manifolds-0.6.1.0: Data/Manifold/PseudoAffine.hs
-- |
-- Module : Data.Manifold.PseudoAffine
-- Copyright : (c) Justus Sagemüller 2015
-- License : GPL v3
--
-- Maintainer : (@) jsag $ hvl.no
-- Stability : experimental
-- Portability : portable
--
-- This is the second prototype of a manifold class. It appears to give considerable
-- advantages over 'Data.Manifold.Manifold', so that class will probably soon be replaced
-- with the one we define here (though 'PseudoAffine' does not follow the standard notion
-- of a manifold very closely, it should work quite equivalently for pretty much all
-- Haskell types that qualify as manifolds).
--
-- Manifolds are interesting as objects of various categories, from continuous to
-- diffeomorphic. At the moment, we mainly focus on /region-wise differentiable functions/,
-- which are a promising compromise between flexibility of definition and provability of
-- analytic properties. In particular, they are well-suited for visualisation purposes.
--
-- The classes in this module are mostly aimed at manifolds /without boundary/.
-- Manifolds with boundary (which we call @MWBound@, never /manifold/!)
-- are more or less treated as a disjoint sum of the interior and the boundary.
-- To understand how this module works, best first forget about boundaries – in this case,
-- @'Interior' x ~ x@, 'fromInterior' and 'toInterior' are trivial, and
-- '.+~|', '|-~.' and 'betweenBounds' are irrelevant.
-- The manifold structure of the boundary itself is not considered at all here.
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE LiberalTypeSynonyms #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UnicodeSyntax #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE CPP #-}
module Data.Manifold.PseudoAffine (
-- * Manifold class
Manifold
, Semimanifold(..), Needle'
, PseudoAffine(..)
, LinearManifold, ScalarManifold
, Num', RealFrac', RealFloat'
, Num'', RealFrac'', RealFloat''
-- * Type definitions
-- ** Needles
, Local(..)
#if !MIN_VERSION_manifolds_core(0,6,0)
, (!+~^)
#endif
-- ** Metrics
, Metric, Metric'
, RieMetric, RieMetric'
-- ** Constraints
, SemimanifoldWitness(..)
, PseudoAffineWitness(..)
, DualNeedleWitness
, WithField
, LocallyScalable
-- ** Local functions
, LocalLinear, LocalBilinear, LocalAffine
-- * Misc
, alerpB, palerp, palerpB, LocallyCoercible(..), CanonicalDiffeomorphism(..)
, ImpliesMetric(..), coerceMetric, coerceMetric'
, Connected (..)
) where
import Math.Manifold.Core.PseudoAffine
import Data.Manifold.WithBoundary.Class
import Data.Maybe
import Data.Fixed
import Data.VectorSpace
import Linear.V0
import Linear.V1
import Linear.V2
import Linear.V3
import Linear.V4
import qualified Linear.Affine as LinAff
import Data.Embedding
import Data.LinearMap
import Data.VectorSpace.Free
import Math.LinearMap.Category
import Data.AffineSpace
import Data.Tagged
import Data.Manifold.Types.Primitive
import qualified Prelude as Hask
import qualified Control.Applicative as Hask
import Control.Category.Constrained.Prelude hiding ((^))
import Control.Arrow.Constrained
import Control.Monad.Constrained
import Data.Foldable.Constrained
import Control.Lens (Lens', lens, (^.), (&), (%~), (.~))
import Data.CallStack (HasCallStack)
import GHC.Exts (Constraint)
-- | See 'Semimanifold' and 'PseudoAffine' for the methods.
-- As a 'Manifold' we understand a pseudo-affine space whose 'Needle'
-- space is a well-behaved vector space that is isomorphic to
-- all of the manifold's tangent spaces.
