1 / d2) a
    (/) = liftD2 (Prelude./) (Prelude./) (Name./)

    -- | Raises a 'Quantity' or 'Unit' to an integer power.
    --
    -- Because the power chosen impacts the 'Dimension' of the result, it is necessary to supply a type-level representation
    -- of the exponent in the form of a 'Proxy' to some 'TypeInt'. Convenience values 'pos1', 'pos2', 'neg1', ...
    -- are supplied by the "Numeric.NumType.DK.Integers" module. The most commonly used ones are
    -- also reexported by "Numeric.Units.Dimensional.Prelude".
    --
    -- The intimidating type signature captures the similarity between these operations
    -- and ensures that composite 'Unit's are 'NotPrefixable'.
    (^) :: (Fractional a, KnownTypeInt i, KnownVariant v, KnownVariant (Weaken v))
        => Dimensional v d1 a -> Proxy i -> Dimensional (Weaken v) (d1 ^ i) a
    x ^ n = let n' = (toNum n) :: Int
             in liftD (Prelude.^^ n') (Prelude.^^ n') (Name.^ n') x

    {-
    A special case is that dimensionless quantities are not restricted
    to integer exponents. This is accommodated by the '**' operator
    defined later.

    = Quantity operations =

    Some additional operations obviously only make sense for quantities.
    Of these, negation, addition and subtraction are particularly simple
    as they are done in a single physical dimension.
    -}

    -- | Negates the value of a 'Quantity'.
    negate :: Num a => Quantity d a -> Quantity d a
    negate = liftQ Prelude.negate

    -- | Adds two 'Quantity's.
    (+) :: Num a => Quantity d a -> Quantity d a -> Quantity d a
    (+) = liftQ2 (Prelude.+)

    -- | Subtracts one 'Quantity' from another.
    (-) :: Num a => Quantity d a -> Quantity d a -> Quantity d a
    (-) = liftQ2 (Prelude.-)

    -- | Takes the absolute value of a 'Quantity'.
    abs :: Num a => Quantity d a -> Quantity d a
    abs = liftQ Prelude.abs

    {-
    Roots of arbitrary (integral) degree. Appears to occasionally be useful
    for units as well as quantities.
    -}

    -- | Computes the nth root of a 'Quantity' using 'Prelude.**'.
    --
    -- The 'Root' type family will prevent application of this operator where the result would have a fractional dimension or where n is zero.
    --
    -- Because the root chosen impacts the 'Dimension' of the result, it is necessary to supply a type-level representation
    -- of the root in the form of a 'Proxy' to some 'TypeInt'. Convenience values 'pos1', 'pos2', 'neg1', ...
    -- are supplied by the "Numeric.NumType.DK.Integers" module. The most commonly used ones are
    -- also reexported by "Numeric.Units.Dimensional.Prelude".
    --
    -- Also available in operator form, see '^/'.
    nroot :: (KnownTypeInt n, Floating a)
          => Proxy n -> Quantity d a -> Quantity (Root d n) a
    nroot n = let n' = 1 Prelude./ toNum n
               in liftQ (Prelude.** n')

    {-
    We provide short-hands for the square and cubic roots.
    -}

    -- | Computes the square root of a 'Quantity' using 'Prelude.**'.
    --
    -- The 'Root' type family will prevent application where the supplied quantity does not have a square dimension.
    --
    -- prop> sqrt x == nroot pos2 x
    sqrt :: Floating a => Quantity d a -> Quantity (Root d 'Pos2) a
    sqrt = nroot pos2

    -- | Computes the cube root of a 'Quantity' using 'Prelude.**'.
    --
    -- The 'Root' type family will prevent application where the supplied quantity does not have a cubic dimension.
    --
    -- prop> cbrt x == nroot pos3 x
    cbrt :: Floating a => Quantity d a -> Quantity (Root d 'Pos3) a
    cbrt = nroot pos3

    {-
    We also provide an operator alternative to nroot for those that
    prefer such.
    -}

    -- | Computes the nth root of a 'Quantity' using 'Prelude.**'.
    --
    -- The 'Root' type family will prevent application of this operator where the result would have a fractional dimension or where n is zero.
    --
    -- Because the root chosen impacts the 'Dimension' of the result, it is necessary to supply a type-level representation
    -- of the root in the form of a 'Proxy' to some 'TypeInt'. Convenience values 'pos1', 'pos2', 'neg1', ...
    -- are supplied by the "Numeric.NumType.DK.Integers" module. The most commonly used ones are
    -- also reexported by "Numeric.Units.Dimensional.Prelude".
    --
    -- Also available in prefix form, see 'nroot'.
    (^/) :: (KnownTypeInt n, Floating a)
         => Quantity d a -> Proxy n -> Quantity (Root d n) a
    (^/) = flip nroot

    {- $collections
    Here we define operators and functions to make working with homogenuous
    lists of dimensionals more convenient.