-- It must also be an instance of the 'SemimanifoldWithBoundary' class
-- with explicitly empty boundary (in other words, with /no/ boundary).
class (OpenManifold m, ProjectableBoundary m, LSpace (Needle m))
=> Manifold m where
instance (OpenManifold m, ProjectableBoundary m, LSpace (Needle m))
=> Manifold m
-- | Instances of this class must be diffeomorphic manifolds, and even have
-- /canonically isomorphic/ tangent spaces, so that
-- @'fromPackedVector' . 'asPackedVector' :: 'Needle' x -> 'Needle' ξ@
-- defines a meaningful “representational identity“ between these spaces.
class ( Semimanifold x, Semimanifold ξ, LSpace (Needle x), LSpace (Needle ξ)
, Scalar (Needle x) ~ Scalar (Needle ξ) )
=> LocallyCoercible x ξ where
-- | Must be compatible with the isomorphism on the tangent spaces, i.e.
-- @
-- locallyTrivialDiffeomorphism (p .+~^ v)
-- ≡ locallyTrivialDiffeomorphism p .+~^ 'coerceNeedle' v
-- @
locallyTrivialDiffeomorphism :: x -> ξ
coerceNeedle :: Hask.Functor p => p (x,ξ) -> (Needle x -+> Needle ξ)
coerceNeedle' :: Hask.Functor p => p (x,ξ) -> (Needle' x -+> Needle' ξ)
coerceNorm :: Hask.Functor p => p (x,ξ) -> Metric x -> Metric ξ
coerceNorm p = case ( oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x
, dualSpaceWitness :: DualSpaceWitness (Needle x)
, dualSpaceWitness :: DualSpaceWitness (Needle ξ) ) of
(CanonicalDiffeomorphism, DualSpaceWitness, DualSpaceWitness)
-> case ( coerceNeedle (swap<$>p), coerceNeedle' p ) of
(f, f') -> \(Norm n) -> Norm $ f' . n . f
coerceVariance :: Hask.Functor p => p (x,ξ) -> Metric' x -> Metric' ξ
coerceVariance p = case ( oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x
, dualSpaceWitness :: DualSpaceWitness (Needle x)
, dualSpaceWitness :: DualSpaceWitness (Needle ξ) ) of
(CanonicalDiffeomorphism, DualSpaceWitness, DualSpaceWitness)
-> case ( coerceNeedle p, coerceNeedle' (swap<$>p) ) of
(f, f') -> \(Norm n) -> Norm $ f . n . f'
oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x
default oppositeLocalCoercion :: LocallyCoercible ξ x => CanonicalDiffeomorphism ξ x
oppositeLocalCoercion = CanonicalDiffeomorphism
type NumPrime n = (Num' n, Eq n)
#define identityCoercion(c,t) \
instance (c) => LocallyCoercible (t) (t) where { \
locallyTrivialDiffeomorphism = id; \
coerceNeedle _ = id; \
coerceNeedle' _ = id; \
oppositeLocalCoercion = CanonicalDiffeomorphism }
identityCoercion(NumPrime s, ZeroDim s)
identityCoercion(NumPrime s, V0 s)
identityCoercion((), ℝ)
identityCoercion(NumPrime s, V1 s)
identityCoercion((), (ℝ,ℝ))
identityCoercion(NumPrime s, V2 s)
identityCoercion((), (ℝ,(ℝ,ℝ)))
identityCoercion((), ((ℝ,ℝ),ℝ))
identityCoercion(NumPrime s, V3 s)
identityCoercion(NumPrime s, V4 s)
data CanonicalDiffeomorphism a b where
CanonicalDiffeomorphism :: LocallyCoercible a b => CanonicalDiffeomorphism a b
-- | A point on a manifold, as seen from a nearby reference point.
newtype Local x = Local { getLocalOffset :: Needle x }
deriving instance (Show (Needle x)) => Show (Local x)
type LocallyScalable s x = ( PseudoAffine x
, LSpace (Needle x)
, s ~ Scalar (Needle x)
, s ~ Scalar (Needle' x)
, Num' s )
type LocalLinear x y = LinearMap (Scalar (Needle x)) (Needle x) (Needle y)
type LocalBilinear x y = LinearMap (Scalar (Needle x))
(SymmetricTensor (Scalar (Needle x)) (Needle x))
(Needle y)
type LocalAffine x y = (Needle y, LocalLinear x y)
-- | Require some constraint on a manifold, and also fix the type of the manifold's
-- underlying field. For example, @WithField ℝ 'HilbertManifold' v@ constrains
-- @v@ to be a real (i.e., 'Double'-) Hilbert space.
-- Note that for this to compile, you will in
-- general need the @-XLiberalTypeSynonyms@ extension (except if the constraint
-- is an actual type class (like 'Manifold'): only those can always be partially
-- applied, for @type@ constraints this is by default not allowed).
type WithField s c x = ( c x, s ~ Scalar (Needle x), s ~ Scalar (Needle' x) )
-- | A co-needle can be understood as a “paper stack”, with which you can measure
-- the length that a needle reaches in a given direction by counting the number
-- of holes punched through them.
type Needle' x = DualVector (Needle x)
-- | The word “metric” is used in the sense as in general relativity.
-- Actually this is just the type of scalar products on the tangent space.
-- The actual metric is the function @x -> x -> Scalar (Needle x)@ defined by
--
-- @
-- \\p q -> m '|$|' (p.-~!q)
-- @
type Metric x = Norm (Needle x)
type Metric' x = Variance (Needle x)
-- | A Riemannian metric assigns each point on a manifold a scalar product on the tangent space.
-- Note that this association is /not/ continuous, because the charts/tangent spaces in the bundle
-- are a priori disjoint. However, for a proper Riemannian metric, all arising expressions
-- of scalar products from needles between points on the manifold ought to be differentiable.
type RieMetric x = x -> Metric x
type RieMetric' x = x -> Metric' x
coerceMetric :: ∀ x ξ . (LocallyCoercible x ξ, LSpace (Needle ξ))
=> RieMetric ξ -> RieMetric x
coerceMetric = case ( dualSpaceWitness :: DualNeedleWitness x
, dualSpaceWitness :: DualNeedleWitness ξ ) of
(DualSpaceWitness, DualSpaceWitness)
-> \m x -> case m $ locallyTrivialDiffeomorphism x of
Norm sc -> Norm $ bw . sc . fw
where fw = coerceNeedle ([]::[(x,ξ)])
bw = case oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x of
CanonicalDiffeomorphism -> coerceNeedle' ([]::[(ξ,x)])
coerceMetric' :: ∀ x ξ . (LocallyCoercible x ξ, LSpace (Needle ξ))
=> RieMetric' ξ -> RieMetric' x
coerceMetric' = case ( dualSpaceWitness :: DualNeedleWitness x
, dualSpaceWitness :: DualNeedleWitness ξ ) of
(DualSpaceWitness, DualSpaceWitness)
-> \m x -> case m $ locallyTrivialDiffeomorphism x of
Norm sc -> Norm $ bw . sc . fw
where fw = coerceNeedle' ([]::[(x,ξ)])
bw = case oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x of
CanonicalDiffeomorphism -> coerceNeedle ([]::[(ξ,x)])
hugeℝVal :: ℝ
hugeℝVal = 1e+100
#define deriveAffine(c,t) \
instance (c) => Semimanifold (t) where { \
type Needle (t) = Diff (t); \
fromInterior = id; \
toInterior = pure; \
translateP = Tagged (.+^); \
(.+~^) = (.+^) }; \
instance (c) => PseudoAffine (t) where { \
a.-~.b = pure (a.-.b); }
instance (NumPrime s) => LocallyCoercible (ZeroDim s) (V0 s) where
locallyTrivialDiffeomorphism Origin = V0
coerceNeedle _ = LinearFunction $ \Origin -> V0
coerceNeedle' _ = LinearFunction $ \Origin -> V0