    We define two convenience operators for applying units to all
    elements of a functor (e.g. a list).
    -}

    -- | Applies '*~' to all values in a functor.
    (*~~) :: (Functor f, Num a) => f a -> Unit m d a -> f (Quantity d a)
    xs *~~ u = fmap (*~ u) xs

    -- | Applies '/~' to all values in a functor.
    (/~~) :: (Functor f, Fractional a) => f (Quantity d a) -> Unit m d a -> f a
    xs /~~ u = fmap (/~ u) xs

    infixl 7  *~~, /~~

    -- | The sum of all elements in a list.
    sum :: (Num a, Foldable f) => f (Quantity d a) -> Quantity d a
    sum = foldr (+) _0

    -- | The arithmetic mean of all elements in a list.
    mean :: (Fractional a, Foldable f) => f (Quantity d a) -> Quantity d a
    mean = uncurry (/) . foldr accumulate (_0, _0)
      where
        accumulate val (accum, count) = (accum + val, count + _1)

    -- | The length of the foldable data structure as a 'Dimensionless'.
    -- This can be useful for purposes of e.g. calculating averages.
    dimensionlessLength :: (Num a, Foldable f) => f (Dimensional v d a) -> Dimensionless a
    dimensionlessLength x = (fromIntegral $ length x) *~ one
      where
        -- As in base-4.8 Data.Foldable for GHC 7.8 (base-4.6) compatibility.
        -- Once base-4.6. compatibility is abandoned this where clause can
        -- be deleted (and imports adjusted).
        length :: Foldable t => t a -> Int
        length = foldl' (\c _ -> c Prelude.+ 1) 0

    -- | Returns a list of quantities between given bounds.
    nFromTo :: (Fractional a, Integral b) => Quantity d a -- ^ The initial value.
                                          -> Quantity d a -- ^ The final value.
                                          -> b -- ^ The number of intermediate values. If less than one, no intermediate values will result.
                                          -> [Quantity d a]
    nFromTo xi xf n = fmap f [0..n'] ++ [xf]
      where
        n' = max 0 n
        f i = xi + realToFrac (i % succ n') *~ one * (xf - xi)

    {-
    We continue by defining elementary functions on 'Dimensionless'
    that may be obviously useful.
    -}

    exp, log, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh
      :: Floating a => Dimensionless a -> Dimensionless a
    exp   = fmap Prelude.exp
    log   = fmap Prelude.log
    sin   = fmap Prelude.sin
    cos   = fmap Prelude.cos
    tan   = fmap Prelude.tan
    asin  = fmap Prelude.asin
    acos  = fmap Prelude.acos
    atan  = fmap Prelude.atan
    sinh  = fmap Prelude.sinh
    cosh  = fmap Prelude.cosh
    tanh  = fmap Prelude.tanh
    asinh = fmap Prelude.asinh
    acosh = fmap Prelude.acosh
    atanh = fmap Prelude.atanh

    -- | Raises a dimensionless quantity to a floating power using 'Prelude.**'.
    (**) :: Floating a => Dimensionless a -> Dimensionless a -> Dimensionless a
    (**) = liftQ2 (Prelude.**)

    -- | The standard two argument arctangent function.
    -- Since it interprets its two arguments in comparison with one another, the input may have any dimension.
    atan2 :: (RealFloat a) => Quantity d a -> Quantity d a -> Dimensionless a
    atan2 = liftQ2 Prelude.atan2

    {-
    The only unit we will define in this module is 'one'.
    -}

    -- | The unit 'one' has dimension 'DOne' and is the base unit of dimensionless values.
    --
    -- As detailed in 7.10 "Values of quantities expressed simply as numbers:
    -- the unit one, symbol 1" of <#note1 [1]> the unit one generally does not
    -- appear in expressions. However, for us it is necessary to use 'one'
    -- as we would any other unit to perform the "boxing" of dimensionless values.
    one :: Num a => Unit 'NonMetric DOne a
    one = Unit nOne 1 1

    {- $constants
    For convenience we define some constants for small integer values
    that often show up in formulae. We also throw in 'pi' and 'tau' for
    good measure.