instance (NumPrime s) => LocallyCoercible (V0 s) (ZeroDim s) where
locallyTrivialDiffeomorphism V0 = Origin
coerceNeedle _ = LinearFunction $ \V0 -> Origin
coerceNeedle' _ = LinearFunction $ \V0 -> Origin
instance LocallyCoercible ℝ (V1 ℝ) where
locallyTrivialDiffeomorphism = V1
coerceNeedle _ = LinearFunction V1
coerceNeedle' _ = LinearFunction V1
instance LocallyCoercible (V1 ℝ) ℝ where
locallyTrivialDiffeomorphism (V1 n) = n
coerceNeedle _ = LinearFunction $ \(V1 n) -> n
coerceNeedle' _ = LinearFunction $ \(V1 n) -> n
instance LocallyCoercible (ℝ,ℝ) (V2 ℝ) where
locallyTrivialDiffeomorphism = uncurry V2
coerceNeedle _ = LinearFunction $ uncurry V2
coerceNeedle' _ = LinearFunction $ uncurry V2
instance LocallyCoercible (V2 ℝ) (ℝ,ℝ) where
locallyTrivialDiffeomorphism (V2 x y) = (x,y)
coerceNeedle _ = LinearFunction $ \(V2 x y) -> (x,y)
coerceNeedle' _ = LinearFunction $ \(V2 x y) -> (x,y)
instance LocallyCoercible ((ℝ,ℝ),ℝ) (V3 ℝ) where
locallyTrivialDiffeomorphism ((x,y),z) = V3 x y z
coerceNeedle _ = LinearFunction $ \((x,y),z) -> V3 x y z
coerceNeedle' _ = LinearFunction $ \((x,y),z) -> V3 x y z
instance LocallyCoercible (ℝ,(ℝ,ℝ)) (V3 ℝ) where
locallyTrivialDiffeomorphism (x,(y,z)) = V3 x y z
coerceNeedle _ = LinearFunction $ \(x,(y,z)) -> V3 x y z
coerceNeedle' _ = LinearFunction $ \(x,(y,z)) -> V3 x y z
instance LocallyCoercible (V3 ℝ) ((ℝ,ℝ),ℝ) where
locallyTrivialDiffeomorphism (V3 x y z) = ((x,y),z)
coerceNeedle _ = LinearFunction $ \(V3 x y z) -> ((x,y),z)
coerceNeedle' _ = LinearFunction $ \(V3 x y z) -> ((x,y),z)
instance LocallyCoercible (V3 ℝ) (ℝ,(ℝ,ℝ)) where
locallyTrivialDiffeomorphism (V3 x y z) = (x,(y,z))
coerceNeedle _ = LinearFunction $ \(V3 x y z) -> (x,(y,z))
coerceNeedle' _ = LinearFunction $ \(V3 x y z) -> (x,(y,z))
instance LocallyCoercible ((ℝ,ℝ),(ℝ,ℝ)) (V4 ℝ) where
locallyTrivialDiffeomorphism ((x,y),(z,w)) = V4 x y z w
coerceNeedle _ = LinearFunction $ \((x,y),(z,w)) -> V4 x y z w
coerceNeedle' _ = LinearFunction $ \((x,y),(z,w)) -> V4 x y z w
instance LocallyCoercible (V4 ℝ) ((ℝ,ℝ),(ℝ,ℝ)) where
locallyTrivialDiffeomorphism (V4 x y z w) = ((x,y),(z,w))
coerceNeedle _ = LinearFunction $ \(V4 x y z w) -> ((x,y),(z,w))
coerceNeedle' _ = LinearFunction $ \(V4 x y z w) -> ((x,y),(z,w))
instance ∀ a b c .
( Semimanifold a, Semimanifold b, Semimanifold c
, LSpace (Needle a), LSpace (Needle b), LSpace (Needle c)
, Scalar (Needle a) ~ Scalar (Needle b), Scalar (Needle b) ~ Scalar (Needle c)
)
=> LocallyCoercible (a,(b,c)) ((a,b),c) where
locallyTrivialDiffeomorphism = regroup
coerceNeedle _ = regroup
coerceNeedle' _ = case ( dualSpaceWitness @(Needle a)
, dualSpaceWitness @(Needle b)
, dualSpaceWitness @(Needle c) ) of
(DualSpaceWitness, DualSpaceWitness, DualSpaceWitness) -> regroup
oppositeLocalCoercion = CanonicalDiffeomorphism
instance ∀ a b c .