    -}

    -- | The constant for zero is polymorphic, allowing
    -- it to express zero 'Length' or 'Capacitance' or 'Velocity' etc, in addition
    -- to the 'Dimensionless' value zero.
    _0 :: Num a => Quantity d a
    _0 = Quantity 0

    _1, _2, _3, _4, _5, _6, _7, _8, _9 :: (Num a) => Dimensionless a
    _1 = 1 *~ one
    _2 = 2 *~ one
    _3 = 3 *~ one
    _4 = 4 *~ one
    _5 = 5 *~ one
    _6 = 6 *~ one
    _7 = 7 *~ one
    _8 = 8 *~ one
    _9 = 9 *~ one

    pi :: Floating a => Dimensionless a
    pi = Prelude.pi *~ one

    -- | Twice 'pi'.
    --
    -- For background on 'tau' see http://tauday.com/tau-manifesto (but also
    -- feel free to review http://www.thepimanifesto.com).
    tau :: Floating a => Dimensionless a
    tau = _2 * pi

    {- $functor
    We intentionally decline to provide a 'Functor' instance for 'Dimensional' because its use breaks the
    abstraction of physical dimensions.

    If you feel your work requires this instance, it is provided as an orphan in "Numeric.Units.Dimensional.Functor".

    -}

    -- | Convenient conversion between numerical types while retaining dimensional information.
    changeRep :: (KnownVariant v, Real a, Fractional b) => Dimensional v d a -> Dimensional v d b
    changeRep = dmap realToFrac

    -- | Convenient conversion from exactly represented values while retaining dimensional information.
    changeRepApproximate :: (KnownVariant v, Floating b) => Dimensional v d ExactPi -> Dimensional v d b
    changeRepApproximate = dmap approximateValue

    {- $dimension-terms
    To facilitate parsing and pretty-printing functions that may wish to operate on term-level representations of dimension,
    we provide a means for converting from type-level dimensions to term-level dimensions.

    -}

    -- | Forms a new atomic 'Unit' by specifying its 'UnitName' and its definition as a multiple of another 'Unit'.
    --
    -- Use this variant when the scale factor of the resulting unit is irrational or 'Approximate'. See 'mkUnitQ' for when it is rational
    -- and 'mkUnitZ' for when it is an integer.
    --
    -- Note that supplying zero as a definining quantity is invalid, as the library relies
    -- upon units forming a group under multiplication.
    --
    -- Supplying negative defining quantities is allowed and handled gracefully, but is discouraged
    -- on the grounds that it may be unexpected by other readers.
    mkUnitR :: Floating a => UnitName m -> ExactPi -> Unit m1 d a -> Unit m d a
    mkUnitR n s (Unit _ e _) | isExactZero s = error "Supplying zero as a conversion factor is not valid."
                             | otherwise     = Unit n e' x'
      where
        e' = s Prelude.* e
        x' = approximateValue e'

    -- | Forms a new atomic 'Unit' by specifying its 'UnitName' and its definition as a multiple of another 'Unit'.
    --
    -- Use this variant when the scale factor of the resulting unit is rational. See 'mkUnitZ' for when it is an integer
    -- and 'mkUnitR' for the general case.
    --
    -- For more information see 'mkUnitR'.
    mkUnitQ :: Fractional a => UnitName m -> Rational -> Unit m1 d a -> Unit m d a
    mkUnitQ n s (Unit _ e x) | s == 0    = error "Supplying zero as a conversion factor is not valid."
                             | Just x'' <- toExactRational e' = Unit n e' (fromRational x'')
                             | otherwise = Unit n e' x'
      where
        e' = fromRational s Prelude.* e
        x' = fromRational s Prelude.* x

    -- | Forms a new atomic 'Unit' by specifying its 'UnitName' and its definition as a multiple of another 'Unit'.
    --
    -- Use this variant when the scale factor of the resulting unit is an integer. See 'mkUnitQ' for when it is rational
    -- and 'mkUnitR' for the general case.
    --
    -- For more information see 'mkUnitR'.
    mkUnitZ :: Num a => UnitName m -> Integer -> Unit m1 d a -> Unit m d a
    mkUnitZ n s (Unit _ e x) | s == 0    = error "Supplying zero as a conversion factor is not valid."
                             | Just x'' <- toExactInteger e' = Unit n e' (fromInteger x'')
                             | otherwise = Unit n e' x'
      where
        e' = fromInteger s Prelude.* e
        x' = fromInteger s Prelude.* x