( Semimanifold a, Semimanifold b, Semimanifold c
, LSpace (Needle a), LSpace (Needle b), LSpace (Needle c)
, Scalar (Needle a) ~ Scalar (Needle b), Scalar (Needle b) ~ Scalar (Needle c)
)
=> LocallyCoercible ((a,b),c) (a,(b,c)) where
locallyTrivialDiffeomorphism = regroup'
coerceNeedle _ = regroup'
coerceNeedle' _ = case ( dualSpaceWitness @(Needle a)
, dualSpaceWitness @(Needle b)
, dualSpaceWitness @(Needle c) ) of
(DualSpaceWitness, DualSpaceWitness, DualSpaceWitness) -> regroup'
oppositeLocalCoercion = CanonicalDiffeomorphism
instance (LinearSpace (a n), Needle (a n) ~ a n)
=> Semimanifold (LinAff.Point a n) where
type Needle (LinAff.Point a n) = a n
LinAff.P v .+~^ w = LinAff.P $ v ^+^ w
instance (LinearSpace (a n), Needle (a n) ~ a n)
=> PseudoAffine (LinAff.Point a n) where
LinAff.P v .-~. LinAff.P w = return $ v ^-^ w
LinAff.P v .-~! LinAff.P w = v ^-^ w
instance RealFloat' r => Semimanifold (S⁰_ r) where
type Needle (S⁰_ r) = ZeroDim r
p .+~^ Origin = p
p .-~^ Origin = p
instance RealFloat' r => PseudoAffine (S⁰_ r) where
PositiveHalfSphere .-~. PositiveHalfSphere = pure Origin
NegativeHalfSphere .-~. NegativeHalfSphere = pure Origin
_ .-~. _ = Nothing
PositiveHalfSphere .-~! PositiveHalfSphere = Origin
NegativeHalfSphere .-~! NegativeHalfSphere = Origin
_ .-~! _ = error "There is no path between the two 0-dimensional half spheres."
instance RealFloat' r => Semimanifold (S¹_ r) where
type Needle (S¹_ r) = r
S¹Polar φ₀ .+~^ δφ = S¹Polar $ φ'
where φ' = toS¹range $ φ₀ + δφ
semimanifoldWitness = case linearManifoldWitness @r of
LinearManifoldWitness -> SemimanifoldWitness
instance RealFloat' r => PseudoAffine (S¹_ r) where
p .-~. q = pure (p.-~!q)
S¹Polar φ₁ .-~! S¹Polar φ₀
| δφ > pi = δφ - tau
| δφ < (-pi) = δφ + tau
| otherwise = δφ
where δφ = φ₁ - φ₀
instance RealFloat' s => Semimanifold (S²_ s) where
type Needle (S²_ s) = V2 s
(.+~^) = case linearManifoldWitness @s of
LinearManifoldWitness ->
let addS² (S²Polar θ₀ φ₀) 𝐯 = S²Polar θ₁ φ₁
where -- See images/constructions/sphericoords-needles.svg.
S¹Polar γc = coEmbed 𝐯
γ | θ₀ < pi/2 = γc - φ₀
| otherwise = γc + φ₀
d = magnitude 𝐯
S¹Polar φ₁ = S¹Polar φ₀ .+~^ δφ
-- Cartesian coordinates of p₁ in the system whose north pole is p₀
-- with φ₀ as the zero meridian
V3 bx by bz = embed $ S²Polar d γ
sθ₀ = sin θ₀; cθ₀ = cos θ₀
-- Cartesian coordinates of p₁ in the system with the standard north pole,
-- but still φ₀ as the zero meridian
(qx,qz) = ( cθ₀ * bx + sθ₀ * bz
,-sθ₀ * bx + cθ₀ * bz )
qy = by
S²Polar θ₁ δφ = coEmbed $ V3 qx qy qz
in addS²
instance RealFloat' s => PseudoAffine (S²_ s) where
p.-~.q = pure (p.-~!q)
S²Polar θ₁ φ₁ .-~! S²Polar θ₀ φ₀ = d *^ embed(S¹Polar γc)
where -- See images/constructions/sphericoords-needles.svg.
V3 qx qy qz = embed $ S²Polar θ₁ (φ₁-φ₀)
sθ₀ = sin θ₀; cθ₀ = cos θ₀
(bx,bz) = ( cθ₀ * qx - sθ₀ * qz
, sθ₀ * qx + cθ₀ * qz )
by = qy
S²Polar d γ = coEmbed $ V3 bx by bz
γc | θ₀ < pi/2 = γ + φ₀
| otherwise = γ - φ₀
instance Semimanifold ℝP² where
type Needle ℝP² = ℝ²
HemisphereℝP²Polar θ₀ φ₀ .+~^ v
= case S²Polar θ₀ φ₀ .+~^ v of
S²Polar θ₁ φ₁
| θ₁ > pi/2 -> HemisphereℝP²Polar (pi-θ₁) (-φ₁)
| otherwise -> HemisphereℝP²Polar θ₁ φ₁
instance PseudoAffine ℝP² where
p.-~.q = pure (p.-~!q)
HemisphereℝP²Polar θ₁ φ₁ .-~! HemisphereℝP²Polar θ₀ φ₀
= case S²Polar θ₁ φ₁ .-~! S²Polar θ₀ φ₀ of
v -> let r² = magnitudeSq v
in if r²>pi^2/4
then S²Polar (pi-θ₁) (-φ₁) .-~! S²Polar θ₀ φ₀
else v
-- instance (PseudoAffine m, VectorSpace (Needle m), Scalar (Needle m) ~ ℝ)
-- => Semimanifold (CD¹ m) where
-- type Needle (CD¹ m) = (Needle m, ℝ)
-- CD¹ h₀ m₀ .+~^ (h₁δm, δh)
-- = let h₁ = min 1 . max 1e-300 $ h₀+δh; δm = h₁δm^/h₁
-- in CD¹ h₁ (m₀.+~^δm)
-- instance (PseudoAffine m, VectorSpace (Needle m), Scalar (Needle m) ~ ℝ)
-- => PseudoAffine (CD¹ m) where
-- CD¹ h₁ m₁ .-~. CD¹ h₀ m₀
-- = fmap ( \δm -> (h₁*^δm, h₁-h₀) ) $ m₁.-~.m₀
class ImpliesMetric s where
type MetricRequirement s x :: Constraint
type MetricRequirement s x = Semimanifold x
inferMetric :: (MetricRequirement s x, LSpace (Needle x))
=> s x -> Metric x
inferMetric' :: (MetricRequirement s x, LSpace (Needle x))
=> s x -> Metric' x
instance ImpliesMetric Norm where
type MetricRequirement Norm x = (SimpleSpace x, x ~ Needle x)
inferMetric = id
inferMetric' = dualNorm
type DualNeedleWitness x = DualSpaceWitness (Needle x)
#if !MIN_VERSION_manifolds_core(0,6,0)
infixl 6 !+~^
-- | Boundary-unsafe version of `.+~^`.
(!+~^) :: ∀ x . (Semimanifold x, HasCallStack) => x -> Needle x -> x
p!+~^v = case toInterior p of
Just p' -> p'.+~^v
#endif
infix 6 .−.
-- | A connected manifold is one where any point can be reached by translation from
-- any other point.
class (PseudoAffine m) => Connected m where
{-# MINIMAL #-}
-- | Safe version of '(.-~.)'.
(.−.) :: m -> m -> Needle m
(.−.) = (.-~!)
instance Connected ℝ⁰
instance Connected ℝ
instance Connected ℝ¹
instance Connected ℝ²
instance Connected ℝ³
instance Connected ℝ⁴
instance Connected S¹
instance Connected S²
instance Connected ℝP⁰
instance Connected ℝP¹
instance Connected ℝP²
instance (Connected x, Connected y) => Connected (x,y)
instance (Connected x, Connected y, PseudoAffine (FibreBundle x y))
=> Connected (FibreBundle x y)
type LinearManifold m = (LinearSpace m, Manifold m)
type ScalarManifold s = (Num' s, Manifold s, Manifold (ZeroDim s))
type Num'' s = ScalarManifold s
type RealFrac'' s = (RealFrac' s, ScalarManifold s)
type RealFloat'' s = (RealFloat' s, SimpleSpace s, ScalarManifold s)